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Quantum matter without quasiparticles

HARVARD

Talk online: sachdev.physics.harvard.edu

Frontiers in Many Body Physics: Memorial for Lev Petrovich Gor’kov

National High Magnetic Field Laboratory, Tallahassee

Subir SachdevJanuary 13, 2018

Ubiquitous“Strange”,

“Bad”,

or “Incoherent”,

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.

Strange 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!?

Strange metals just got stranger…Scaling between B and T !?

Ba-122

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

Ba-122

Theories of metallic states without quasiparticleswithout disorder

• Breakdown of quasiparticles arises from long-wavelengthcoupling of electrons to some bosonic collective mode.In all cases this can be written in terms of a contin-uum theory with a conserved momentum.The critical theory has zero resistance, eventhough the electron quasiparticles do not ex-ist.

• Need to add irrelevant (umklapp) e↵ects to obtain anon-zero resistivity, but this not yield a large linear-in-T resistivity.

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

The Sachdev-Ye-Kitaev (SYK) model

Pick a set of random positions

Place electrons randomly on some sites

The SYK model

Entangle electrons pairwise randomly

The SYK model

This describes both a strange metal and a black hole!

The SYK model

A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

(See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit;T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981))

The SYK model

H =1

(2N)3/2

NX

i,j,k,`=1

Uij;k` c†i c

†jckc` � µ

X

i

c†i ci

cicj + cjci = 0 , cic†j + c

†jci = �ij

Q =1

N

X

i

c†i ci

Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U2

N ! 1 yields critical strange metal.

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

The SYK model

Feynman graph expansion in Jij.., 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.

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

The SYK model• T = 0 fermion Green’s function is singular:

G(⌧) ⇠ 1p⌧

at large ⌧ .

(Fermi liquids with quasiparticles have

G(⌧) ⇠ 1/⌧)

• T > 0 Green’s function has conformal invariance,and is ‘dephased’ at the characteristic scale⇠ kBT/~,which is independent of U .

G ⇠ e�2⇡ET⌧

✓T

sin(⇡kBT ⌧/~)

◆1/2

E measures particle-hole asymmetry.

A. Georges and O. Parcollet PRB 59, 5341 (1999)S. Sachdev, PRX, 5, 041025 (2015)


G(⌧) ⇠ 1p⌧

at large ⌧ .


G(⌧) ⇠ 1/⌧)


G ⇠ e�2⇡ET⌧

✓T

sin(⇡kBT ⌧/~)

◆1/2



G(⌧) ⇠ 1p⌧

at large ⌧ .


G(⌧) ⇠ 1/⌧)


G ⇠ e�2⇡ET⌧

✓T

sin(⇡kBT ⌧/~)

◆1/2


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

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

The SYK model

So the Green’s functions display thermal ‘damping’ at ascale set by T alone, which is independent of U .

!

�ImGR(!)�ReGR(!)

GR(!)GA(!)

FIG. 1. Plots of the Green’s functions in Eq. (3) for � = 1/4, q = 1, T = 1, A = 1, E = 1/4 with

~ = kB = 1. Note that while neither ImGR(!) or ReGR(!) have any definite properties under ! $ �!,

the product GR(!)GA(!) becomes an even function of ! after a shift by !S = 2⇡qET = ⇡/2.

where SBH is the Bekenstein-Hawking entropy densiy of the AdS2 horizon. Indeed, Eq. (8) is a

general consequence of the classical Maxwell and Einstein equations, and the conformal invariance

of the AdS2 horizon, as we shall show in Section III B. Moreover, a Legendre transform of the

identity in Eq. (8) was established by Sen [15, 16] for a wide class of theories of gravity in the

Wald formalism [17–21], in which SBH is generalized to the Wald entropy.

