theory of integer quantum hall polaritons in graphene

Theory of integer quantum Hall polaritons in graphene

FMD Pellegrino1 lowast L Chirolli2 Rosario Fazio1 V Giovannetti1 and Marco Polini3 4

1NEST Scuola Normale Superiore and Istituto Nanoscienze-CNR I-56126 Pisa Italy2Instituto de Ciencia de Materiales de Madrid (CSIC)Sor Juana Ines de la Cruz 3 E-28049 Madrid Spain

3NEST Istituto Nanoscienze-CNR and Scuola Normale Superiore I-56126 Pisa Italy4Istituto Italiano di Tecnologia Graphene Labs Via Morego 30 I-16163 Genova Italy

We present a theory of the cavity quantum electrodynamics of the graphene cyclotron resonanceBy employing a canonical transformation we derive an effective Hamiltonian for the system com-prised of two neighboring Landau levels dressed by the cavity electromagnetic field (integer quantumHall polaritons) This generalized Dicke Hamiltonian which contains terms that are quadratic inthe electromagnetic field and respects gauge invariance is then used to calculate thermodynamicproperties of the quantum Hall polariton system Finally we demonstrate that the generalized Dickedescription fails when the graphene sheet is heavily doped ie when the Landau level spectrum of2D massless Dirac fermions is approximately harmonic In this case we ldquointegrate outrdquo the Landaulevels in valence band and obtain an effective Hamiltonian for the entire stack of Landau levels inconduction band as dressed by strong light-matter interactions

PACS numbers 7867Wj 4250Pq 7343-f

I INTRODUCTION

Light-matter interactions in graphene a two-dimensional (2D) honeycomb crystal of Carbon atoms1ndash3have been intensively explored in the past decade for bothfundamental and applied purposes4ndash7

Recent experimental advances have made it possible tomonolithically integrate graphene with optical microcav-ities89 paving the way for fundamental studies of cav-ity quantum electrodynamics (QED)10 at the nanome-ter scale with graphene as an active medium Anotherapproach which has been successful11 in achieving theso-called strong-coupling regime of cavity QED10 in con-ventional 2D electron systems in semiconductor quantumwells consists in coupling graphene carriers with the pho-tonic modes of an array of split-ring resonators12

Graphene-based cavity QED offers at least in princi-ple a number of unique advantages First graphene isa highly-tunable active medium since its electrical andheat transport properties can be easily controlled by em-ploying gates1ndash3 Second graphene offers many path-ways to achieve the strong-coupling regime these includei) the exploitation of intrinsic Dirac plasmons67 and ii)the combination of graphene with other plasmonic nanos-tructures13 Third the active medium can be enrichedby employing 2D vertical heterostructures14ndash17 compris-ing graphene as well as other 2D crystalssystems such ashexagonal Boron Nitride18ndash20 transition metal dichalco-genides2122 (eg MoS2 WS2 WSe2) Gallium Arsenidequantum wells2324 etc

A central role in cavity QED is played by the Dickemodel25 which describes a non-dissipative closed sys-tem of identical two-level subsystems interacting witha single-mode radiation field For a sufficiently stronglight-matter coupling constant the thermodynamic limitof the Dicke model exhibits a second-order quantumphase transition to a super-radiant state26 with macro-

scopic photon occupation and coherent atomic polariza-tion

When an external magnetic field is applied to a 2Delectron system transitions between states in full andempty Landau levels (LLs) are dispersionless27ndash29 mim-icking atomic transitions and enabling30 a condensedmatter realization of the Dicke model The light-matterinteraction in the Dicke Hamiltonian is linear in the vec-tor potential Aem of the cavity For condensed matterstates described by parabolic band models a quadraticA2

em term whose strength is related to the systemrsquos Drudeweight and f-sum rule2931 also emerges naturally fromminimal coupling It has long been understood32 thatthe Dicke modelrsquos super-radiant phase transition is sup-pressed when the quadratic terms are retained Demon-strations of this property are often referred to as no-gotheorems

The problem is more subtle in graphene where elec-tronic states near the charge neutrality point are de-scribed in a wide range of energies by a 2D masslessDirac fermion (MDF) Hamiltonian23 The MDF Hamil-tonian contains only one power of momentum p minimalcoupling applied to this Hamiltonian does not generatea term proportional to A2

em The authors of Ref 33demonstrated that in the strong coupling regime themodel for the cavity QED of the graphene cyclotron res-onance must be supplemented by a quadratic term inthe cavity photon field that is dynamically generated byinter-band transitions and again implies a no-go theoremThe terms proportional toA2

em in the theory of the cavityQED of the graphene cyclotron resonance were derived inRef 33 by using as a guiding principle gauge invarianceand by treating inter-band transitions in the frameworkof second-order perturbation theory

The main scope of this Article is to lay down a for-mal theory of the cavity QED of the graphene cyclotronresonance The key point is that one must derive a low-

2

energy effective Hamiltonian by taking into account thecoupling of the two-level systems which are resonant withthe cavity photon field to all non-resonant states Thiscoupling is crucially important in the strong-couplingregime where all the terms that are proportional toA2

emwhich are generated by our renormalization proceduremust be taken into account Indeed these guaranteegauge invariance as well as a no-go theorem for the occur-rence of a super-radiant phase transition therefore cor-roborating the findings of Ref 33 Finally we go beyondthis generalized Dicke description by demonstrating thatit is not adequate to describe the strong-coupling regimeof the cavity QED of the graphene cyclotron resonancein the limit of high doping In this case we derive anddiscuss a renormalized Hamiltonian for the entire stackof LLs in conduction band as dressed by the cavity elec-tromagnetic field

Our Article is organized as following In Section IIwe employ a canonical transformation34ndash36 to derive aneffective low-energy Hamiltonianmdashsee Eq (49)mdashfor thecavity QED of the graphene cyclotron resonance and dis-cuss the limits of its validity We analyze in great detailthe invariance of this effective Hamiltonian with respectto gauge transformations by employing a linear-responsetheory formalism In Sect III we use a functional-integralformalism to study the thermodynamic properties ofthe system described by the effective Hamiltonian InSect IV we transcend the generalized Dicke descriptionof Sect II and present a renormalized HamiltonianmdashseeEq (119)mdashthat enables the study of the strong-couplinglimit of the cavity QED of the graphene cyclotron reso-nance in the limit of high doping Finally in Sect V wereport a summary of our main findings and conclusions

II GENERALIZED DICKE HAMILTONIAN

In this Section we derive an effective low-energy Hamil-tonian for the cavity QED of the graphene cyclotron res-onance

A Landau levels in graphene

At low energies charge carriers in graphene are mod-eled by the usual single-channel massless Dirac fermionHamiltonian23

HD = vDσ middot p (1)

where vD asymp 106 ms is the Dirac velocity Here σ =(σx σy) is a 2D vector of Pauli matrices acting on sub-lattice degrees-of-freedom and p = minusi~nablar is the 2D mo-mentum measured from one of the two corners (valleys)of the Brillouin zone

A quantizing magnetic field B = Bz perpendicular tothe graphene sheet is coupled to the electronic degrees-of-freedom by replacing the canonical momentum p in

Eq (1) with the kinetic momentum Π = p + eA0cwhere A0 is the vector potential that describes the staticmagnetic field B The corresponding Hamiltonian is

H0 = vDσ middotΠ (2)

We work in the Landau gauge A0 = minusByx In thisgauge the canonical momentum along the x directionpx coincides with magnetic translation operator29 alongthe same direction and it commutes with the Hamilto-nian H0 Thus the eigenvalues of px are good quantumnumbers A complete set of eigenfunctions of the Hamil-tonian H0 in Eq (2) is provided by the two componentpseudospinors37

〈r|λ n k〉 =eikxradic

2L

(wminusnφnminus1(y minus `2Bk)λw+nφn(y minus `2Bk)

) (3)

where λ = +1 (minus1) denotes conduction (valence) bandlevels n isin N is the Landau level (LL) index and k is theeigenvalue of the magnetic translation operator in the xdirection In Eq (3)

wplusmnn =radic

1plusmn δn0 (4)

guarantees that the pseudospinor corresponding to then = 0 LL has weight only on one sublattice Furthermoreφn(y) with n = 0 1 2 are the normalized eigenfunc-tions of a 1D harmonic oscillator with frequency equalto the MDF cyclotron frequency ωc =

radic2vD`B Here

`B =radic~c(eB) 25 nm

radicB[Tesla] is the magnetic

lengthThe spectrum of the Hamiltonian (2) has the well-

known form37

ελn = λ~ωc

radicn (5)

Each LL has a degeneracy N = NfS(2π`2B) where Nf =

4 is the spin-valley degeneracy and S = L2 is the samplearea

B Total Hamiltonian

We now couple the 2D electron system described bythe Hamiltonian (2) to a single photon mode in a cavityWe denote by the symbol Aem the vector potential thatdescribes the cavity photon mode Carriers in grapheneare coupled to the cavity electromagnetic field via theminimal substitution

Πrarr Πprime = Π +e

cAem (6)

The cavity vector potential Aem will be treated withinthe dipole approximation We can neglect the spatialdependence of the electromagnetic field in the cavity be-cause the photon wavelength is much larger than anyother length scale of the system

3

Introducing photon annihilation a and creation opera-tors adagger we can write

Aem =

radic2π~c2εωV

eem(a+ adagger) (7)

where eem is a unit vector describing the polarization ofthe electromagnetic field ω is the photon frequency εis the cavity dielectric constant and V = LzL

2 is thevolume of the cavity Here Lz L is the length of thecavity in the z direction

The total Hamiltonian reads

H = Hem +H0 +Hint (8)

where the first term is the cavity photon Hamiltonian

the second term is the MDF Hamiltonian in the pres-ence of a quantizing magnetic field ie Eq (2) and thethird term describes the coupling between MDFs and thecavity photon mode More explicitly

Hem = ~ω(adaggera+

1

2

) (9)

H0 =sum

λnk

ελncdaggerλnkcλnk (10)

and

Hint =gradicN

sum

λλprimennprimek

(λwλne

minusemδnprimen+1 + λprimewλprimenprimee

+emδnprimenminus1

) (a+ adagger

)cdaggerλprimenprimekcλnk (11)

In Eqs (10)-(11) cdaggerλnk (cλnk) creates (annihilates) anelectron with band index λ LL index n and wave numberk Finally

g equiv ~ωc

radice2

2εLz~ω (12)

and eplusmnem = exemplusmnieyem exem and eyem being the componentsof the polarization vector eem

We consider the integer quantum Hall regime in whicha given number of LLs are fully occupied and the re-maining ones are empty Since the MDF Hamiltonianis particle-hole symmetric we can consider without lossof generality the situation in which graphene is n-dopedand the Fermi energy lies in conduction band (λ = +)We denote by n = M the highest occupied LL The low-est empty LL is therefore n = M+1 and the Fermi energylies in the middle between n = M and n = M + 1 ie

EM equiv1

2~ωc(radicM + 1 +

radicM) (13)

C Canonical transformation

The aim of this Section is to present a systematic pro-cedure that allows us to derive an effective low-energyHamiltonian for the LL doublet n = MM +1 as dressedby light-matter interactions We are interested in thecase in which the cavity photon is nearly resonant withthe transition between the two conduction-band LLsn = MM + 1

