subir sachdev february 5, 2018 simons foundation, new...

Ultra-quantum metals

HARVARD

Subir SachdevFebruary 5, 2018

Simons Foundation, New York

Ubiquitous“Strange”,

“Bad”,

“Incoherent”,

or “Ultra-quantum”metal has a resistivity, ⇢, which obeys

⇢ ⇠ T ,

and

in some cases ⇢ � h/e2

(in two dimensions),where h/e2 is the quantum unit of resistance.

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Ultra-quantum metals just got stranger…

I. M. Hayes et. al., Nat. Phys. 2016

Ba-122

P. Giraldo-Gallo et. al., arXiv:1705.05806

LSCO

B-linear magnetoresistance!?

Ultra-quantum metals just got stranger…Scaling between B and T !?

Ba-122

I. M. Hayes et. al., Nat. Phys. 2016

Ba-122

Interactions between L. Balents, J. McGreevy,

S. Sachdev, T. Senthil(partly nucleated by student, Aavishkar Patel)

• Quantum spin liquids and the metal-insulator transition in dopedsemiconductors, Andrew C. Potter, Maissam Barkeshli, John McGreevy,T. Senthil, Phys. Rev. Lett. 109, 077205 (2012).

• A strongly correlated metal built from Sachdev-Ye-Kitaev mod-els, Xue-Yang Song, Chao-Ming Jian, Leon Balents, Phys. Rev. Lett.119, 216601 (2017).

• Magnetotransport in a model of a disordered strange metal,Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev,arXiv:1712.05026

• Translationally invariant non-Fermi liquid metals with criticalFermi-surfaces: Solvable models, Debanjan Chowdhury, Yochai Wer-man, Erez Berg, T. Senthil, arXiv:1801.06178

The Sachdev-Ye-Kitaev (SYK) model

Pick a set of random positions

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi

fifj + fjfi = 0 , fif†j + f

†j fi = �ij

Q =1

N

X

i

f†i fi

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Place electrons randomly on some sites

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi


Entangle electrons pairwise randomly

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi


This describes both a strange metal and a black hole!

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` f†i f

†j fkf` � µ

X

i

f†i fi


†j fi = �ij

Q =1

N

X

i

f†i fi


Feynman graph expansion in Uijk`, and graph-by-graph average,yields exact equations in the large N limit:

G(i!) =1

i! + µ� ⌃(i!), ⌃(⌧) = �U2G2(⌧)G(�⌧)

G(⌧ = 0�) = Q.

Low frequency analysis shows that the solutions must be gaplessand obey

⌃(z) = µ� 1

A

pz + . . . , G(z) =

Apz

where A = e�i⇡/4(⇡/U2)1/4 at half-filling. The ground state is anon-Fermi liquid, with a continuously variable density Q.

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S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

The SYK model

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

The SYK model

X

Feynman graph expansion in Uijk`, and graph-by-graph average,yields exact equations in the large N limit:

G(i!) =1

i! + µ� ⌃(i!), ⌃(⌧) = �U2G2(⌧)G(�⌧)

G(⌧ = 0�) = Q.

Low frequency analysis shows that the solutions must be gaplessand obey

⌃(z) = µ� 1

A

pz + . . . , G(z) =

Apz

where A = e�i⇡/4(⇡/U2)1/4 at half-filling. The ground state is anon-Fermi liquid, with a continuously variable density Q.

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Many-bodylevel spacing ⇠2�N = e�N ln 2

W. Fu and S. Sachdev, PRB 94, 035135 (2016)

Non-quasiparticleexcitations withspacing ⇠ e�Ns0

There are 2N many body levelswith energy E, which do not

admit a quasiparticledecomposition. Shown are all

values of E for a single cluster ofsize N = 12. The T ! 0 statehas an entropy SGPS = Ns0

with

s0 =G

⇡+

ln(2)

4= 0.464848 . . .

< ln 2

where G is Catalan’s constant,for the half-filled case Q = 1/2.

The SYK model

GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

Many-bodylevel spacing ⇠2�N = e�N ln 2

W. Fu and S. Sachdev, PRB 94, 035135 (2016)

Non-quasiparticleexcitations withspacing ⇠ e�Ns0

There are 2N many body levelswith energy E, which do not

admit a quasiparticledecomposition. Shown are all

values of E for a single cluster ofsize N = 12. The T ! 0 statehas an entropy SGPS = Ns0

with

s0 =G

⇡+

ln(2)

4= 0.464848 . . .

< ln 2

where G is Catalan’s constant,for the half-filled case Q = 1/2.