The main result of this paper is the identical forms of the relationship Eq. (7) for the statistical

entropy of the SY state, and Eq. (8) for the Bekenstein-Hawking entropy of AdS2 horizons. This

result is strong evidence that there is a gravity dual of the SY state with a AdS2 horizon. Con-

versely, assuming the existence of a gravity dual, Eqs. (7) and (8) show that such a correspondence

is consistent only if the black hole entropy has the Bekenstein-Hawking value, and endow the black

hole entropy with a statistical interpretation [30].

It is important to keep in mind that (as we mentioned earlier) the models considered here have

a di↵erent ‘equation of state’ relating E to Q: this is specified for the SY state in Eq. (A5), for the

planar black hole in Eq. (58), and for the spherical black hole in Eq. (B8).

The holographic link between the SY state and the AdS2 horizons of charged black branes has

been conjectured earlier [22–24], based upon the presence of a non-vanishing zero temperature

entropy density and the conformal structure of correlators. The results above sharpen this link by

establishing a precise quantitative connection for the Bekenstein-Hawking entropy [31, 32] of the

black hole with the UV complete computation on the microscopic degrees of freedom of the SY

5

Green’s functions away from half-filling

Building a metal

......

......

H =X

x

X

i<j,k<l

Uijkl,xc†ixc

†jxckxclx +

X

hxx0i

X

i,j

tij,xx0c†i,xcj,x0

A strongly correlated metal built from Sachdev-Ye-Kitaev models

Xue-Yang Song,1, 2 Chao-Ming Jian,2, 3 and Leon Balents2

1International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China

2Kavli Institute of Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

3Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA

(Dated: May 23, 2017)

Prominent systems like the high-Tc cuprates and heavy fermions display intriguing features going beyondthe quasiparticle description. The Sachdev-Ye-Kitaev(SYK) model describes a 0 + 1D quantum cluster withrandom all-to-all four-fermion interactions among N Fermion modes which becomes exactly solvable as N !

1, exhibiting a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absenceof quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site quadratic

hopping. Combining the imaginary time path integral with real time path integral formulation, we obtain aheavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperatureLandau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We find linear intemperature resistivity in the incoherent regime, and a Lorentz ratio L ⌘

⇢T

varies between two universal valuesas a function of temperature. Our work exemplifies an analytically controlled study of a strongly correlatedmetal.

Introduction - Strongly correlated metals comprise an en-during puzzle at the heart of condensed matter physics. Com-monly a highly renormalized heavy Fermi liquid occurs be-low a small coherence scale, while at higher temperatures abroad incoherent regime pertains in which quasi-particle de-scription fails[1–9]. Despite the ubiquity of this phenomenol-ogy, strong correlations and quantum fluctuations make itchallenging to study. The exactly soluble SYK models pro-vide a powerful framework to study such physics. The most-studied SYK4 model, a 0 + 1D quantum cluster of N Ma-jorana fermion modes with random all-to-all four-fermioninteractions[10–18] has been generalized to SYKq modelswith q-fermion interactions. Subsequent works[19, 20] ex-tended the SYK model to higher spatial dimensions by cou-pling a lattice of SYK4 quantum clusters by additional four-fermion “pair hopping” interactions. They obtained electricaland thermal conductivities completely governed by di↵usivemodes and nearly temperature-independent behavior owing tothe identical scaling of the inter-dot and intra-dot couplings.

Here, we take one step closer to realism by considering alattice of complex-fermion SYK clusters with SYK4 intra-cluster interaction of strength U0 and random inter-cluster“SYK2” two-fermion hopping of strength t0[21–26]. Un-like the previous higher dimensional SYK models where lo-cal quantum criticality governs the entire low temperaturephysics, here as we vary the temperature, two distinctivemetallic behaviors appear, resembling the previously men-tioned heavy fermion systems. We assume t0 ⌧ U0, whichimplies strong interactions, and focus on the correlated regimeT ⌧ U0. We show the system has a coherence temperature

scale Ec ⌘ t20/U0[21, 27, 28] between a heavy Fermi liquid

and an incoherent metal. For T < Ec, the SYK2 induces aFermi liquid, which is however highly renormalized by thestrong interactions. For T > Ec, the system enters the incoher-ent metal regime and the resistivity ⇢ depends linearly on tem-perature. These results are strikingly similar to those of Par-collet and Georges[29], who studied a variant SYK model ob-tained in a double limit of infinite dimension and large N. Ourmodel is simpler, and does not require infinite dimensions. Wealso obtain further results on the thermal conductivity , en-tropy density and Lorentz ratio[30, 31] in this crossover. Thiswork bridges traditional Fermi liquid theory and the hydrody-namical description of an incoherent metallic system.