~ω asymp ΩM equiv ~ωc(radicM + 1minus

radicM) (14)

We anticipate33 that the effective Hamiltonian will bedifferent from the bare Dicke Hamiltonian that one ob-tains from Eqs (9) (10) and (11) by selecting λ = +1and n = MM + 1 ie

HDicke = Hem +

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN

(a+ adagger)(eminusemτ+k + e+

emτminusk )

] (15)

Here38 11k τzk τplusmnk with k = 1 N is a set of Pauli ma-

trices that act in the 2N -fold degenerate subspace of theLL doublet n = MM + 1 11k being the 2 times 2 identityand τplusmnk equiv (τxk plusmn iτyk ) 2 More precisely the final resultof the canonical transformation yields a generalized DickeHamiltonian of the formmdashsee Eq (49)

HGDH = HDicke + ∆M (a+ adagger)2

+

Nsum

k=1

[κ

N (a+ adagger)211k minusκz

N (a+ adagger)2τzk

]

(16)

We notice that HGDH differs from the bare Dicke Hamil-tonian (15) because of the presence of three terms thatare quadratic in the operator a+adagger and that renormalizeboth Hem and the light-matter interaction HamiltonianMicroscopic expressions for the parameters ∆M κ andκz are derived below

We denote by the symbol SM the subspace of thefermionic Hilbert space spanned by the two LLs which areresonantly coupled to the cavity field ie n = MM + 1and lay on opposite sides of the Fermi energy The sym-bol SN on the other hand denotes the subspace of the

4

fermionic Hilbert space which is comprised of all LLs butn = MM + 1 We employ a canonical transformationwith the aim of decoupling the LL doublet n = MM+1from the SN sector (see Refs 34ndash36 and also Chapter 8in Ref 29)

Before proceeding further it is convenient to rewritethe Hamiltonian (8) in the following manner

H = Hem +H0 + VD + VO (17)

where Hem and H0 have been introduced in Eqs (9) and(10) respectively whereas the light-matter interactionHamiltonian Hint has been written as the sum of twoterms i) VD which connects states either belonging tothe subspace SM or to the subspace SN and ii) VO whichconnects states belonging to different subspaces There-fore VD is a block-diagonal operator with one block re-ferring to the SM subspace and the other one to the SNsubspace In the same representation H0 is trivially ablock-diagonal operator since it is a diagonal operatorand Hem is also a block-diagonal operator since it con-tains only photonic creation and annihilation operatorsand therefore acts as the identity operator with respectto fermionic labels On the other hand VO is a block-off-diagonal operator in the same representation

We now introduce an unitary transformation

U = eS (18)

where S is its anti-Hermitian generator The transformedHamiltonian reads

Hprime = eSHeminusS (19)

The spirit of the canonical transformation34ndash36 is totransform the original Hamiltonian H onto an Hamil-tonian Hprime that has no block-off-diagonal terms A neces-sary condition to achieve this is that the generator S bea block-off-diagonal operator

The operator S can be found with the desired levelof accuracy by following a perturbative approach Weuse the Baker-Campbell-Hausdorff formula to rewriteEq (19)

Hprime = H+ [SH] +1

2[S [SH]] + (20)

where [AB] denotes the commutator between the twooperators A and B

We now expand the generator S in a power series

S =

infinsum

j=1

S(j) (21)

where S(j) is proportional to (g0)j ie the j-th powerof a suitable dimensionless coupling constant that is con-trolled by the strength g of light-matter interactionsmdashseeEq (24) below

After inserting Eq (21) in Eq (20) we require thateach term of the expansion cancels the corresponding

1 2 3 4 5 6 7 8 9 10

M

0

1

2

3

4

5

g 0

a)

1 5 10 15 20 25 30

ε

2

4

6

8

10

Mm

ax

b)

FIG 1 Panel a) Dependence of the dimensionless interactionparameter g0 as defined in Eq (24) on the Landau levelindex M in the resonant case ie ~ω = ΩM Different curvescorrespond to different values of the cavity dielectric constantε = 1 (solid line) ε = 5 (dashed line) and ε = 15 (dash-dottedline) Panel b) Since g0 increases as a function of M for a fixedvalue of ε we can define the maximum value Mmax of M upto which g0 lt 1 We plot Mmax as a function of the cavitydielectric constant ε

block-off-diagonal term order by order in the perturba-tive expansion in powers of g0 This approach leads to ahierarchy of equations one for each order in perturbationtheory

For example the equation for the generator S(1) up tofirst order in g0 reads as follows

[S(1)H0 +Hem] + VO = 0 (22)

The transformed Hamiltonian is given by the followingexpression

Hprime = Hem +H0 + VD +1

2[S(1) VO] +O(g3

0) (23)

We emphasize that Hprime is correct up to second order ing0

The expansion parameter g0 is defined by

g0 equiv maxmisinSM nisinSN

(∣∣∣∣g

~ω minus |εmn|

∣∣∣∣) (24)

5

where εmn equiv εm minus εn is the difference between the en-ergies of two LLs From the definition of g0 we clearlysee that the canonical transformation cannot be appliedif the photon cavity is resonant with a transition betweena LL belonging to the subspace SM and one belongingto the subspace SN As stated above we are interestedin the case in which the cavity photon is nearly resonantwith the transition between the two LLs in the subspaceSM ie ~ω asymp ΩM Leaving aside the case M = 0 whichneeds a separate treatment the anharmonicity of the LLspectrum in graphene Eq (5) ensures that the samecavity photon cannot be resonant with other transitions

In particular in the resonant case we obtain g0 =g[~ωc(

radicM + 2+

radicMminus2

radicM + 1)] If we consider a half-

wavelength cavity we have ω = πc(Lzradicε) and conse-

quently g = ~ωc

radicα(2π

radicε) where α = e2(~c) sim 1137

is the QED fine structure constant Fig 1a) shows a plotof g0 evaluated at ~ω = ΩM as a function of the LL in-dex M and for different values of the dielectric constantε The procedure outlined in this Section is rigorouslyjustified for g0 lt 1 In this regime the LL anharmonicityis larger than the light-matter coupling g Fig 1b) showsthat for a given value of the cavity dielectric constant εthe inequality g0 lt 1 is satisfied up to maximum value ofM denoted by the symbol Mmax and that one can pushthe limit of validity of this approach to higher values ofM by increasing the value of ε

In Sections II D-II E we derive the desired low-energyeffective Hamiltonian by using the canonical transforma-tion approach described in this Section The procedureis carried out in three steps i) we first decouple the sub-space SN from the subspace SM by applying the canon-ical transformation S up to first order in the small pa-rameter g0mdashEq (22) ii) we then use a different canonicaltransformation to take care of inter-band transitions be-tween LLs belonging to the subspace SN iii) finally wetake into account Pauli blocking

D Explicit form of the canonical transformationup to order g0

Following the notation of Sect II C we start from theoriginal Hamiltonian in Eq (17) Here H0 which hasbeen introduced in Eq (10) refers to bare electrons in thepresence of a quantizing magnetic field and it is diagonalwith respect to spin projection valley index and theeigenvalue of the magnetic translation operator in thex direction It does not couple states belonging to thesubspace SM with states belonging to the subspace SN

H0 =sum

misinSM

εmcdaggermcm +

sum

nisinSN

εncdaggerncn (25)

Here cdaggerm and cdaggern (cm and cn) are fermionic creation (an-nihilation) operators for a bare electron We emphasize

that in this Section the indices m and n are collectivelabels for the spin projection along the z axis the valleyindex the eigenvalue of the magnetic translation opera-tor in the x direction the intra-band LL integer labeland the conductionvalence band label

The Hamiltonian that couples electronic degrees-of-freedom with the electromagnetic field is written as asum of a block-diagonal term VD and a block-off-diagonalterm VO

VD =sum

mmprimeisinSM

gmmprimeradicN(a+ adagger

)cdaggermcmprime

+sum

nnprimeisinSN

gnnprimeradicN(a+ adagger

)cdaggerncnprime (26)

and

VO =sum

misinSM nisinSN

[gmnradicN(a+ adagger

)cdaggermcn

+gnmradicN(a+ adagger

)cdaggerncm

] (27)

In Eqs (26)-(27) we have introduced

gmn = δkkprime(λwλne

minusemδmn+1 + λprimewλprimeme

+emδmnminus1

)

(28)where n (m) is the collective label n λ k (m λprime kprime) Eachof these three numbers represents an intra-band LL la-bel (n m) a band index (λ λprime) and a collective label(k kprime) comprising the eigenvalue of the magnetic trans-lation operator in the x direction together with the spinprojection along the along the z axis and the valley in-dex

By solving Eq (22) we obtain an explicit expressionfor the anti-Hermitian generator S up to first order in g0

S(1) =sum

misinSM nisinSN

(gmnradicNAωcdaggermcn minus

gnmradicNcdaggerncmAdaggerω

)

(29)where we have introduced the operator

Aω equiva

εmn minus ~ω+

adagger

εmn + ~ω (30)

Given the first-order generator S(1) the commutator[S(1) VO] generates a new block-diagonal term Usingthe dipole selection rules the commutator reads

[S(1) VO] = 2(a+ adagger

)2 sum

misinSM nisinSN

εmnε2mn minus (~ω)2

times gmngnmN

(cdaggermcm minus cdaggerncn

)+ Bω (31)

where

6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN

gmngnmε2mn minus ~2ω2

(cdaggermcm + cdaggerncn

)+

sum

mmprimeisinSM

sum

nnprimeisinSN

[gmprimenprimegmnε2mn minus ~2ω2

cdaggermprimecnprimecdaggermcn +

gnprimemprimegmnε2mn minus ~2ω2

cdaggernprimecmprimecdaggermcn +

gmprimenprimegnmε2mn minus ~2ω2

cdaggermprimecnprimecdaggerncm +

gnprimemprimegnmε2mn minus ~2ω2

cdaggernprimecmprimecdaggerncm

] (32)

Note that B0 = 0 More importantly the operator Bωbeing proportional to Nminus1 is negligible in the limit of amacroscopic LL degeneracy (N 1)

Using Eq (23) and neglecting terms that are O(g30) we

finally find the effective Hamiltonian Hprime which is correctup to order g2

0

Hprime = Hem +HM +HN (33)

Here HM is the sum of N independent contributions

one for each value of k = 1 N ie HM =sumNk=1Hk

with

Hk = EM11k +ΩM2τzk +

gradicN


emτminusk )

minus κz

N (a+ adagger)2τzk +κ

N (a+ adagger)211k (34)

where EM and ΩM have been introduced earlier inEqs (13) and (14) respectively

The quadratic terms in the electromagnetic fieldie the terms in the second line of Eq (34) stem fromthe canonical transformation In Eq (34) we have intro-duced

κz equiv κzs minus κzd (35)

where the first term is independent of the cavity photon

frequency while the second term that we define ldquodynam-icalrdquo explicitly depends on the cavity photon frequency

κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~

radicM + 1[ω2 minus (4M + 5)ω2

c ]

[(2M + 3)ω2c minus ω2]2 minus 4(M + 1)(M + 2)ω4

c

+

radicM [ω2 minus (4M minus 3)ω2

c ]

[(2M minus 1)ω2c minus ω2]2 minus 4M(M minus 1)ω4

c

(37)