The SYK model

For SYK models withN = 2 supersymmetry, GPS=BPS

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The SYK model

• The last property indicates ⌧eq ⇠ ~/(kBT ), and thishas been found in a recent numerical study.

• Low energy, many-body density of states⇢(E) ⇠ eNs0 sinh(

p2(E � E0)N�)

• Low temperature entropy S = Ns0 +N�T + . . ..

• T = 0 fermion Green’s function G(⌧) ⇠ ⌧�1/2 atlarge ⌧ . (Fermi liquids with quasiparticles have G(⌧) ⇠1/⌧)

• T > 0 Green’s function has conformal invarianceG ⇠ (T/ sin(⇡kBT ⌧/~))1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999)

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arXiv:1706.07803

D. Stanford and E. Witten, 1703.04612A. M. Garica-Garcia, J.J.M. Verbaarschot, 1701.06593

D. Bagrets, A. Altland, and A. Kamenev, 1607.00694

Black holes

• Black holes have an entropyand a temperature, TH =~c3/(8⇡GMkB).

• The entropy is proportionalto their surface area.

• They relax to thermal equi-librium in a time⇠ ~/(kBTH).

LIGOSeptember 14, 2015

• The ring-down is predicted by General Relativity to happen in a

time8⇡GM

c3⇠ 8 milliseconds. Curiously this happens to equal

~kBTH

: so the ring down can also be viewed as the approach of a

quantum system to thermal equilibrium at the fastest possible rate.!

The SYK model

• The last property indicates ⌧eq ⇠ ~/(kBT ), and thishas been found in a recent numerical study.

• Low energy, many-body density of states⇢(E) ⇠ eNs0 sinh(

p2(E � E0)N�)

• Low temperature entropy S = Ns0 +N�T + . . ..

• T = 0 fermion Green’s function G(⌧) ⇠ ⌧�1/2 atlarge ⌧ . (Fermi liquids with quasiparticles have G(⌧) ⇠1/⌧)

• T > 0 Green’s function has conformal invarianceG ⇠ (T/ sin(⇡kBT ⌧/~))1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999)

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arXiv:1706.07803

D. Stanford and E. Witten, 1703.04612A. M. Garica-Garcia, J.J.M. Verbaarschot, 1701.06593

D. Bagrets, A. Altland, and A. Kamenev, 1607.00694

⇣ ~x

SYK and black holesBlack holehorizon

Black holes with a near-horizon AdS2 geometry (described by quantum gravity in 1+1 spacetime

dimensions) match the properties of the 0+1 dimensional SYK model in the previous

slide: Ns0 is the Bekenstein-Hawking entropyS. Sachdev, PRL 105, 151602 (2010); A. Kitaev (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arXiv:1606.01857

Mobile electrons (c) interacting with SYK quantum dots (f) with exchange interactions.

This yields the first model agreeing with magnetotransport in strange metals !

fc

Debanjan Chowdhury, Yochai Werman, Erez Berg, T. Senthil, arXiv:1801.06178

Infecting a Fermi liquid and making it SYK


This yields the first model agreeing with magnetotransport in strange metals !

fc


Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev, arXiv:1712.05026


This yields the first model agreeing with magnetotransport in strange metals ! Large N solution (with or without microscopic disorder) yields a

‘marginal Fermi liquid’ metal, with conductivities of the form:

�xx(B, T ) =1

T�L

✓B

T

◆

�xy(B, T ) =B

T 2�H

✓B

T

◆

where the scaling functions interpolate as

�L,H(b ! 0) ⇠ constant ; �L,H(b ! 1) ⇠ 1/b2

This solution exhibits B/T scaling, but the magnetoresistance ⇢xxsaturates for B � T .

<latexit sha1_base64="NvHRmARtpt4GwjsHXqwwoa/+rOw=">AAAEdXicjVPfSyRHEB7XTfQ2vzR5PDiKuDEKss5ucv6440AMBB8kGFhzguNtenpqZorr6R67a3SXYf/Puz8i73kK6dldUYMH6Zcuurq+qvrqq7hU5DgMPy61ltuffb6y+qzzxZdfff3N2vq3fzhTWYnn0ihjL2LhUJHGcyZWeFFaFEWs8G38/pfG//YGrSOjhzwp8aoQmaaUpGD/NFpvXUfakE5Qc+dU2Ayh+1sXnFFV44etW+IcjIXmNhVDQdIaJ01JEhJ 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</latexit><latexit sha1_base64="NvHRmARtpt4GwjsHXqwwoa/+rOw=">AAAEdXicjVPfSyRHEB7XTfQ2vzR5PDiKuDEKss5ucv6440AMBB8kGFhzguNtenpqZorr6R67a3SXYf/Puz8i73kK6dldUYMH6Zcuurq+qvrqq7hU5DgMPy61ltuffb6y+qzzxZdfff3N2vq3fzhTWYnn0ihjL2LhUJHGcyZWeFFaFEWs8G38/pfG//YGrSOjhzwp8aoQmaaUpGD/NFpvXUfakE5Qc+dU2Ayh+1sXnFFV44etW+IcjIXmNhVDQdIaJ01JEhJ 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</latexit>