SYK model and Imaginary-time formulation - We considera d-dimensional array of quantum dots, each with N speciesof fermions labeled by i, j, k · · · ,

H =X

x

X

i< j,k<l

Ui jkl,xc†

ixc†

jxc

kxc

lx+X

hxx0i

X

i, j

ti j,xx0c†

i,xcj,x0 (1)

where Ui jkl,x = U⇤

kli j,x and ti j,xx0 = t⇤

ji,x0x are random zero meancomplex variables drawn from Gaussian distribution whosevariances |Ui jkl,x|

2 = 2U20/N

3 and |ti j,x,x0 |2 = t

20/N.

In the imaginary time formalism, one studies the partitionfunction Z = Tr e

��(H�µN), with N =P

i,x c†

i,xci,x, written as

a path integral over Grassman fields cix⌧, cix⌧. Owing to theself-averaging established for the SYK model at large N, it issu�cient to study Z =

R[dc][dc]e�S c , with (repeated species

indices are summed over)

S c =X

x

Z �

0d⌧ cix⌧(@⌧ � µ)cix⌧ �

Z �

0d⌧1d⌧2

hX

x

U20

4N3 cix⌧1 c jx⌧1 ckx⌧1 clx⌧1 clx⌧2 ckx⌧2 c jx⌧2 cix⌧2 +X

hxx0i

t20

Ncix⌧1 c jx0⌧1 c jx0⌧2 cix⌧2

i. (2)

The basic features can be determined by a simple power- counting. Considering for simplicity µ = 0, starting from

Ut

Sachdev-Ye-Kitaev modelA toy exactly soluble model

of a non-Fermi liquid

Like a strongly interacting quantum dot or atom with complicated Kanamori

interactions between many “orbitals”

H =X

i<j,k<l

Uijkl c†i c

†jckcl

|Uijkl|2 =2U2

N3

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1. arXiv:1707.02308 [pdf, ps, other]Title: Fracton topological phases from strongly coupled spin chainsAuthors: Gábor B. Halász, Timothy H. Hsieh, Leon BalentsComments: 5+1 pages, 4 figuresSubjects: Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph)

2. arXiv:1705.00117 [pdf, other]Title: A strongly correlated metal built from Sachdev-Ye-Kitaev modelsAuthors: Xue-Yang Song, Chao-Ming Jian, Leon BalentsComments: 17 pages, 6 figuresSubjects: Strongly Correlated Electrons (cond-mat.str-el)

3. arXiv:1703.08910 [pdf, other]Title: Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn

Sn/GeAuthors: Jianpeng Liu, Leon BalentsSubjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl-sci)

4. arXiv:1611.00646 [pdf, ps, other]Title: Spin-Orbit Dimers and Non-Collinear Phases in Cubic Double PerovskitesAuthors: Judit Romhányi, Leon Balents, George Jackeli

3

d1

See also A. Georges and O. Parcollet PRB 59, 5341 (1999)






(Dated: May 8, 2017)

Strongly correlated metals comprise an enduring puzzle at the heart of condensed matter physics.Commonly a highly renormalized heavy Fermi liquid occurs below a small coherence scale, while athigher temperatures a broad incoherent regime pertains in which quasi-particle description fails. Despitethe ubiquity of this phenomenology, strong correlations and quantum fluctuations make it challenging tostudy. The Sachdev-Ye-Kitaev(SYK) model describes a 0 + 1D quantum cluster with random all-to-allfour-fermion interactions among N Fermion modes which becomes exactly solvable as N ! 1, exhibitinga zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absence of quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site quadratic hopping.Combining the imaginary time path integral with real time path integral formulation, we obtain a heavyFermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperatureLandau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We findlinear in temperature resistivity in the incoherent regime, and a Lorentz ratio L ⌘

⇢T

varies between twouniversal values as a function of temperature. Our work exemplifies an analytically controlled study of astrongly correlated metal.