Note that κzd = 0 for ω = 0 Finally

κ =ω2

ωc

g2

~


c ]


c

minusradicM [ω2 minus (4M minus 3)ω2

c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)

The second term in Eq (33) reads as following

HN =sum

nisinSN

[εn +

sum

misinSM

εnmε2nm minus (~ω)2

(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN



E Elimination of the off-diagonal terms in HN andPauli blocking

The Hamiltonian (33) is not yet the desired resultie an effective Hamiltonian for the n = MM + 1 dou-blet Indeed HN contains fermionic operators that acton the subspace SN

In particular we note that the last term in Eq (39) isan off-diagonal contribution in the labels n nprime isin SN Weutilize a suitable canonical transformation that elimi-nates this term For the sake of simplicity we here report

only the final result We find a renormalized Hamiltonianoperating on the subspace SN which is diagonal in thelabels n nprime isin SN

HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`

εn`gn`g`nε2n` minus ~2ω2

cdaggerncn (40)

where the index ` runs over all LLsSince the Dirac model applies over a large but finite en-

ergy region we must regularize33 Eq (40) by employing

7

a cut-off νmax Moreover we treat the fermionic portionof the renormalized Hamiltonian (40) as a mean field forthe photons We therefore replace

cdaggerncn rarr nF(εn) equiv 1

exp [(εn minus microe)(kBT )] + 1 (41)

where microe is the chemical potential of the electronic sys-tem The accuracy of this mean-field treatment will bejustified below in Sect IV

In the low-temperature limit

kBT |εn minus microe| foralln isin SN (42)

we can replace the Fermi-Dirac function in Eq (41) witha Fermi step

We are therefore led to define the prefactor of the (a+adagger)2 term in Eq (40) as

∆M (νmax) =sum

nisinSN

sum

`


Θ(microe minus εn) (43)

where the sums are regularized with the cut-off νmaxMore explicitly for every M 6= 0 we have

∆M (νmax) = minus2εmaxg2

~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2

Ω2Mminus1 minus ~2ω2

(44)

where εmax equiv ~ωcradicνmax and

IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2

ω2 minus ω2c (radic`+radic`+ 1)2

+(radic`minusradic`minus 1)ω2

ω2 minus ω2c (radic`minus 1 +

radic`)2

] (45)

As explained in Refs 3339 we must regularize the ex-pression in Eq (44) by subtracting the cut-off dependentterm minus2εmax g

2(~2ω2c ) After applying this regulariza-

tion one can take the limit νmax rarrinfin discovering thatthe quantity

∆M equiv limνmaxrarrinfin

[∆M (νmax) + 2εmax

g2

~2ω2c

]

=g2

~ωcIinfinMminus1 minus

g2

ΩMminus1

ω2

Ω2Mminus1 minus ω2

(46)

with

IinfinM equiv limνmaxrarrinfin

IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)

is well definedDiscarding constant terms29 (ie terms that do not

contain the photon field operators a and adagger) the renor-malized Hamiltonian (40) becomes

HprimeN = ∆M

(a+ adagger

)2 (48)

We stress that ∆M as defined in Eq (46) depends bothon the LL label M and the photon frequency ω and thatit vanishes in the static ω = 0 limit

F Final result for the effective Hamiltonian

In summary the correct low-energy Hamiltonian isgiven by Hprime as in Eq (33) with HN replaced by HprimeNin Eq (48) ie

HGDH equiv Hem + ∆M (a+ adagger)2 +

Nsum

k=1

Hk (49)

where Hk has been defined in Eq (34) and without lossof generality we have chosen a specific polarization of theelectromagnetic field ie eem = ux

Eq (49) is the first important result of this Articleand represents a low-energy effective Hamiltonian for thecavity QED of the graphene cyclotron resonance It isevident that HGDH differs from the bare Dicke Hamilto-nian (15) since it contains terms that are quadratic inthe electromagnetic field We will therefore refer to thelow-energy effective Hamiltonian (49) as to generalizedDicke Hamiltonian (GDH)

As discussed earlier and as illustrated in Fig 1b) theGDH (49) is rigorously justified only for a finite intervalof values of M which depends on the cavity dielectricconstant For example for ε = 15 Eq (49) is justi-fied in the interval 0 lt M le 8 This implies that forthis value of ε the description of the cavity QED of thegraphene cyclotron resonance in terms of the GDH maybreak down for M ge 9 Below we discuss an alterna-tive approach which is valid for arbitrarily large valuesof the highest-occupied LL index M and transcends thedescription based on the GDH

For future purposes it is useful to highlight the follow-ing identity

∆M =g2

ΩM+

g2

~ωcIinfinM minus κz minus κ (50)

and the following inequality

FM (ω) le IinfinM le FM+1(ω) (51)

which is valid ω le ωc

radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8

For large M one therefore finds

IinfinM ω

2ωclog

(2EM minus ~ω2EM + ~ω

) (53)

In the resonant ~ω = ΩM case the quantities κz andκ defined earlier in Eqs (35)-(38) reduce to

κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)

G Linear-response theory analysis

In this Section we demonstrate that the GDH (49) isgauge invariant

To this end we treat the cavity electromagnetic fieldas a weak perturbation with respect to the MDF Hamil-tonian in the presence of a quantizing magnetic fieldThe cavity electromagnetic field induces a matter cur-rent that can be calculated by the powerful means oflinear response theory2931 In particular the physicalmatter current in response to the electromagnetic fieldis composed by paramagnetic and diamagnetic contribu-tions2931

It is easy to demonstrate that the paramagnetic re-sponse function of a system described by the GDH (49)to the electromagnetic field is given by

χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω

= g2 2ΩM~2ω2 minus Ω2

M

tanh

(βΩM

4

) (56)

where τxtot =sumNk=1 τ

xk and β = 1(kBT ) In Eq (56) we

have introduced the Kubo product29

〈〈AB〉〉ω equiv minusi

~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)

where 〈〉 denotes a thermal average and A(t) is theoperator A in the Heisenberg representation ie A(t) equivexp(iHGDHt)A exp(minusiHGDHt)

Similarly the diamagnetic response function is givenby

χD(ω) =2

N 〈〈κ11tot minus κzτztot〉〉ω + 2∆M

= 2κ+ 2∆M + 2κz tanh

(βΩM

4

) (58)

where τztot =sumNk=1 τ

zk and 11tot =

sumNk=1 11k

The diamagnetic response function χD(ω) can berewritten in a compact form as

χD(ω) = 2Ωg (59)

where

Ωg = Ωg(β) equiv g2

ΩM+

g2

~ωcIinfinM

minus κz [1minus tanh (βΩM4)] (60)

In writing Eqs (59)-(60) we have used the mathematicalidentity (50)

Therefore the physical current-current response func-tion is the sum of these two contributions

χJ(ω) = χP(ω) + χD(ω)


M

tanh

(βΩM

4

)+ 2Ωg (61)

In the static ω = 0 limit we have

χP(ω rarr 0) = minus 2g2

ΩMtanh

(βΩM

4

)(62)

and

χD(ω rarr 0) = 2κzs tanh

(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)

Paramagnetic and diamagnetic contributions inEqs (62)-(63) are equal in magnitude and oppositein sign Hence a quasi-homogeneous vector potentialdoes not induce any response in the static limit in thislimit the vector potential represents a pure gauge andcannot induce any physical effect unless gauge invarianceis broken2931

Alert readers will note that the paramagnetic contri-bution to the physical current-current response functiondominates over the diamagnetic contribution in the res-onant limit ~ω rarr ΩM Indeed χP(ω) has a pole at~ω rarr ΩM while χD(ω) is finite at the same frequencyAs we will see below in Sect III however the quadraticterms in the photon field in Eq (49) which yield a finitediamagnetic response are absolutely crucial to ensurethermodynamic stability of the system

In passing we notice that the current-current responsefunction in Eq (61) has the following asymptotic behav-ior

χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)

in the limit of zero temperature and for M such thatΩM ~ω lt 2EM Eq (64) is formally identical to thecurrent-current response function of a doped graphenesheet in the absence of a quantizing magnetic field39 pro-vided that one replaces EM with the Fermi energy microe

9

H Comparison with the findings of Ref 33

For the sake of completeness we now compare the mainresult obtained so far ie the GDH (49) with the resultsof Ref 33

We start by recalling the effective Hamiltonian thatwas derived in Ref 33 In the notation of this Article itreads

Heff = ~ω(adaggera+

1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN

(a+ adagger)τxk minusκzsN (a+ adagger)2τzk

] (65)

Note that the term proportional to (a+adagger)2 in the previ-ous equation contains the Pauli matrix minusτzk this correctsa mistake that was made in Ref 33

Although the Hamiltonian (65) respects gauge invari-ance in the sense of Sect II G it misses dynamical con-tributions that are naturally captured by the canonicaltransformation The GDH Hamiltonian (49) indeed re-duces to Eq (65) when the dynamical contributions κzdκ and ∆M are set to zero We remind the reader thatin the static ω rarr 0 limit κzd κ∆M rarr 0

III THERMODYNAMICS OF THE GDH

In this Section we present a thorough analysis of thethermodynamic properties of the GDH (49)

The partition function Z in the grand-canonical en-semble reads

Z = Tr[eminusβ(HeffminusmicrophNphminusmicroeNe)

] (66)

where Nph (Ne) is the photon (electron) number and microph

(microe) is the chemical potential of the photonic (electronic)system Here we assume that the chemical potential ofthe electronic system is fixed at EM while the chemicalpotential of the photons is set to zero

In order to evaluate the grand-canonical partition func-tion we use the functional integral formalism40 In thisformalism the grand-canonical partition function Z iswritten as a functional integral over bosonic and Grass-mann fields

Z =

intD[φlowast(τ) φ(τ)]

intD[ξlowastjk(τ) ξjk(τ)]

times eminusS[φlowast(τ)φ(τ)ξlowastjk(τ)ξjk(τ)] (67)

Here φlowast(τ) φ(τ) represent bosonic fields which are de-fined on the imaginary-time interval [0 β] and repeatedperiodically elsewhere whereas ξjk(τ) ξlowastjk(τ) are Grass-mann fermionic fields which are anti-periodic in the sameimaginary-time interval In Eq (67) k = 1 N and jlabels the eigenvalues of the 2times2 matrix τz ie j = plusmn1Finally the Euclidean action S reads

S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ) + ∆M [φlowast(τ) + φ(τ)]

2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN

[φlowast(τ) + φ(τ)] τxjjprime + [φlowast(τ) + φ(τ)]2

(κ

N δjjprime minusκz

N τzjjprime

)ξjprimek(τ) (68)

A Static path approximation

The simplest approximation to evaluate the grand-canonical partition function Z in Eq (67) is the so-calledldquostatic path approximationrdquo (SPA) In the SPA the de-pendence of the bosonic fields φlowast(τ) φ(τ) on imaginarytime is neglected Therefore quantum fluctuations of theelectromagnetic field are absent in the SPA The SPA isa good approximation when the average photon numberis macroscopic ie when it is O(N ) This is preciselywhat occurs in a super-radiant phase