Exact numerical solution of charge-transport equations in a random-resistor network. (M. M. Parish and P. Littlewood, Nature 426, 162 (2003))

• Current path length increases linearly with B at large B due to local Hall effect, which causes the global resistance to increase linearly with B at large B.

Need mesoscopic disorder to obtain linear-in-B magnetoresistanceInfecting a Fermi liquid and making it SYK

~ 50 T (a = 3.82 A)

nb/na = 0.8

�b/�a = 0.8

�a = 0.1kBT

(B = 0.0025)

(T = t/100)

t/100



Scaling between B and T

• This simple two-component model describes

a new state of matter which is realized by

electrons in the presence of strong interac-

tions and disorder.

• Can such a model be realized as a fixed-point

of a generic theory of strongly-interacting

electrons in the presence of disorder?

• Can we start from a single-band Hubbard

model with (or without) disorder, and end

up with such two-band fixed point, with emer-

gent local conservation laws?<latexit sha1_base64="MuUuoyJNBgbjqJKic9mrUaxqEOE=">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</latexit><latexit sha1_base64="MuUuoyJNBgbjqJKic9mrUaxqEOE=">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</latexit>

VOLUME 68, NUMBER 9 PHYSICAL REVIEW LETTERS 2 MARCH 1992

10'0.

10~ Si: P, B; n=2.6x10,n/n&=0. 58Si: P; n=2.9x10,n/pc=0. 78

10

102 10~ ~~0 '~

g++C~ ~

10—

1.8

10 2—00~

Og O

OO/w

0

9o00

10-1

10 10-' [

10-"10 ~

10

0.11P 2

[

10-1 10"

T(K)FIG. 2. Comparison of measured normalized susceptibilities

g/gc„, ;; (circles) of insulating Si:P and Si:P,B samples withtheoretical calculation (dashed lines) described in the text.

T(K)F1G. 1. Temperature dependence of normalized susceptibili-

ty g/gp, „~; of three Si:P,B samples with diA'erent normalizedelectron densities, n/n, =0.58, I. I, and 1.8. Solid lines throughdata are a guide to the eye.

terms. The ground state is tested for stability againstone- and two-electron hops. The susceptibility is thencalculated in the manner of BL using the antiferromag-netic spin- & Heisenberg exchange Hamiltonian:

H QJ(rj)S; S/,In Fig. 1 we show the enhancement of the susceptiblity

g (relative to gp.„„~;=3npa/2kaTF) as a function of tem-perature for all three compensated samples. These dataare qualitatively similar to the uncompensated Si:P data[4], i.e., the susceptibility increases towards lower tem-peratures approximately as a power law @~T '. Asshown in Fig. 2, this temperature dependence is observedover our entire temperature range for insulating samples.In this figure we have compared the normalized suscepti-bilities g/gc„„, ec T' ' (gc«,„=npa/3kaT) of compen-sated and uncompensated Si:P and find, using least-squares fits, that the exponent a=0.75+ 0.05 for Si:P,Bis somewhat larger than the value of 0.62 ~0.03 for Si:P.The dashed lines in Fig. 2 represent a quantitativetheoretical calculation of the susceptibility using no ad-justable parameters as explained below.The susceptibility of uncompensated Si:P for n &n,

was explained by Bhatt and Lee (BL) using a quantumspin- —, random Heisenberg antiferromagnetic Hamiltoni-an [23]. We have performed a similar computer calcula-tion of the susceptibility of a model appropriate for acompensated doped semiconductor deep in the insulatingphase. The model consists of distributing donor and ac-ceptor sites at random in a 3D continuum. The negative-ly charged acceptors provide a fixed random Coulomb po-tential while the electrons are allowed to occupy thedonor sites with the lowest-self-consistent energies,neglecting quantum-mechanical (hopping, exchange)