Prominent systems like the high-Tc cuprates and heavyfermions display intriguing features going beyond the quasi-particle description[1–9]. The exactly soluble SYK modelsprovide a powerful framework to study such physics. Themost-studied SYK4 model, a 0 + 1D quantum cluster of N

Majorana fermion modes with random all-to-all four-fermioninteractions[10–18] has been generalized to SYKq modelswith q-fermion interactions. Subsequent works[19, 20] ex-tended the SYK model to higher spatial dimensions by cou-pling a lattice of SYK4 quantum clusters by additional four-fermion “pair hopping” interactions. They obtained electricaland thermal conductivities completely governed by di↵usivemodes and nearly temperature-independent behavior owing tothe identical scaling of the inter-dot and intra-dot couplings.

Here, we take one step closer to realism by considering alattice of complex-fermion SYK clusters with SYK4 intra-cluster interaction of strength U0 and random inter-cluster“SYK2” two-fermion hopping of strength t0[21–25]. Un-like the previous higher dimensional SYK models where lo-cal quantum criticality governs the entire low temperaturephysics, here as we vary the temperature, two distinctivemetallic behaviors appear, resembling the previously men-tioned heavy fermion systems. We assume t0 ⌧ U0, whichimplies strong interactions, and focus on the correlated regimeT ⌧ U0. We show the system has a coherence temperaturescale Ec ⌘ t

20/U0[21, 26, 27] between a heavy Fermi liquid

and an incoherent metal. For T < Ec, the SYK2 induces a

Fermi liquid, which is however highly renormalized by thestrong interactions. For T > Ec, the system enters the incoher-ent metal regime and the resistivity ⇢ depends linearly on tem-perature. These results are strikingly similar to those of Par-collet and Georges[28], who studied a variant SYK model ob-tained in a double limit of infinite dimension and large N. Ourmodel is simpler, and does not require infinite dimensions. Wealso obtain further results on the thermal conductivity , en-tropy density and Lorentz ratio[29, 30] in this crossover. Thiswork bridges traditional Fermi liquid theory and the hydrody-namical description of an incoherent metallic system.


H =X

x

X

i< j,k<l

Ui jkl,xc†

ixc†

jxc

kxc

lx+X

hxx0i

X

i, j

ti j,xx0c†

i,xcj,x0 (1)




2 = 2U20/N

3 and |ti j,x,x0 |2 = t

20/N.



i,x c†

i,xci,x, written as




S c =X

x

Z �

0d⌧ cix⌧(@⌧ � µ)cix⌧ �

Z �

0d⌧1d⌧2

hX

x

U20


hxx0i

t20


i. (2)


arX

iv:1

705.

0011

7v2

[con

d-m

at.st

r-el

] 5

May

201

7

Building a metal

......

......

H =X

x

X

i<j,k<l

Uijkl,xc†ixc

†jxckxclx +

X

hxx0i

X

i,j

tij,xx0c†i,xcj,x0






(Dated: May 23, 2017)

Prominent systems like the high-Tc cuprates and heavy fermions display intriguing features going beyondthe quasiparticle description. The Sachdev-Ye-Kitaev(SYK) model describes a 0 + 1D quantum cluster withrandom all-to-all four-fermion interactions among N Fermion modes which becomes exactly solvable as N !

1, exhibiting a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absenceof quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site quadratic

hopping. Combining the imaginary time path integral with real time path integral formulation, we obtain aheavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperatureLandau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We find linear intemperature resistivity in the incoherent regime, and a Lorentz ratio L ⌘

⇢T

varies between two universal valuesas a function of temperature. Our work exemplifies an analytically controlled study of a strongly correlatedmetal.