The gran-canonical partition function in the SPA reads

ZSPA equivintdφlowastdφ

2πi


times eminusS[φlowastφξlowastjk(τ)ξjk(τ)] (69)

where φlowast and φ are just complex numbers and not fluc-tuating fields

Carrying out the integral over the Grassmann fieldsξlowastjk(τ) ξjk(τ) and over =m(φ) we find

ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)

where x = lte(φ)radicN and

10

Φ(x) = minusβ(~ω + 4∆M + 4κ)x2 + log

2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)

In the limit N 1 the integral in Eq (70) can be calcu-lated by employing the steepest descent method40 ie

ZSPA radic

2

β~ω|Φprimeprime(x0)|eNΦ(x0) (72)

Here x0 denotes a maximum ie

Φprime(x0) equiv dΦ(x)

dx

∣∣∣∣x=x0

= 0 (73)

and

Φprimeprime(x0) equiv d2Φ(x)

dx2

∣∣∣∣x=x0

lt 0 (74)

We now look for solutions of the saddle-point equation(73)

Since Φ(x) depends on x through x2mdashsee Eq (71)mdashx0 = 0 is always an extremum of Φ(x) Physically thesolution x0 = 0 corresponds to the ldquonormal phaserdquo inwhich the number of photons vanishes in the thermody-namic limit We study the nature of this extremum byevaluating Φprimeprime(0) Straightforward algebraic manipula-tions yield

Φprimeprime(0) = minus2β~ω + 4g2IinfinM (~ωc)

+ 4(g2ΩM minus κz)[1minus tanh(βΩM4)](75)

Since g2ΩM gt κz and g(~ωc) lt 1 the quantity Φprimeprime(0)can satisfy Φprimeprime(0) ge 0 if and only if the dimensionlessfunction

fM (ω) equiv minus4ωc

ωIinfinM (76)

is larger than unity Note that fM (ω) is independent ofthe cavity dielectric constant ε Since we are interested inthe resonant regime we can set ω = ΩM~ in Eq (76) Inthis case fM becomes a function of the LL label M onlyFig 2 illustrates the dependence of fM = fM (ω = ΩM~)on M We clearly see that fM (ω = ΩM~) lt 1 for everyM We can therefore conclude that x0 = 0 is always amaximum ie Φprimeprime(0) lt 0

In what follows we investigate the possibility of hav-ing a super-radiant phase ie a phase with a macro-scopic number of photons in the thermodynamic limitThis phase corresponds to the existence of a maximumof Φ(x) located at a non-zero value of the order parameterx We will show that if g0 lt 1 no such extremum existsThis implies that the GDH (49) is not unstable towardsa super-radiant state in the regime where its derivationbased on the canonical transformation (Sect II) is rigor-ously justified

1 Absence of a super-radiant phase

We now prove that the saddle-point equation (73) doesnot admit any solution at x0 6= 0 To this end we writeexp[NΦ(x)] as a sum of functions which are all concavedownwards and have a maximum at x0 = 0 This is easilyaccomplished by exploiting the binomial theorem

(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)

Using Eq (77) in Eq (70) we find

eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where

Φ`(x) = log(2)minus β(~ω + 4∆M + 4κ)x2

+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)

Let us now study the solutions of the equation

dΦ`(x)

dx= 0 (81)

We first notice that Eq (81) admits always the trivialsolution x = 0 because Φ`(x) depends on x only throughx2 We now investigate whether solutions exist at non-zero values of x The trivial x = 0 solution can be easilydiscarded by taking the first derivative of Φ`(x) with re-spect to x2 Requiring that this vanishes is equivalent tofinding the solutions of the following equation

~ω + 4∆M + 4κ

[1minus N minus `N tanh

(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)

Eq (82) can also be written as following

c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M

FIG 2 Dependence of the function fM defined in Eq (76)on the LL index M

where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with

t(`)1 (x) equiv 1minus N minus `N tanh

(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)

From the form of Eq (83) one clearly sees that in orderto find a solution of Eq (81) at finite x one of the coef-

ficients c(`)4 (x) c

(`)2 (x) and c

(`)0 (x) must change sign for

one value of ` and x 6= 0

It is easy to see that the functions c(`)n (x) with n = 0 2

and 4 are positive definite for any temperature and anyvalue of x unless the following inequality is satisfied

fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)

FIG 3 Panel a) The quantity Mcr is plotted as a function ofthe cavity dielectric constant ε We remind the reader thatfor M gt Mcr the condition fM (ω = ΩM~) lt 0 is satisfiedNote that Mcr gt Mmaxmdashsee Fig 1b) Panel b) The ratioΦ(x)Φ(0) as a function of x for ω = ΩM~ ε = 15 andkBT = 01 ΩM=20 The solid line refers to M = 5 which issmaller than the value of Mmax for ε = 15 while the dashedline refers to M = 20 Mcr gt Mmax In this case the GDH(49) is not applicable

Since we are interested in the resonant regime we canset ω = ΩM~ in Eq (89) In this case fM becomesa function of the LL label M only ie fM = fM (ω =ΩM~) We find that for every value of ε there is a valueMcr of the LL index label M such that the inequality inEq (89) is satisfied for M gt Mcr Fig 3a) illustrates thedependence of Mcr on ε By comparison with Fig 1b) weclearly see that Mcr gt Mmax We therefore conclude thatthe necessary condition for the occurrence of solutions ofEq (81) at finite x ie fM lt 0 cannot be achievedwithin the limit of validity of the derivation of the GDH(49) ie for M lt Mmax

We have therefore demonstrated that for M lt MmaxΦ`(x) has no estremum at x 6= 0 for every value of `Exploiting the binomial representation in Eq (78) wenotice that the function exp[NΦ(x)] can be written as asum of concave downwards functions which have a maxi-mum at x = 0 Therefore Φ(x) is also concave downwardsand has only one maximum at x = 0 The function Φ(x)

12

has neither a global nor a local maximum at x 6= 0 Thisimplies the impossibility to have a transition to a super-radiant phase

Fig 3b) shows the quantity Φ(x) as a function of x fortwo values of the LL index M M lt Mmax (solid line)where the GDH (49) is rigorously justified and M Mcr

(dashed line) well beyond the limit of validity of theGDH In both cases we see that Φ(x) has a maximum atx = 0 as demonstrated earlier For M lt Mmax no otherextremum of Φ(x) is present In the case M Mcrithowever the function Φ(x) presents a minimum at x 6= 0and diverges for x 1 More precisely its is possibleto show that Φ(x 1) rarr minusβfMx2 It follows thatthe partition function ZSPA in Eq (69) is ill-defined forM Mcr gt Mmax The ldquocatastrophicrdquo growth Φ(x 1) rarr minusβfMx2 for large x stems from the application ofthe GDH (49) well beyond its limit of validity ie forM gt Mcrit gt Mmax where fM lt 0

Sect IV will be devoted to the presentation of a theorythat transcends the GDH and that is valid also for M Mmax

2 The partition function in the SPA

We can now finalize the calculation of the partitionfunction in the SPA by following the steepest descentmethod (72) We expand Φ(x) around the maximum atx = 0 as

Φ(x) Φ(0) + Φprimeprime(0)x2

2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)

Using Eqs (75) and (91) in Eq (72) we find

ZSPA Z(2)

free

β~ωg (92)

where

Z(2)free equiv [1 + exp (βΩM2)]N [1 + exp (minusβΩM2)]N (93)

and

ωg = ωg(β) equivω[ω + 4g2IinfinM (~2ωc) + 4(g2ΩM minus κz)

times [1minus tanh(βΩM4)]~]12 (94)

The quantity Z(2)free is easily recognized to be the grand-

canonical partition function of the LL doublet n =MM + 1 in the absence of the cavity photon field

It is also possible to evaluate the photon occupation

number n(SPA)ph in the SPA

n(SPA)ph = minuspart logZSPA

part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph

FIG 4 The (red) circles denote the logarithm of the photon

occupation number n(SPA)ph as a function of the LL label M

for ~ω = ΩMmdashsee Eq (95) The (green) triangles denote theSPA photon occupation number evaluated at g = 0 Eq (96)and for ~ω = ΩM In this plot kBT = 01 ΩM=8 and ε = 15

which is formally identical to the SPA occupation num-ber of a photon gas that does not interact with matter(ie g = 0)

minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)

provided that one replaces ω rarr ωg In Fig 4 we com-

pare the photon occupation number n(SPA)ph evaluated on

resonance ~ω = ΩM with the SPA occupation numberof the photon gas evaluated at g = 0 Eq (96) We seethat light-matter interactions do not cause any signifi-cant modification of the photon occupation number withrespect to the g = 0 case We therefore do not see anysign of a super-radiant phase

3 Super-radiance in the absence of the quadratic terms

We now show that a super-radiant phase transition canoccur when the quadratic terms in the photon field areneglected26

In this case a maximum of Φ(x) at x0 6= 0 can occurif2641

~ωΩM4g2

lt 1 (97)

This implies that choosing a suitable cavity dielectricconstant for a given M or a value of the LL index Mfor a given ε a super-radiant phase transition is pos-sible Consider for instance a half-wavelength cavityand set ~ω = ΩM where ω = πc(Lz

radicε) In this case

g = ~ωc

radicα(2π

radicε) and the critical condition (97) be-

comesradicM + 1 +

radicM gt 2π

radicεα A super-radiant

phase transition is therefore possible41 for large enoughvalues of M

13

If the condition (97) is satisfied the maximum of Φ(x)appears at

x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)

in the zero-temperature limit Hence one can gainenergy when the photon occupation number becomes

macroscopic n(SPA)ph = x2

0N These are artefacts stemming from the neglect of

quadratic terms in the photon field

B The impact of quantum fluctuations of theelectromagnetic field

Within the SPA we have demonstrated that thesaddle-point equation (73) admits only the ldquotrivialrdquo so-

lution x = 0 ie lte(φ) = 0 for any value of the tem-perature T In this Section we present a careful studyof the impact of imaginary-time (ie quantum) fluctu-ations of the photonic field φ(τ) around φ = 0 on thethermodynamic properties of the effective Hamiltonian(49) In other words we want to verify whether the nor-mal phase is robust with respect to quantum fluctuationsof the electromagnetic field

We rewrite the Euclidean action S in Eq (68) in thefollowing form

S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastkj(τ)[minusGminus1

0 (τ) + Σ(τ)]jjprimeξkjprime(τ) (99)

where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN

[φlowast(τ) + φ(τ)] τx (102)

and

Σ2 = [φlowast(τ) + φ(τ)]2

(κ

N 11minus κz

N τz) (103)

The key point now is to realize that the fermionic partof the action can be integrated out exactly since it corre-sponds to a Gaussian functional integral The resultingeffective action is

Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)

+ ∆M [φlowast(τ) + φ(τ)]2

minus Tr

[log(minusGminus1

0 + Σ)]

(104)

where the symbol ldquoTrrdquo means a trace over all degrees-of-freedom including the imaginary time

In order to study the effect of Gaussian fluctuations weexpand the last term in the effective action Seff in powersof Σ up to second order in the bosonic fields φlowast(τ) φ(τ)In order to do so we employ the identity

Tr[log(minusGminus1

0 + Σ)]

= Tr[log(minusGminus1

0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)

Neglecting terms of order φ3(τ) we therefore find

Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define

S(2)fluct equiv Tr [G0Σ2] +

1

2Tr [G0Σ1G0Σ1] (107)