where the sum over i and j includes the electron occupieddonor sites. For the exchange constant we use the asymp-totic hydrogenic result [24] J(r) =Jo(r/a) / exp( —2r/a), where a 16 A (n, / a=0.25 for Si:P) and Jo=I40K. The high-temperature curvature of the theoreticallines in Fig. 2 is due to the asymptotic formula chosen forJ(r). This formula underestimates J at small r and theseare the values relevant at high temperatures. A theoreti-cal estimate of the exponent a, obtained from the low-temperature behavior of the dashed lines in Fig. 2, isfound to be slightly larger in the compensated case. Thisis due to the rearrangement of the electron occupieddonor sites, which results for the compensated case in adistribution differing from the Poisson distribution atshort distances. In summary, the theory with no adjust-able parameters is in remarkable agreement with the ex-perirnental results for the insulating phase.The difference between Si:P and Si:P,B is more

dramatic on the metallic side of the MI transition —thesusceptibility enhancement is unexpectedly large inSi:P,B at the lowest temperatures even for the very rnetal-lic sample n/n, =1.8. As shown in Fig. 3, comparing Si:Pand Si:P,B samples with similar values of n/n, =1.1, th.ecompensated system shows a factor of 3 to 5 larger localmoment fraction than the uncompensated one for T & 0.1K. This is in contrast to the theoretical results of Milo-vanovic, Sachdev, and Bhatt [16) who find for the disor-dered Hubbard model that the fraction of local moments

1419

• Electrons in doped silicon appear to sepa-rate into two components: localized spin mo-ments and itinerant electrons

M. J. Hirsch, D.F. Holcomb, R.N. Bhatt, and M.A. PaalanenPRL 68, 1418 (1992)

M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989)A.C. Potter, M. Barkeshli, J. McGreevy, T. Senthil, PRL 109, 077205 (2012)

M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989)

Magnetic excitations are localized

(eigenmodes of the spin susceptibility)

Charged fermionic excitations are

extended(eigenmodes of the

electron Hamiltonian)

Many-body quantum chaos and transport

• Using holographic analogies, Shenker, Stanford, Mal-

dacena introduced the “Lyapunov time”, ⌧L, the time

over which a generic many-body quantum system

loses memory of its initial state (defined via an “out-

of-time-order” (OTOC) correlator), and established

a shortest-possible time to reach quantum chaos

⌧L � ~2⇡kBT

A related bound was proposed earlier (Sachdev, 1999).

The SYK model, and black holes in Einstein gravity,

saturate this bound.<latexit sha1_base64="Fo91DwskZs4PR/LZhPDfdYlE7SI=">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</latexit><latexit sha1_base64="Fo91DwskZs4PR/LZhPDfdYlE7SI=">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</latexit>

Many-body quantum chaos and transport

A. A. Patel and S. Sachdev, PNAS 114, 1844 (2017)M. Blake, R. A. Davison, and S. Sachdev, PRD 96, 106008 (2017)

• Much theoretical work has examined connections be-tween this chaos bound, and measurable transportco-e�cients. The most robust connection is to thethermal di↵usivity , DT

DT ⇠ v2B⌧L

where vB is the “butterfly velocity” at which chaospropagates.

<latexit sha1_base64="rHTlzneykPpQWlJ1lMQohAwSJP8=">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</latexit><latexit sha1_base64="rHTlzneykPpQWlJ1lMQohAwSJP8=">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</latexit>

(Kapitulnik)

Many-body quantum chaos and transportMIT-CTP/4975

January 3, 2018

A quantum hydrodynamical description for scrambling and

many-body chaos

Mike Blake, Hyunseok Lee, and Hong Liu

Center for Theoretical Physics,

Massachusetts Institute of Technology,

Cambridge, MA 02139

Abstract

Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems

and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into

manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit

calculations in specific models. In this paper we propose a unified description of the exponential

growth and ballistic butterfly spreading of OTOCs across di↵erent systems using a newly formulated

“quantum hydrodynamics,” which is valid at finite ~ and to all orders in derivatives. The scrambling

of a generic few-body operator in a chaotic system is described as building up a “hydrodynamic

cloud,” and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic

action. The shift symmetry also shields correlation functions of the energy density and flux, and

time ordered correlation functions of generic operators from exponential growth, while leads to

chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping

of a pole at special values of complex frequency and momentum in two-point functions of energy

density and flux. This pole-skipping phenomenon may be considered as a “smoking gun” for the

hydrodynamic origin of the chaotic mode.

1

arX

iv:1

801.

0001

0v1

[hep

-th]

29 D

ec 2

017 arXiv:1801.00010

• A direct connection was established between the en-

ergy di↵usion mode and the OTOCs measuring chaos.