Introduction - Strongly correlated metals comprise an en-during puzzle at the heart of condensed matter physics. Com-monly a highly renormalized heavy Fermi liquid occurs be-low a small coherence scale, while at higher temperatures abroad incoherent regime pertains in which quasi-particle de-scription fails[1–9]. Despite the ubiquity of this phenomenol-ogy, strong correlations and quantum fluctuations make itchallenging to study. The exactly soluble SYK models pro-vide a powerful framework to study such physics. The most-studied SYK4 model, a 0 + 1D quantum cluster of N Ma-jorana fermion modes with random all-to-all four-fermioninteractions[10–18] has been generalized to SYKq modelswith q-fermion interactions. Subsequent works[19, 20] ex-tended the SYK model to higher spatial dimensions by cou-pling a lattice of SYK4 quantum clusters by additional four-fermion “pair hopping” interactions. They obtained electricaland thermal conductivities completely governed by di↵usivemodes and nearly temperature-independent behavior owing tothe identical scaling of the inter-dot and intra-dot couplings.

Here, we take one step closer to realism by considering alattice of complex-fermion SYK clusters with SYK4 intra-cluster interaction of strength U0 and random inter-cluster“SYK2” two-fermion hopping of strength t0[21–26]. Un-like the previous higher dimensional SYK models where lo-cal quantum criticality governs the entire low temperaturephysics, here as we vary the temperature, two distinctivemetallic behaviors appear, resembling the previously men-tioned heavy fermion systems. We assume t0 ⌧ U0, whichimplies strong interactions, and focus on the correlated regimeT ⌧ U0. We show the system has a coherence temperature

scale Ec ⌘ t20/U0[21, 27, 28] between a heavy Fermi liquid

and an incoherent metal. For T < Ec, the SYK2 induces aFermi liquid, which is however highly renormalized by thestrong interactions. For T > Ec, the system enters the incoher-ent metal regime and the resistivity ⇢ depends linearly on tem-perature. These results are strikingly similar to those of Par-collet and Georges[29], who studied a variant SYK model ob-tained in a double limit of infinite dimension and large N. Ourmodel is simpler, and does not require infinite dimensions. Wealso obtain further results on the thermal conductivity , en-tropy density and Lorentz ratio[30, 31] in this crossover. Thiswork bridges traditional Fermi liquid theory and the hydrody-namical description of an incoherent metallic system.


H =X

x

X

i< j,k<l

Ui jkl,xc†

ixc†

jxc

kxc

lx+X

hxx0i

X

i, j

ti j,xx0c†

i,xcj,x0 (1)




2 = 2U20/N

3 and |ti j,x,x0 |2 = t

20/N.



i,x c†

i,xci,x, written as




S c =X

x

Z �

0d⌧ cix⌧(@⌧ � µ)cix⌧ �

Z �

0d⌧1d⌧2

hX

x

U20


hxx0i

t20


i. (2)


Ut





All papers










3

d1


Self-consistent equations

2

t0 = 0, the U20 term is invariant under ⌧! b⌧ and c! b

�1/4c,

c ! b�1/4

c, fixing the scaling dimension � = 1/4 of thefermion fields. Under this scaling c@⌧c term is irrelevant.Yet upon addition of two-fermion coupling, under rescaling,t20 ! bt

20, so two-fermion coupling is a relevant perturba-

tion. By standard reasoning, this implies a cross-over fromthe SYK4-like model to another regime at the energy scalewhere the hopping perturbation becomes dominant, which isEc = t

20/U0. We expect the renormalization flow is to the

SYK2 regime. Indeed keeping the SYK2 term invariant fixes

� = 1/2, and U20 ! b

�1U

20 is irrelevant. Since the SYK2

Hamiltonian (i.e.,U0 = 0) is quadratic, the disordered freefermion model supports quasi-particles and defines a Fermiliquid limit. For t0 ⌧ U0, Ec defines a crossover scale be-tween SYK4-like non-Fermi liquid and the low temperatureFermi liquid.