The first term in the previous equation is non-zero be-cause Σ2 is quadratic in the bosonic fields

Hence the grand-canonical partition function in theGaussian approximation reads

14

ZG Z(2)free

intD[φlowast(τ) φ(τ)]e

minusint β

0

dτφlowast(τ) (partpartτ + ~ω)φ(τ) + ∆M [φlowast(τ) + φ(τ)]

2+ S(2)

fluct(φlowast(τ) φ(τ))

(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph


occupation number n(G)ph as a function of the LL label M for

~ω = ΩMmdashsee Eq (117) The (green) triangles denote theBose-Einstein thermal factor nB(ΩM ) In this plot kBT =01 ΩM=8 and ε = 15

where Z(2)free has been defined earlier in Eq (93) We can

now calculate the bosonic functional integral on the right-hand side of Eq (108) since it is a Gaussian functionalintegral This is most easily done by using the Matsubararepresentation of the photonic field

φ(τ) =1radicβ

+infinsum

m=minusinfineminusiωmτφm (109)

where ωm = 2πmβ with m isin N We find

ZG ZSPA

int infinprod

m=1

dϕlowastmdϕm2πiβ

eminussum

m ϕdaggermmiddotSmmiddotϕm (110)

where ϕm = (φm φlowastminusm)T and ZSPA has been defined

earlier in Eq (92)

To evaluate the integral on the right-hand side ofEq (110) we need the determinant of the matrix SmFor each positive integer m this reads as follows

Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)

We again analyze the resonant case ~ω = ΩM It is easyto demonstrate that the function fM in Eq (76) needsto be larger than unity to drive at least one of the de-terminants Sm to a negative value But we have alreadyverified that fM lt 1 for every Mmdashsee Fig 2 Hence wehave found that the normal phase is robust with respectto quantum fluctuations of the electromagnetic field

The partition function (110) can be written as

ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)

We now exploit the identity

1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)

where Zho(ω) is the partition function of an harmonicoscillator with characteristic frequency ω

We therefore conclude that the grand-canonical parti-tion function in the Gaussian approximation is given bythe following expression

ZG ZSPA(β~ω+)(β~ωminus)

βΩM

Zho(ω+)Zho(ωminus)

Zho(ΩM~) (114)

where

15

~ωplusmn =

radicradicradicradic~ω (~ω + 4Ωg) + Ω2M

2plusmn

radic[~ω (~ω + 4Ωg)minus Ω2

M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)

The quantity Ωg has been introduced earlier in Eq (60)and is proportional to diamagnetic response functionχD(ω) Physically the quantities ωplusmn represent thefrequencies of the two integer quantum Hall polaritonmodes The quantity Ωg encodes all the contributions to

the polariton modes that stem from quadratic correctionsin the photon fields which are present in the low-energyeffective Hamiltonian (49)

Neglecting these terms results in the following integerquantum Hall polariton frequencies41

~ωplusmn|Ωg=0 =

radicradicradicradic~2ω2 + Ω2M

2plusmn

radic(~2ω2 minus Ω2

M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0

FIG 6 Dependence on the LL index M of the smallnessparameter g0 as defined in Eq (118) and evaluated at ~ω =ΩM Different curves correspond to different values of thedielectric constant ε = 1 (solid line) ε = 5 (dashed line)and ε = 15 (dash-dotted line)

With the partition function at our disposal we can

evaluate the photon occupation number n(G)ph in the pres-

ence of Gaussian fluctuations of the electromagnetic fieldWe find

n(G)ph = minuspart logZG

part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)

where n(SPA)ph has been defined in Eq (95) and nB(ω) =

[exp (β~ω)minus 1]minus1 is the Bose-Einstein thermal factorIn Fig 5 we compare the photon occupation number

n(G)ph evaluated on resonance ie for ~ω = ΩM with

the Bose-Einstein function nB(ΩM ) From this figurewe clearly see the photon occupation number obtainedfrom Eq (117) is comparable with the non-interactingphoton thermal occupation number No evidence of a

super-radiant phase transition is seen Comparing n(G)ph

in Fig 5 with n(SPA)ph in Fig 4 we immediately see that

the SPA which treats quasi-classically the electromag-netic field overestimates the photon occupation numberWe have therefore verified that quantum fluctuations ofthe electromagnetic do not drive the system towards asuper-radiant phase and that on the contrary suppressthe photon occupation number

IV BEYOND THE GDH

As we have discussed above the description of the cav-ity QED of the graphene cyclotron resonance in terms ofthe GDH is not valid for M Mmax where Mmax hasbeen illustrated in Fig 1b) In this Section we presenta theory that transcends the GDH and that is valid forevery M

We again employ a canonical transformation but thistime we use it to ldquointegrate outrdquo the entire valence bandremaining with an effective Hamiltonian for the entireconduction band as dressed by strong light-matter inter-actions With the notation of Sect II C we denote bySM the Hilbert subspace spanned by LLs in conductionband including the zero-energy (m = 0) LL whereas SNdenotes the Hilbert subspace spanned by LLs in valenceband In this case the dimensionless parameter that con-trols the validity of the canonical transformation is

g0 =g

|~ωc minus ~ω| (118)

16

Fig 6 shows g0 for ~ω = ΩM as a function of the LL labelM We clearly see that g0 lt 1 for any positive M andthat g0 decreases as M increases Hence the approachof this Section allows us to study the cavity QED of thegraphene cyclotron resonance well beyond the regime of

M values where the modeling described in Sect II works

Following the approach summarized in Sect II C wefind the following effective Hamiltonian for the conduc-tion band


1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger

) (cdaggernkcn+1k + cdaggern+1kcnk

)+κnN(a+ adagger

)2cdaggernkcnk

] (119)

where once again we have chosen without loss of gener-ality a specific polarization of the electromagnetic fieldie eem = ux For the sake of simplicity we havedropped the label ldquo+rdquo from the fermionic field opera-

tors c+nk and cdagger+nk Eq (119) is the second importantresult of this Article

In Eq (119)

κn =(w+ng)

2

~ωc

(radicn+radicn+ 1)ω2

c

(radicn+radicn+ 1)2ω2

c minus ω2

+(wminusng)

2

~ωc

(radicn+radicnminus 1)ω2

c

(radicn+radicnminus 1)2ω2

c minus ω2 (120)

which is finite in the static ω rarr 0 limit and

∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)

where Iinfin0 can be simply obtained by setting M = 0 inEq (47) The quantity ∆ in Eq (121) vanishes in thestatic limit The quantities wplusmnn in Eq (120) have beenintroduced earlier in Eq (4)

A Thermodynamic properties of the effectiveHamiltonian for the entire conduction band

mean-field theory

Starting from the effective Hamiltonian in Eq (119)we evaluate the grand-canonical partition function Zby using again the functional integral formalism Inorder to decouple the electronic system from the elec-tromagnetic field we introduce four complex auxiliaryfields ie ylowast(τ) y(τ) and zlowast(τ) z(τ) via the Hubbard-Stratonovich transformation40

Z =

intD[ylowast(τ) y(τ)]

intD[zlowast(τ) z(τ)]



times exp

[minusradicN g

int β

0

dτ |y(τ)|2 minusN g2

~ωc

int β

0

dτ |z(τ)|2 minus SF minus SB

] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ) + gy(τ) [φlowast(τ) + φ(τ)] +

[g2

~ωcz(τ) + ∆

][φlowast(τ) + φ(τ)]

2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part

partτ+ (ε+n minus EM )minus κnzlowast(τ)

]ξnk(τ)

minus w+ngylowast(τ)

[ξlowastnk(τ)ξn+1k(τ) + ξlowastn+1k(τ)ξnk(τ)

] (123b)

The previous expression for Z is formally exact and con-tains only terms that are quadratic in the fermionboson

fields In the following we apply the SPA for the aux-

17

iliary complex fields by neglecting their imaginary-timedependence and the steepest descent method with re-spect to the auxiliary fields In order to find the sad-dle point we have to deform the contours of integrationwith respect to the static auxiliary fields in the complexplane42

We find that the saddle point is located at

ylowast = minus 1radicN〈a+ adagger〉MF (124a)

y =sum

kn

w+nradicN〈cdaggernkcn+1k + cdaggern+1kcnk〉MF (124b)

zlowast = minus 1

N 〈(a+ adagger)2〉MF (124c)

z =~ωc

g2

sum

nk

κnN 〈c

daggernkcnk〉MF (124d)

where the grand-canonical ensemble averages 〈〉MF areevaluated with respect to the following mean-field Hamil-tonian

HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc

daggernkcnk minus gw+ny

lowast(cdaggernkcn+1k

+ cdaggern+1kcnk

)minus κnzlowastcdaggernkcnk

] (127)

Starting from the bosonic quadratic Hamiltonian HB weobtain the following relations between the mean fields

ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1

N minus (ylowast)2 (128b)

where

ω = ω(z) equivradicω

(ω +

4∆

~+

4g2z

~2ωc

) (129)

Since the LL degeneracy is macroscopic ie N 1 inEq (128b) we can neglect the first term on the right-hand side and write zlowast minus(ylowast)2 The correspondingmean-field fermionic Hamiltonian (127) becomes

HF sum

nk

[ε+nc



+ cdaggern+1kcnk

)+ κn (ylowast)2 cdaggernkcnk

] (130)

For any ω gt 0 each eigenstate of the mean-field Hamil-tonian in Eq (130) has an energy that is a monotonicallyincreasing function of |ylowast| and has a minimum at ylowast = 0Thus the self-consistent problem has the following solu-tion

ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)

where nF(x) = [eβ(xminusEM ) + 1]minus1 is the Fermi-Dirac ther-mal factor We emphasize that the solution (131b)-(131d) of the mean-field problem posed by the Hamil-tonian (125) is an a posteriori check of the mean-fieldtreatment we adopted in Eq (41) of Sect II E

By using the steepest descent method we can explic-itly write the grand-canonical function as

Z ZMF equiv Z(infin)free Zho(ω) (132)

where Zho(ω) has been introduced in Eq (113) ω isgiven by Eq (129) evaluated at z as from Eqs (131b)-(131d) and

Z(infin)free equiv

infinprod

n=0

[1 + eβ(EMminusε+n)

]N (133)

Note that Z(infin)free is the grand-canonical partition function

of the multi-level system n = 0 1 2 in the absence ofthe cavity photon field

B Gaussian fluctuations beyond mean-field theory

In this Section we investigate the stability of the mean-field solution given in the Sect IV A by calculatingthe fluctuations of the Hubbard-Stratonovich auxiliaryfields42 To this end we expand the grand-canonical par-tition function in Eq (122) around its saddle point up toquadratic order

Following a procedure analogous to the one sketchedin Sect III B we find

Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0

w2+n [nF(ε+n)minus nF(ε+n+1)]Gm(Ωn) (135)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)