This assumes ‘dominance’ of the energy di↵usion mode

up to energies of order kBT , which is beyond the

hydrodynamic regime. Such a dominance holds in

SYK models and holographic theories, and can be ex-

pected more generally in ultra-quantum metals with-

out quasiparticle excitations.<latexit sha1_base64="c70WV0dS5ti9hJ5MdniNCK/7mMQ=">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</latexit><latexit sha1_base64="c70WV0dS5ti9hJ5MdniNCK/7mMQ=">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</latexit>

Non-zero density of fermions strongly coupled to gauge fields,(in the presence of disorder)


(Son)

• Half-filled Landau level

• Surfaces of (correlated) topological insula-

tors

• Strange metal state of high temperature

superconductors

• Pseudogap state of hole-doped cuprates<latexit sha1_base64="psKis3wp903mnHhacG4I6ePd0io=">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</latexit><latexit sha1_base64="psKis3wp903mnHhacG4I6ePd0io=">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</latexit>

(Son)




tors


superconductors


(Son)

SM

FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

SM

FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

Deconfinement transition of a gauge theory ?




tors


superconductors


(Son)


• Distinct phases of the gauge theory

can be distinguished even in the

presence of disorder

• Is the usual confinement/Higgs/deconfinement

criterion general enough in the

presence of disorder?

• Novel critical points or phases?<latexit sha1_base64="8BJCuuLNfkN/Rzb5/MvHSG1uERo=">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</latexit><latexit sha1_base64="8BJCuuLNfkN/Rzb5/MvHSG1uERo=">AAADF3icfVJNb9NAEF2brxI+msCRy4oIiVPqpEpaTlTAoVxQkQitFEfRej22V13vWrvr0GD1j3DgCj+DE+LKkV/BX2A2MaggYE5PM2/mzbzdpJLCuij6FoSXLl+5em3reufGzVu3t7u9O6+trg2HKddSm5OEWZBCwdQJJ+GkMsDKRMJxcvrU14+XYKzQ6pVbVTAvWa5EJjhzmFr0gl6cQC5UIxyU4i2cd2KPOs9QWijuaFXgdEt1Rl0BNGd1Dh5ps6KcKZoATdfUvBa2gJTCEhQVynPimOIuFhQH3488bVIwrcJzu55Y25pJyrXK8IISlNs5FHlud1K4kKPcYI/BjWkOCgx2gNJ1Xvxf6XEr9UIvQa5n4N2SVloohyeZ9jikgUp/ObDo9qNBNJ6M9napB6PJeIJgfzcaRhEdDqJ19EkbR4vu9zjVvPaLcsmsnQ2jys0bZlBOekdrCxXjpyyHGULFSrDzZv125/QBZlKa4TKZ9of67MWOhpXWrsoEmSVzhf2z5pN/q81ql+3PG6Gq2qEvG6GsltRp6j8CmmSAO7lCwDbWUF4wwzgajZMs4F9SuSua2MGZeyNS1GlGg7FQ3qGfNtB/g+lo8GgQvRz1D560Vm2Re+Q+eUiGZI8ckENyRKaEB2fB++BD8DF8F34KP4dfNtQwaHvukt8i/PoD4SH/eA==</latexit>

A solvable modelFractionalize the electron (ci↵, i = 1 . . . N , ↵ = 1 . . .M) into an“orthogonal fermion” fi↵ and an Ising spin �

zi = ±1:

ci↵ = �zi fi↵

This introduces a Z2 gauge invariance

�zi ! ⌘i�

zi , fi↵ ! ⌘ifi↵

The solvable model is (closely related to) the N ! 1, M ! 1 limit of

H =X

i

⇣�µf

†i↵fi↵ � g�

xi

⌘

+1pNM

X

i,j

tij�zi �

zj f

†i↵fj↵ +

1pNM

X

i,j

Jijf†i↵fi�f

†j�fj↵

where tij and Jij are random numbers.What is the fate of the Z2 gauge theory as a function of the coupling g?

(Senthil, Metlitski, Vishwanath, Sachdev…)

A solvable model

gc

(Wenbo Fu, Yingfei Gu, SS, G. Tarnopolsky)

There is no confining/Higgs phase, but there is a phase transi-

tion between a deconfined phase and a “critical-Higgs” phase.<latexit sha1_base64="UIpqfLi2sV3P+6x1XEPMiMn1yKw=">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</latexit><latexit sha1_base64="UIpqfLi2sV3P+6x1XEPMiMn1yKw=">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</latexit>

1/g<latexit sha1_base64="2MRNkH2IvJCtmqnORG3Ix6N58gQ=">AAACCHicdVDLTgJBEJz1ifhCPXqZCCae1gVR9Eb04lETV0iAkNmhgYmzs5uZXpVs+AIPXvUzPBmv/oVf4S84ICZqtJJOKlXd6e4KYikMet6bMzU9Mzs3n1nILi4tr6zm1tYvTZRoDj6PZKTrATMghQIfBUqoxxpYGEioBVcnI792DdqISF3gIIZWyHpKdAVnaCW/UNztFdq5vOfuH5YOygfUc70xLCnvVSoVjxYnSp5McNbOvTc7EU9CUMglM6ZR9GJspUyj4BKG2WZiIGb8ivWgYaliIZhWOj52SLet0qHdSNtSSMfq94mUhcYMwsB2hgz75rc3Ev/yGgl2D1upUHGCoPjnom4iKUZ09DntCA0c5cASxrWwt1LeZ5pxtPlkmwZseKqH/bSJcIs3omP3pGV3X6ihTegrBvo/8Uvukeudl/LV40lUGbJJtsgOKZIKqZJTckZ8wokg9+SBPDp3zpPz7Lx8tk45k5kN8gPO6wf5QZo+</latexit><latexit sha1_base64="2MRNkH2IvJCtmqnORG3Ix6N58gQ=">AAACCHicdVDLTgJBEJz1ifhCPXqZCCae1gVR9Eb04lETV0iAkNmhgYmzs5uZXpVs+AIPXvUzPBmv/oVf4S84ICZqtJJOKlXd6e4KYikMet6bMzU9Mzs3n1nILi4tr6zm1tYvTZRoDj6PZKTrATMghQIfBUqoxxpYGEioBVcnI792DdqISF3gIIZWyHpKdAVnaCW/UNztFdq5vOfuH5YOygfUc70xLCnvVSoVjxYnSp5McNbOvTc7EU9CUMglM6ZR9GJspUyj4BKG2WZiIGb8ivWgYaliIZhWOj52SLet0qHdSNtSSMfq94mUhcYMwsB2hgz75rc3Ev/yGgl2D1upUHGCoPjnom4iKUZ09DntCA0c5cASxrWwt1LeZ5pxtPlkmwZseKqH/bSJcIs3omP3pGV3X6ihTegrBvo/8Uvukeudl/LV40lUGbJJtsgOKZIKqZJTckZ8wokg9+SBPDp3zpPz7Lx8tk45k5kN8gPO6wf5QZo+</latexit>

Critical-Higgs phase.

f gapless and critical.

�zgapless and critical.

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Deconfined phase:a ‘critical orthogonal metal’.

f gapless and critical.�z gapped.

<latexit sha1_base64="du47f7VJqKJ5iJVhrhYDNbqcKp0=">AAAChHicdVFdb9MwFHXCx0bHR/l64sWiQ/BClHbtOnhAE/DA45Aom1SXcuPcJNYcO7KdjRJV/E5+AM/8Bdy0lQDBlSwdnXPuPdJxUklhXRx/D8IrV69d39m90dm7eev2ne7dex+trg3HCddSm7MELEqhcOKEk3hWGYQykXianL9Z6acXaKzQ6oNbVDgrIVciExycp+bdbyzBXKiGo3Jolp23yLXK/LGUVoW/+5IxCvQzN8L5HU m1cYXOtfKwRAfyacRYZz/bpzlUEq2loFK6dbcasyIv4dPX1lJhGnUYqnQbOO/24mh0NDgcHtI4itvxYHgwHo9j2t8wPbKZk3n3J0s1r0u/ziVYO+3HlZs1YHyixGWH1RYr4OeQ49RDBSXaWdMWtaRPPJPSTBv/lKMt+/tGA6W1izLxzhJcYf/WVuS/tGntsqNZI1RVO1R8HZTVkjpNV63TVBjkTi48gHU7lBdggPsO/CWL/uNU7oqGOfziLkXqc5phNBJq1dC2Bvp/MBlEL6L4/aB3/HpT1S55RB6TZ6RPxuSYvCMnZEI4+RHsBQ+Ch+FO+Dw8CEdraxhsdu6TPyZ89QtEKsPi</latexit><latexit sha1_base64="du47f7VJqKJ5iJVhrhYDNbqcKp0=">AAAChHicdVFdb9MwFHXCx0bHR/l64sWiQ/BClHbtOnhAE/DA45Aom1SXcuPcJNYcO7KdjRJV/E5+AM/8Bdy0lQDBlSwdnXPuPdJxUklhXRx/D8IrV69d39m90dm7eev2ne7dex+trg3HCddSm7MELEqhcOKEk3hWGYQykXianL9Z6acXaKzQ6oNbVDgrIVciExycp+bdbyzBXKiGo3Jolp23yLXK/LGUVoW/+5IxCvQzN8L5HU m1cYXOtfKwRAfyacRYZz/bpzlUEq2loFK6dbcasyIv4dPX1lJhGnUYqnQbOO/24mh0NDgcHtI4itvxYHgwHo9j2t8wPbKZk3n3J0s1r0u/ziVYO+3HlZs1YHyixGWH1RYr4OeQ49RDBSXaWdMWtaRPPJPSTBv/lKMt+/tGA6W1izLxzhJcYf/WVuS/tGntsqNZI1RVO1R8HZTVkjpNV63TVBjkTi48gHU7lBdggPsO/CWL/uNU7oqGOfziLkXqc5phNBJq1dC2Bvp/MBlEL6L4/aB3/HpT1S55RB6TZ6RPxuSYvCMnZEI4+RHsBQ+Ch+FO+Dw8CEdraxhsdu6TPyZ89QtEKsPi</latexit>