FIG 7 Upper and lower integer quantum Hall polaritonbranches as a function of the LL label M Filled (red) circlesdenote the upper polariton branch ~ω+ in units of ΩM andevaluated on resonance ~ω = ΩM Similarly filled (blue) tri-angles denote the lower polariton branch ~ωminus in units of ΩM In this plot T = 0 and ε = 1 The results in panel a) havebeen obtained by including the contribution to the polaritonmodes that is due to quadratic terms in the electromagneticfieldmdashEq (115) On the other hand in panel b) the quantityΩg due to quadratic terms in the electromagnetic field is arti-ficially set to zeromdashEq (116) In this case the lower polaritonbranch ωminus softens at a sufficiently large value of M (M 53for ε = 1) signaling an artificial second-order phase transitionto a super-radiant phase

where ω has been defined after Eq (132) Ωn equiv ε+n+1minusε+n and Gm(Ω) = 2Ω[(iωm)2minusΩ2] with ωm = 2πmβ

In the low-temperature limit kBT ΩM and for M gt0

ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω

ωg2Gm(~ω)Gm(ΩM ) (137)

In writing Eq (137) we have used that the Fermi energylies between the M -th and (M+1)-th LL ie that EM =

~ωc(radicM + 1 +

radicM)2

It is easy to see that D0 gt 0 if and only if fM lt 1where fM is defined in Eq (76) This condition hasalready been discussed in Sect III A 1 and is always sat-isfied Moreover since Dm gt D0 for any positive integerm no instability of the mean-field state occurs Hencewe have demonstrated that the mean-field state is robustwith respect to Gaussian fluctuations of the Hubbard-Stratonovich fields

The grand-canonical partition function can be writtenin the low-temperature limit as

Z =Z(infin)

free

β~ωg(β~ω+)(β~ωminus)

βΩM


Zho(ΩM~) (138)

where ωplusmn are the frequencies of the integer quantum Hall polaritons in the low-temperature limit βΩM 1mdash

19

Eq (115) with the replacement tanh(βΩM4)rarr 1 Sim-ilarly ωg is defined in Eq (94) and needs here to beevaluated in the low- temperature limit βΩM 1 ie

ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)

Figs 7a)-b) illustrate the dependence of the frequen-cies ωplusmn on M In particular panel b) shows that thefrequency of the lower polariton branch ωminus vanishes inthe case in which quadratic terms in the photon fields areneglected ie when Ωg is set to zeromdashEq (116) For thevalues of the parameters chosen in this figure this occursat M 53 The softening of the lower polariton branchsignals the occurrence of an artificial second-order super-radiant phase transition at a large but finite value of M In Fig 7a) we see that for any M the polariton branchesωplusmn evaluated for Ωg 6= 0 are positive definite In partic-ular Fig 7a) shows that the frequency of the lower po-lariton ωminus is a monotonically decreasing function of M using the definition (115) we find that ωminus rarr Ω2

M(2~g)for M 1 This result ensures that there is no finite Mat which ωminus crosses zero if Ωg 6= 0 In summary we haveverified that there is no occurrence of super-radiant phasetransitions in the cavity QED of the graphene cyclotronresonance This statement is true also for large values ofthe highest occupied LL M where the two-level systemdescription adopted in Sect II fails and one has to resortto the multi-level effective Hamiltonian in Eq (119)

Finally we highlight that the partition function inEq (138) formally coincides with the partition functionof the two-level system effective model Eq (114) pro-

vided that Z(infin)free is replaced by Z(2)

free

V SUMMARY AND CONCLUSIONS

In this Article we have presented a theory of the cavityQED of the graphene cyclotron resonance

We have first employed a canonical transformation toderive an effective Hamiltonian for the system comprisedof two neighboring Landau levels dressed by the cavityelectromagnetic field (integer quantum Hall polaritons)The final result is in Eq (49) This effective Hamiltonianwhich we have termed ldquogeneralized Dicke Hamiltonianrdquorespects gauge invariance and contains terms that are

quadratic in the electromagnetic field We have then usedEq (49) and a functional integral formalism to calculatethermodynamic properties of the integer quantum Hallpolariton system We have corroborated the results ofRef 33 by confirming that no super-radiant phase tran-sitions are possible in the cavity QED of the graphenecyclotron resonance

Starting from a careful analysis of the smallness pa-rameter g0 of the canonical transformation Eq (24) wehave proved that the generalized Dicke Hamiltonian de-scription fails for sufficiently large value of the highest-occupied Landau level index Mmdashsee Sect III A 1 Themaximum value Mmax of M up to which the derivationof the generalized Dicke Hamiltonian is reliable dependson the value of the cavity dielectric constant ε as illus-trated in Fig 1b) For M gt Mmax one has to transcendthe generalized Dicke Hamiltonian description In thiscase we have used a canonical transformation to projectout the entire stack of Landau levels belonging to thevalence band The end result of this approach is an ef-fective Hamiltonian for the entire stack of Landau levelsin conduction band as dressed by light-matter interac-tions This result is reported in Eq (119)

In this Article we have discarded electron-electron in-teractions which play a very important role in low-dimensional electron systems and in particular inthe quantum Hall regime where the kinetic energy isquenched and interactions are dominant Future workwill be devoted to understand the role of electron-electroninteractions in the theory of quantum Hall polaritons43

Acknowledgments

It is a pleasure to thank Allan MacDonald for manyenlightening conversations We acknowledge support bythe EC under Graphene Flagship (contract no CNECT-ICT-604391) (MP) the European Research Council Ad-vanced Grant (contract no 290846) (LC) the Ital-ian Ministry of Education University and Research(MIUR) through the programs ldquoFIRB IDEASrdquo - ProjectESQUI (Grant No RBID08B3FM) (VG) ldquoFIRB -Futuro in Ricerca 2010rdquo - Project PLASMOGRAPH(Grant No RBFR10M5BT) (MP) and PRIN Grant No2010LLKJBX (RF) and a 2012 SNS Internal Project(VG)

lowast Electronic address francescopellegrinosnsit1 AK Geim and KS Novoselov Nature Mater 6 183

(2007)2 AH Castro Neto F Guinea NMR Peres KS

Novoselov and AK Geim Rev Mod Phys 81 109(2009)

3 MI Katsnelson Graphene Carbon in Two Dimensions(Cambridge University Press Cambridge 2012)

4 F Bonaccorso Z Sun T Hasan and AC Ferrari NaturePhoton 4 611 (2010)

5 NMR Peres Rev Mod Phys 82 2673 (2010)6 FHL Koppens DE Chang and FJ Garcıa de Abajo

Nano Lett 11 3370 (2011)7 AN Grigorenko M Polini and KS Novoselov Nature

Photon 6 749 (2012)8 M Engel M Steiner A Lombardo AC Ferrari H v

20

Loehneysen P Avouris and R Krupke Nature Commun3 906 (2012)

9 M Furchi A Urich A Pospischil G Lilley K Unter-rainer H Detz P Klang AM Andrews W Schrenk GStrasser and T Mueller Nano Lett 12 2773 (2012)

10 JM Raimond M Brune and S Haroche Rev ModPhys 73 565 (2001) H Mabuchi and AC Doherty Sci-ence 298 1372 (2002) H Walther BTH Varcoe B-GEnglert and T Becker Rep Prog Phys 69 1325 (2006)

11 G Scalari C Maissen D Turcinkova D Hagenmuller SDe Liberato C Ciuti C Reichl D Schuh W Wegschei-der M Beck and J Faist Science 335 1323 (2012)

12 F Valmorra G Scalari C Maissen W Fu CSchonenberger JW Choi HG Park M Beck and JFaist Nano Lett 13 3193 (2013)

13 See eg TJ Echtermeyer L Britnell PK Jasnos ALombardo RV Gorbachev AN Grigorenko AK GeimAC Ferrari and KS Novoselov Nature Commun 2 458(2011)

14 KS Novoselov Rev Mod Phys 83 837 (2011)15 KS Novoselov and AH Castro Neto Phys Scr T146

014006 (2012)16 F Bonaccorso A Lombardo T Hasan Z Sun L

Colombo and AC Ferrari Mater Today 15 564 (2012)17 AK Geim and IV Grigorieva Nature 499 419 (2013)18 LA Ponomarenko AK Geim AA Zhukov R Jalil SV

Morozov KS Novoselov IV Grigorieva EH Hill VVCheianov VI Falrsquoko K Watanabe T Taniguchi andRV Gorbachev Nature Phys 7 958 (2011)

19 RV Gorbachev AK Geim MI Katsnelson KSNovoselov T Tudorovskiy IV Grigorieva AH MacDon-ald SV Morozov K Watanabe T Taniguchi and LAPonomarenko Nature Phys 8 896 (2012)

20 L Britnell RV Gorbachev R Jalil BD Belle FSchedin A Mishchenko T Georgiou MI Katsnelson LEaves SV Morozov NMR Peres J Leist AK GeimKS Novoselov and LA Ponomarenko Science 335 947(2012)

21 QH Wang K Kalantar-Zadeh A Kis JN Coleman andMS Strano Nature Nanotech 7 699 (2012)

22 L Britnell RM Ribeiro A Eckmann R Jalil B DBelle A Mishchenko Y-J Kim RV Gorbachev TGeorgiou SV Morozov AN Grigorenko AK Geim CCasiraghi AH Castro Neto and KS Novoselov Science340 1311 (2013)

23 A Principi M Carrega R Asgari V Pellegrini and MPolini Phys Rev B 86 085421 (2012)

24 A Gamucci D Spirito M Carrega B Karmakar ALombardo M Bruna AC Ferrari LN Pfeiffer KWWest M Polini and V Pellegrini arXiv14010902 (2014)

25 RH Dicke Phys Rev 93 99 (1954)

26 K Hepp and EH Lieb Ann Phys (NY) 76 360 (1973)YK Wang and FT Hioe Phys Rev A 7 831 (1973) KHepp and EH Lieb ibid 8 2517 (1973)

27 RE Prange and SM Girvin The Quantum Hall Effect(Springer-Verlag New York 1990)

28 AH MacDonald Introduction to the Physics of the Quan-tum Hall Regime in Proceedings of the Les Houches Sum-mer School on Mesoscopic Physics edited by E Akker-mans G Montambeaux and JL Pichard (Elsevier Am-sterdam 1995)

29 GF Giuliani and G Vignale Quantum Theory of theElectron Liquid (Cambridge University Press Cambridge2005)

30 D Hagenmuller S De Liberato and C Ciuti Phys RevB 81 235303 (2010)

31 D Pines and P Nozieres The Theory of Quantum Liquids(WA Benjamin Inc New York 1966)

32 K Rzazewski K Wodkiewicz and W Zakowicz PhysRev Lett 35 432 (1975) I Bialynicki-Birula and KRzazewski Phys Rev A 19 301 (1979) K GawedzkiK Rzazewski ibid 23 2134 (1981)

33 L Chirolli M Polini V Giovannetti and AH MacDon-ald Phys Rev Lett 109 267404 (2012)

34 DR Hamann and AW Overhauser Phys Rev 143 183(1966)

35 JR Schrieffer and PA Wolff Phys Rev 149 491 (1966)36 S Bravyi DP DiVincenzo and D Loss Ann Phys

(NY) 326 2793 (2011)37 MO Goerbig Rev Mod Phys 83 1193 (2011)38 By direct comparison of Eqs (9)-(11) with Eq (15) we see

that the Pauli matrices we have introduced are a shorthandfor the following combinations of creationdestruction op-erators