Critical-Higgs point.



A solvable model

gc

(Wenbo Fu, Yingfei Gu, SS, G. Tarnopolsky)

There is no confining/Higgs phase, but there is a phase transi-

tion between a deconfined phase and a “critical-Higgs” phase.<latexit sha1_base64="UIpqfLi2sV3P+6x1XEPMiMn1yKw=">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</latexit><latexit sha1_base64="UIpqfLi2sV3P+6x1XEPMiMn1yKw=">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</latexit>

1/g<latexit sha1_base64="2MRNkH2IvJCtmqnORG3Ix6N58gQ=">AAACCHicdVDLTgJBEJz1ifhCPXqZCCae1gVR9Eb04lETV0iAkNmhgYmzs5uZXpVs+AIPXvUzPBmv/oVf4S84ICZqtJJOKlXd6e4KYikMet6bMzU9Mzs3n1nILi4tr6zm1tYvTZRoDj6PZKTrATMghQIfBUqoxxpYGEioBVcnI792DdqISF3gIIZWyHpKdAVnaCW/UNztFdq5vOfuH5YOygfUc70xLCnvVSoVjxYnSp5McNbOvTc7EU9CUMglM6ZR9GJspUyj4BKG2WZiIGb8ivWgYaliIZhWOj52SLet0qHdSNtSSMfq94mUhcYMwsB2hgz75rc3Ev/yGgl2D1upUHGCoPjnom4iKUZ09DntCA0c5cASxrWwt1LeZ5pxtPlkmwZseKqH/bSJcIs3omP3pGV3X6ihTegrBvo/8Uvukeudl/LV40lUGbJJtsgOKZIKqZJTckZ8wokg9+SBPDp3zpPz7Lx8tk45k5kN8gPO6wf5QZo+</latexit><latexit sha1_base64="2MRNkH2IvJCtmqnORG3Ix6N58gQ=">AAACCHicdVDLTgJBEJz1ifhCPXqZCCae1gVR9Eb04lETV0iAkNmhgYmzs5uZXpVs+AIPXvUzPBmv/oVf4S84ICZqtJJOKlXd6e4KYikMet6bMzU9Mzs3n1nILi4tr6zm1tYvTZRoDj6PZKTrATMghQIfBUqoxxpYGEioBVcnI792DdqISF3gIIZWyHpKdAVnaCW/UNztFdq5vOfuH5YOygfUc70xLCnvVSoVjxYnSp5McNbOvTc7EU9CUMglM6ZR9GJspUyj4BKG2WZiIGb8ivWgYaliIZhWOj52SLet0qHdSNtSSMfq94mUhcYMwsB2hgz75rc3Ev/yGgl2D1upUHGCoPjnom4iKUZ09DntCA0c5cASxrWwt1LeZ5pxtPlkmwZseKqH/bSJcIs3omP3pGV3X6ihTegrBvo/8Uvukeudl/LV40lUGbJJtsgOKZIKqZJTckZ8wokg9+SBPDp3zpPz7Lx8tk45k5kN8gPO6wf5QZo+</latexit>

Critical-Higgs phase.



<latexit sha1_base64="Z/w7RVojjt1g12b0LgXYpNIAWlc=">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</latexit><latexit sha1_base64="Z/w7RVojjt1g12b0LgXYpNIAWlc=">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</latexit>

Deconfined phase:a ‘critical orthogonal metal’.

f gapless and critical.�z gapped.

<latexit sha1_base64="du47f7VJqKJ5iJVhrhYDNbqcKp0=">AAAChHicdVFdb9MwFHXCx0bHR/l64sWiQ/BClHbtOnhAE/DA45Aom1SXcuPcJNYcO7KdjRJV/E5+AM/8Bdy0lQDBlSwdnXPuPdJxUklhXRx/D8IrV69d39m90dm7eev2ne7dex+trg3HCddSm7MELEqhcOKEk3hWGYQykXianL9Z6acXaKzQ6oNbVDgrIVciExycp+bdbyzBXKiGo3Jolp23yLXK/LGUVoW/+5IxCvQzN8L5HU m1cYXOtfKwRAfyacRYZz/bpzlUEq2loFK6dbcasyIv4dPX1lJhGnUYqnQbOO/24mh0NDgcHtI4itvxYHgwHo9j2t8wPbKZk3n3J0s1r0u/ziVYO+3HlZs1YHyixGWH1RYr4OeQ49RDBSXaWdMWtaRPPJPSTBv/lKMt+/tGA6W1izLxzhJcYf/WVuS/tGntsqNZI1RVO1R8HZTVkjpNV63TVBjkTi48gHU7lBdggPsO/CWL/uNU7oqGOfziLkXqc5phNBJq1dC2Bvp/MBlEL6L4/aB3/HpT1S55RB6TZ6RPxuSYvCMnZEI4+RHsBQ+Ch+FO+Dw8CEdraxhsdu6TPyZ89QtEKsPi</latexit><latexit sha1_base64="du47f7VJqKJ5iJVhrhYDNbqcKp0=">AAAChHicdVFdb9MwFHXCx0bHR/l64sWiQ/BClHbtOnhAE/DA45Aom1SXcuPcJNYcO7KdjRJV/E5+AM/8Bdy0lQDBlSwdnXPuPdJxUklhXRx/D8IrV69d39m90dm7eev2ne7dex+trg3HCddSm7MELEqhcOKEk3hWGYQykXianL9Z6acXaKzQ6oNbVDgrIVciExycp+bdbyzBXKiGo3Jolp23yLXK/LGUVoW/+5IxCvQzN8L5HU m1cYXOtfKwRAfyacRYZz/bpzlUEq2loFK6dbcasyIv4dPX1lJhGnUYqnQbOO/24mh0NDgcHtI4itvxYHgwHo9j2t8wPbKZk3n3J0s1r0u/ziVYO+3HlZs1YHyixGWH1RYr4OeQ49RDBSXaWdMWtaRPPJPSTBv/lKMt+/tGA6W1izLxzhJcYf/WVuS/tGntsqNZI1RVO1R8HZTVkjpNV63TVBjkTi48gHU7lBdggPsO/CWL/uNU7oqGOfziLkXqc5phNBJq1dC2Bvp/MBlEL6L4/aB3/HpT1S55RB6TZ6RPxuSYvCMnZEI4+RHsBQ+Ch+FO+Dw8CEdraxhsdu6TPyZ89QtEKsPi</latexit>

Critical-Higgs point.



Distinct exponents

Thoughts on field theories of ultra-quantum metals

• Breakdown of quasiparticles requires strong coupling

to a low energy collective mode

• In all known cases, we can write down the singular

processes in terms of a continuum field theory of the

fermions near the Fermi surface coupled to the col-

lective mode.

• In all known cases, the continuum critical theory has

a conserved total (pseudo-) momentum, ~P , which

commutes with the Hamiltonian. This momentum

may not be equal to the crystal momentum of the

underlying lattice model.<latexit sha1_base64="g0am6ho2E7vX53EPw3XYkYAmO98=">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</latexit><latexit sha1_base64="g0am6ho2E7vX53EPw3XYkYAmO98=">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</latexit>


• As long as � ~J,~P 6= 0 (where ~J is the electrical cur-rent) the d.c. resistivity of the critical theory isexactly zero. This is the case even though the elec-tron self energy can be highly singular and there areno fermionic quasiparticles (many well-known paperson non-Fermi liquid transport ignore this point.)

• We need to include additional (dangerously) irrele-vant umklapp corrections to obtain a non-zero resis-tivity. Because these additional corrections are ir-relevant, it is di�cult to see how they can induce alinear-in-T resistivity.

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Theories of metallic states without quasiparticlesin the presence of disorder

• Well-known perturbative theory of disordered met-als has 2 classes of known fixed points, the insulatorat strong disorder, and the metal at weak disorder.The latter state has long-lived, extended quasiparti-cle excitations (which are not plane waves).

• Needed: a metallic fixed point at intermedi-ate disorder and strong interactions withoutquasiparticle excitations. Although disorder ispresent, it largely self-averages at long scales.

• SYK models


subir sachdev february 5, 2018 simons foundation, new...

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