11k = cdagger+M+1kc+M+1k + cdagger+Mkc+Mk

τzk = cdagger+M+1kc+M+1k minus cdagger+Mkc+Mk

τ+k = cdagger+M+1kc+Mk

τminusk = cdagger+Mkc+M+1k

39 A Principi M Polini and G Vignale Phys Rev B 80075418 (2009)

40 JW Negele and H Orland Quantum Many-Particle Sys-tems (Westview Press Boulder 1988)

41 D Hagenmuller and C Ciuti Phys Rev Lett 109 267403(2012)

42 A Auerbach and BE Larson Phys Rev B 43 7800(1991)

43 FMD Pellegrino M Polini V Giovannetti R Fazioand AH MacDonald to be published

2

energy effective Hamiltonian by taking into account thecoupling of the two-level systems which are resonant withthe cavity photon field to all non-resonant states Thiscoupling is crucially important in the strong-couplingregime where all the terms that are proportional toA2

emwhich are generated by our renormalization proceduremust be taken into account Indeed these guaranteegauge invariance as well as a no-go theorem for the occur-rence of a super-radiant phase transition therefore cor-roborating the findings of Ref 33 Finally we go beyondthis generalized Dicke description by demonstrating thatit is not adequate to describe the strong-coupling regimeof the cavity QED of the graphene cyclotron resonancein the limit of high doping In this case we derive anddiscuss a renormalized Hamiltonian for the entire stackof LLs in conduction band as dressed by the cavity elec-tromagnetic field

Our Article is organized as following In Section IIwe employ a canonical transformation34ndash36 to derive aneffective low-energy Hamiltonianmdashsee Eq (49)mdashfor thecavity QED of the graphene cyclotron resonance and dis-cuss the limits of its validity We analyze in great detailthe invariance of this effective Hamiltonian with respectto gauge transformations by employing a linear-responsetheory formalism In Sect III we use a functional-integralformalism to study the thermodynamic properties ofthe system described by the effective Hamiltonian InSect IV we transcend the generalized Dicke descriptionof Sect II and present a renormalized HamiltonianmdashseeEq (119)mdashthat enables the study of the strong-couplinglimit of the cavity QED of the graphene cyclotron reso-nance in the limit of high doping Finally in Sect V wereport a summary of our main findings and conclusions

II GENERALIZED DICKE HAMILTONIAN

In this Section we derive an effective low-energy Hamil-tonian for the cavity QED of the graphene cyclotron res-onance

A Landau levels in graphene

At low energies charge carriers in graphene are mod-eled by the usual single-channel massless Dirac fermionHamiltonian23

HD = vDσ middot p (1)

where vD asymp 106 ms is the Dirac velocity Here σ =(σx σy) is a 2D vector of Pauli matrices acting on sub-lattice degrees-of-freedom and p = minusi~nablar is the 2D mo-mentum measured from one of the two corners (valleys)of the Brillouin zone

A quantizing magnetic field B = Bz perpendicular tothe graphene sheet is coupled to the electronic degrees-of-freedom by replacing the canonical momentum p in

Eq (1) with the kinetic momentum Π = p + eA0cwhere A0 is the vector potential that describes the staticmagnetic field B The corresponding Hamiltonian is

H0 = vDσ middotΠ (2)

We work in the Landau gauge A0 = minusByx In thisgauge the canonical momentum along the x directionpx coincides with magnetic translation operator29 alongthe same direction and it commutes with the Hamilto-nian H0 Thus the eigenvalues of px are good quantumnumbers A complete set of eigenfunctions of the Hamil-tonian H0 in Eq (2) is provided by the two componentpseudospinors37

〈r|λ n k〉 =eikxradic

2L

(wminusnφnminus1(y minus `2Bk)λw+nφn(y minus `2Bk)

) (3)

where λ = +1 (minus1) denotes conduction (valence) bandlevels n isin N is the Landau level (LL) index and k is theeigenvalue of the magnetic translation operator in the xdirection In Eq (3)

wplusmnn =radic

1plusmn δn0 (4)

guarantees that the pseudospinor corresponding to then = 0 LL has weight only on one sublattice Furthermoreφn(y) with n = 0 1 2 are the normalized eigenfunc-tions of a 1D harmonic oscillator with frequency equalto the MDF cyclotron frequency ωc =

radic2vD`B Here

`B =radic~c(eB) 25 nm

radicB[Tesla] is the magnetic

lengthThe spectrum of the Hamiltonian (2) has the well-

known form37

ελn = λ~ωc

radicn (5)

Each LL has a degeneracy N = NfS(2π`2B) where Nf =

4 is the spin-valley degeneracy and S = L2 is the samplearea

B Total Hamiltonian

We now couple the 2D electron system described bythe Hamiltonian (2) to a single photon mode in a cavityWe denote by the symbol Aem the vector potential thatdescribes the cavity photon mode Carriers in grapheneare coupled to the cavity electromagnetic field via theminimal substitution

Πrarr Πprime = Π +e

cAem (6)

The cavity vector potential Aem will be treated withinthe dipole approximation We can neglect the spatialdependence of the electromagnetic field in the cavity be-cause the photon wavelength is much larger than anyother length scale of the system

3


Aem =

radic2π~c2εωV

eem(a+ adagger) (7)







Hem = ~ω(adaggera+

1

2

) (9)

H0 =sum

λnk


and

Hint =gradicN

sum

λλprimennprimek

(λwλne


+emδnprimenminus1

) (a+ adagger



g equiv ~ωc

radice2

2εLz~ω (12)



EM equiv1

2~ωc(radicM + 1 +

radicM) (13)




radicM) (14)


HDicke = Hem +

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


emτminusk )

] (15)




+

Nsum

k=1

[κ


N (a+ adagger)2τzk

]

(16)



4



H = Hem +H0 + VD + VO (17)



U = eS (18)





Hprime = H+ [SH] +1

2[S [SH]] + (20)



S =

infinsum

j=1

S(j) (21)



1 2 3 4 5 6 7 8 9 10

M

0

1

2

3

4

5

g 0

a)

1 5 10 15 20 25 30

ε

2

4

6

8

10

Mm

ax

b)




[S(1)H0 +Hem] + VO = 0 (22)



2[S(1) VO] +O(g3

0) (23)




(∣∣∣∣g

~ω minus |εmn|

∣∣∣∣) (24)

5



radicM + 2+

radicMminus2



quently g = ~ωc

radicα(2π






H0 =sum

misinSM

εmcdaggermcm +

sum

nisinSN

εncdaggerncn (25)




VD =sum

mmprimeisinSM


)cdaggermcmprime

+sum

nnprimeisinSN



and

VO =sum

misinSM nisinSN


)cdaggermcn


)cdaggerncm

] (27)




+emδmnminus1

)



S(1) =sum

misinSM nisinSN



)


Aω equiva

εmn minus ~ω+

adagger

εmn + ~ω (30)



)2 sum

misinSM nisinSN


times gmngnmN


)+ Bω (31)

where

6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN



)+

sum

mmprimeisinSM

sum

nnprimeisinSN









] (32)




0




with


gradicN


emτminusk )

minus κz








κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~


c ]


c

+


c ]


c

(37)


κ =ω2

ωc

g2

~


c ]


c


c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)


HN =sum

nisinSN

[εn +

sum

misinSM


(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN







HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`


cdaggerncn (40)



7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































3


Aem =

radic2π~c2εωV

eem(a+ adagger) (7)







Hem = ~ω(adaggera+

1

2

) (9)

H0 =sum

λnk


and

Hint =gradicN

sum

λλprimennprimek

(λwλne


+emδnprimenminus1

) (a+ adagger



g equiv ~ωc

radice2

2εLz~ω (12)



EM equiv1

2~ωc(radicM + 1 +

radicM) (13)




radicM) (14)


HDicke = Hem +

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


emτminusk )

] (15)




+

Nsum

k=1

[κ


N (a+ adagger)2τzk

]

(16)



4



H = Hem +H0 + VD + VO (17)



U = eS (18)





Hprime = H+ [SH] +1

2[S [SH]] + (20)



S =

infinsum

j=1

S(j) (21)



1 2 3 4 5 6 7 8 9 10

M

0

1

2

3

4

5

g 0

a)

1 5 10 15 20 25 30

ε

2

4

6

8

10

Mm

ax

b)




[S(1)H0 +Hem] + VO = 0 (22)



2[S(1) VO] +O(g3

0) (23)




(∣∣∣∣g

~ω minus |εmn|

∣∣∣∣) (24)

5



radicM + 2+

radicMminus2



quently g = ~ωc

radicα(2π






H0 =sum

misinSM

εmcdaggermcm +

sum

nisinSN

εncdaggerncn (25)




VD =sum

mmprimeisinSM


)cdaggermcmprime

+sum

nnprimeisinSN



and

VO =sum

misinSM nisinSN


)cdaggermcn


)cdaggerncm

] (27)




+emδmnminus1

)



S(1) =sum

misinSM nisinSN



)


Aω equiva

εmn minus ~ω+

adagger

εmn + ~ω (30)



)2 sum

misinSM nisinSN


times gmngnmN


)+ Bω (31)

where

6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN



)+

sum

mmprimeisinSM

sum

nnprimeisinSN









] (32)




0




with


gradicN


emτminusk )

minus κz








κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~


c ]


c

+


c ]


c

(37)


κ =ω2

ωc

g2

~


c ]


c


c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)


HN =sum

nisinSN

[εn +

sum

misinSM


(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN







HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`


cdaggerncn (40)



7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































4



H = Hem +H0 + VD + VO (17)



U = eS (18)





Hprime = H+ [SH] +1

2[S [SH]] + (20)



S =

infinsum

j=1

S(j) (21)



1 2 3 4 5 6 7 8 9 10

M

0

1

2

3

4

5

g 0

a)

1 5 10 15 20 25 30

ε

2

4

6

8

10

Mm

ax

b)




[S(1)H0 +Hem] + VO = 0 (22)



2[S(1) VO] +O(g3

0) (23)




(∣∣∣∣g

~ω minus |εmn|

∣∣∣∣) (24)

5



radicM + 2+

radicMminus2



quently g = ~ωc

radicα(2π






H0 =sum

misinSM

εmcdaggermcm +

sum

nisinSN

εncdaggerncn (25)




VD =sum

mmprimeisinSM


)cdaggermcmprime

+sum

nnprimeisinSN



and

VO =sum

misinSM nisinSN


)cdaggermcn


)cdaggerncm

] (27)




+emδmnminus1

)



S(1) =sum

misinSM nisinSN



)


Aω equiva

εmn minus ~ω+

adagger

εmn + ~ω (30)



)2 sum

misinSM nisinSN


times gmngnmN


)+ Bω (31)

where

6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN



)+

sum

mmprimeisinSM

sum

nnprimeisinSN









] (32)




0




with


gradicN


emτminusk )

minus κz








κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~


c ]


c

+


c ]


c

(37)


κ =ω2

ωc

g2

~


c ]


c


c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)


HN =sum

nisinSN

[εn +

sum

misinSM


(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN







HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`


cdaggerncn (40)



7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































5



radicM + 2+

radicMminus2



quently g = ~ωc

radicα(2π






H0 =sum

misinSM

εmcdaggermcm +

sum

nisinSN

εncdaggerncn (25)




VD =sum

mmprimeisinSM


)cdaggermcmprime

+sum

nnprimeisinSN



and

VO =sum

misinSM nisinSN


)cdaggermcn


)cdaggerncm

] (27)




+emδmnminus1

)



S(1) =sum

misinSM nisinSN



)


Aω equiva

εmn minus ~ω+

adagger

εmn + ~ω (30)



)2 sum

misinSM nisinSN


times gmngnmN


)+ Bω (31)

where

6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN



)+

sum

mmprimeisinSM

sum

nnprimeisinSN









] (32)




0




with


gradicN


emτminusk )

minus κz








κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~


c ]


c

+


c ]


c

(37)


κ =ω2

ωc

g2

~


c ]


c


c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)


HN =sum

nisinSN

[εn +

sum

misinSM


(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN







HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`


cdaggerncn (40)



7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































6

Bω =2~ωN

[a adagger

] sum

misinSM nisinSN



)+

sum

mmprimeisinSM

sum

nnprimeisinSN









] (32)




0




with


gradicN


emτminusk )

minus κz








κzs =g2

ΩM(36)

and

κzd =ω2

ωc

g2

~


c ]


c

+


c ]


c

(37)


κ =ω2

ωc

g2

~


c ]


c


c ]


c

+

radicM + 1minus

radicM

(radicM + 1 +

radicM)2ω2

c minus ω2

(38)


HN =sum

nisinSN

[εn +

sum

misinSM


(a+ adagger

)2 gmngnmN

]cdaggerncn +

sum

nnprimeisinSN







HprimeN =sum

nisinSN

εncdaggerncn

+(a+ adagger

)2 sum

nisinSN

sum

`


cdaggerncn (40)



7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































7









∆M (νmax) =sum

nisinSN

sum

`





~2ω2c

+g2

~ωcIMminus1(νmax)

minus g2

ΩMminus1

~2ω2


(44)


IMminus1(νmax) =

νmaxsum

`=M

[(radic`+ 1minus

radic`)ω2




radic`)2

] (45)






g2

~2ω2c

]

=g2


g2

ΩMminus1

ω2


(46)

with


IM (νmax)

=

infinsum

`=M+1

[(radic`+ 1minus

radic`)ω2




radic`)2

] (47)



HprimeN = ∆M

(a+ adagger

)2 (48)





Nsum

k=1

Hk (49)





∆M =g2

ΩM+

g2





radicM Here

FM (ω) equiv ω

2ωclog

(2ωc

radicM minus ω

2ωc

radicM + ω

)(52)

8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































8


IinfinM ω

2ωclog


) (53)


κz = minus g2

~ωc

1

2

radicM (54)

and

κ =g2

~ωc

[(M + 1)

radicM + 1 +

(M minus 1

2

)radicM

+1

4radicM(M + 1)(

radicM + 1 +

radicM)3

] (55)





χP(ω) =g2

N 〈〈τxtot τ

xtot〉〉ω


M

tanh

(βΩM

4

) (56)





~

int infin

0

dt ei(ω+i0+)t〈[A(t) B]〉 (57)



χD(ω) =2



(βΩM

4

) (58)


zk and 11tot =

sumNk=1 11k


χD(ω) = 2Ωg (59)

where


ΩM+

g2

~ωcIinfinM






M

tanh

(βΩM

4

)+ 2Ωg (61)



ΩMtanh

(βΩM

4

)(62)

and


(βΩM

4

)

=2g2

ΩMtanh

(βΩM

4

) (63)




χJ(ω)rarr g2

~2ω2c

[2EM +

~ω2

log


)] (64)


9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































9





1

2

)+

Nsum

k=1

[EM11k +

ΩM2τzk

+gradicN


] (65)







] (66)




Z =





S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0

dτ ξlowastjk(τ)

(part

partτ11jjprime +

ΩM2τzjjprime

)ξjprimek(τ) +

sum

kjjprime

int β

0

dτ ξlowastjk(τ)

gradicN


(κ


N τzjjprime

)ξjprimek(τ) (68)





2πi





ZSPA =

radicN

πβ~ω

int infin

0

dx eNΦ(x) (70)


10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































10


2 cosh

βΩM

2

radic(1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2

+ 2 cosh

(4βκx2

) (71)


ZSPA radic

2




dx

∣∣∣∣x=x0

= 0 (73)

and


dx2

∣∣∣∣x=x0

lt 0 (74)







ωIinfinM (76)





(A+B)n =

nsum

m=0

(nm

)AnminusmBm (77)


eNΦ(x) =

Nsum

`=0

(N`

)eNΦ`(x) (78)

where


+N minus `N log

[cosh

(4βκx2

)]

+`

N log

[cosh

(βΩM

2χ(x)

)] (79)

with

χ(x) equivradic(

1minus 8κz

ΩMx2

)2

+16g2

Ω2M

x2 (80)


dΦ`(x)

dx= 0 (81)


~ω + 4∆M + 4κ


(4βκx2

)]

=`

NΩM2

tanh

[βΩM

2χ(x)

]dχ(x)

d(x2) (82)


c(`)4 (x)x4 + c

(`)2 (x)x2 + c

(`)0 (x) = 0 (83)

11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































11

5 10 15 20 25 30 35 40

M

00

01

02

03

04

05

06

07

08f M


where

c(`)0 (x) = [~ω + 4∆M + 4κt

(`)1 (x)]2

minus[

4`

N

(g2

ΩMminus κz

)t2(x)

]2

(84)

c(`)2 (x) =

16(g2ΩM minus κz

)

Ω2M

[~ω + 4∆M + 4κt

(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (85)

and

c(`)4 (x) =

(8κz

ΩM

)2[

~ω + 4∆M + 4κt(`)1 (x)

]2

minus[

4`

N κzt2(x)

]2 (86)

with


(4βκx2

)(87)

and

t2(x) equiv tanh

[βΩM

2χ(x)

] (88)



(`)2 (x) and c





fM (ω) equiv ~ω + 4∆M lt 0 (89)

1 5 10 15 20 25 30

ε

10

11

12

13

14

15

16

17

Mcr

a)

00 02 04 06 08 10

x

minus10

minus05

00

05

10

15

20

Φ(x

)Φ

(0)

b)




12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































12








2 (90)

where

Φ(0) = log [2 + 2 cosh (βΩM2)] (91)


ZSPA Z(2)

free

β~ωg (92)

where


and








part(β~ω)=

1

β~ωg (95)

1 2 3 4 5 6 7 8

M

minus15

minus14

minus13

minus12

minus11

minus10

minus09

log

10n

(SP

A)

ph





minuspart logZSPA

part(β~ω)

∣∣∣∣g=0

=1

β~ω (96)







~ωΩM4g2

lt 1 (97)



g = ~ωc

radicα(2π


comesradicM + 1 +

radicM gt 2π



13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































13


x0 =g

~ω

[1minus

(~ωΩM

4g2

)2]12

(98)









S =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


2

+sum

kjjprime

int β

0



where

minusGminus10 =

part

partτ11 +

ΩM2τz (100)

Σ = Σ1 + Σ2 (101)

Σ1 =gradicN


and


(κ

N 11minus κz

N τz) (103)


Seff =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0 + Σ)]

(104)



Tr[log(minusGminus1

0 + Σ)]


0

)]

minus Tr

infinsum

n=1

(G0Σ)n

n (105)


Seff int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω

)φ(τ)


minus Tr

[log(minusGminus1

0

)]

+ Tr [G0Σ2] +1

2Tr [G0Σ1G0Σ1] (106)

We define


1

2Tr [G0Σ1G0Σ1] (107)



14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































14

ZG Z(2)free


minusint β

0


2+ S(2)


(108)

1 2 3 4 5 6 7 8

M

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

log

10n

(G)

ph






φ(τ) =1radicβ

+infinsum



ZG ZSPA

int infinprod

m=1


eminussum



earlier in Eq (92)


Det(Sm) = ω2m + ~2ω2 + 4~ω

[κ+ ∆M +

(κz minus g2 ΩM

ω2m + Ω2

M

)tanh

(βΩM

4

)] (111)



ZG ZSPA

infinprod

m=1

1

β2Det (Sm) (112)


1

β~ω

infinprod

m=1

1

β2(ω2m + ~2ω2)

=1

2 sinh(β~ω2)

equiv Zho(ω) (113)




βΩM


Zho(ΩM~) (114)

where

15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































15

~ωplusmn =


2plusmn


M ]2

4+ 4~ωΩMg2 tanh (βΩM4) (115)




~ωplusmn|Ωg=0 =


2plusmn


M )2

4+ 4~ωΩMg2 tanh (βΩM4) (116)

10 20 30 40 50 60

M

000

001

002

003

004

005

006

g 0






part(β~ω)= n

(SPA)ph

+sum

s=plusmn

[nB(ωs)minus

1

β~ωs

]partωspartω

(117)








IV BEYOND THE GDH



g0 =g


16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































16





1

2

)+ ∆

(a+ adagger

)2

+sum

nisinNk

[ε+nc

daggernkcnk +

w+ngradicN

(a+ adagger


)+κnN(a+ adagger

)2cdaggernkcnk

] (119)



In Eq (119)

κn =(w+ng)

2

~ωc


c


c minus ω2

+(wminusng)

2

~ωc


c


c minus ω2 (120)


∆ = minus g2

~ωc

ω2

ω2c minus ω2

+g2

~ωcIinfin0 (121)



mean-field theory


Z =





times exp

[minusradicN g

int β

0


~ωc

int β

0


] (122)

where

SB =

int β

0

dτ

φlowast(τ)

(part

partτ+ ~ω


[g2

~ωcz(τ) + ∆


2

(123a)

SF =sum

kn

int β

0

dτ

ξlowastnk(τ)

[part


]ξnk(τ)



] (123b)



17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































17




y =sum

kn


zlowast = minus 1


z =~ωc

g2

sum

nk

κnN 〈c



HMF = HB +HF (125)

Here

HB = ~ω(adaggera+

1

2

)+ gy

(a+ adagger

)

+

(g2

~ωcz + ∆

)(a+ adagger

)2(126)

and

HF =sum

nk

[ε+nc



+ cdaggern+1kcnk


] (127)


ylowast =ωg

~ω2

2yradicN

(128a)

zlowast = minusωω

2nB(~ω) + 1


where


(ω +

4∆

~+

4g2z

~2ωc

) (129)


HF sum

nk

[ε+nc



+ cdaggern+1kcnk


] (130)


ylowast = 0 (131a)

y = 0 (131b)

zlowast = 0 (131c)

z =~ωc

g2

sum

n

κnnF(ε+n) (131d)





Z(infin)free equiv

infinprod

n=0


]N (133)






Z ZMFradicD0

infinprod

m=1

1

Dm (134)

where

18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































18

Dm = 1minus ω

ωg2Gm(~ω)

infinsum

n=0


10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

a)

10 20 30 40 50 60

M

00

02

04

06

08

10

12

14

hωplusmnΩ

M

b)




ω radicω

[ω + 4

(g2

~ΩM+

g2

~2ωcIinfinM)]

(136)

and

Dm 1minus ω



~ωc(radicM + 1 +

radicM)2



Z =Z(infin)

free


βΩM


Zho(ΩM~) (138)


19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































19


ωg radicω

[ω + 4

g2

~2ωcIinfinM] (139)





free







Acknowledgments










20







































20







































theory of integer quantum hall polaritons in graphene

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