At the level of thermodynamics, this crossover can be rig-orously established using imaginary time formalism. Intro-ducing a composite field Gx(⌧, ⌧0) = �1

N

Pi cix⌧cix⌧0 and a La-

grange multiplier ⌃x(⌧, ⌧0) enforcing the previous identity, oneobtains Z =

R[dG][d⌃]e�NS , with the action

S = �X

x

ln det⇥(@⌧ � µ)�(⌧1 � ⌧2) + ⌃x(⌧1, ⌧2)

⇤+

Z �

0d⌧1d⌧2

⇣�

X

x

266664U

20

4Gx(⌧1, ⌧2)2

Gx(⌧2, ⌧1)2 + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1)377775

+t20

X

hxx0i

Gx0 (⌧1, ⌧2)Gx(⌧2, ⌧1)⌘. (3)

The large N limit is controlled by the saddle point conditions�S/�G = �S/�⌃ = 0, satisfied by Gx(⌧, ⌧0) = G(⌧ � ⌧0),⌃x(⌧, ⌧0) = ⌃4(⌧ � ⌧0) + zt

20G(⌧ � ⌧0) (z is the coordination

number of the lattice of SYK dots), which obey

G(i!n)�1 = i!n + µ � ⌃4(i!n) � zt20G(i!n),

⌃4(⌧) = �U20G(⌧)2

G(�⌧), (4)

where !n = (2n + 1)⇡/� is the Matsubara frequency. Wesolve them numerically and re-insert into (3) to obtain thefree energy, hence the full thermodynamics[32]. Considerthe entropy S. A key feature of the SYK4 solution is an ex-tensive (/ N) entropy[13] in the T ! 0 limit, an extremenon-Fermi liquid feature. This entropy must be removed overthe narrow temperature window set by the the coherence en-ergy Ec. Consequently, we expect that S/N = S(T/Ec) forT, Ec ⌧ U0, where the universal function S(T = 0) = 0 in-dicating no zero temperature entropy in a Fermi liquid, andS(T ! 1) = 0.4648 · · · , recovering the zero temperatureentropy of the SYK4 model. The universal scaling collapseis confirmed by numerical solution, as shown in Fig. 1. Thisimplies also that the specific heat NC = (T/Ec)S0(T/Ec), andhence the low-temperature Sommerfeld coe�cient

� ⌘ limT!0

C

T=S0(0)Ec

(5)

is large due to the smallness of Ec. Specifically, comparedwith the Sommerfeld coe�cient in the weak interaction limitt0 � U0, which is of order t

�10 , there is an “e↵ective mass

enhancement” of m⇤/m ⇠ t0/Ec ⇠ U0/t0. Thus the low tem-

perature state is a heavy Fermi liquid.To establish that the low temperature state is truly a strongly

renormalized Fermi liquid with large Fermi liquid parame-ters, we compute the compressibility, NK = @N/@µ|

T. Be-

cause the compressibility has a smooth low temperature limit

S(T � �)

FIG. 1. The entropy and specific heat(inset) collapse to universalfunctions of T

Ec, given t0,T ⌧ U0(z = 2). C ! S

0(0)T/Ec asT/Ec ! 0. Solid curves are guides to the eyes.

in SYK4 model, we expect that K is only weakly perturbedby small t0. For t0 ⌧ U0, we indeed have K ⇡ K|t0=0 =c/U0 with the constant c ⇡ 1.04 regardless of T/Ec. Forfree fermions, the compressibility and Sommerfeld coe�cientare both proportional to the single-particle density of states(DOS), and in particular �/K = ⇡2/3 for free fermions. Herewe find �/K = (S0(0)/c)U0/Ec ⇠ (U0/t0)2

� 1. This canonly be reconciled with Fermi liquid theory by introducing alarge Landau interaction parameter. In Fermi liquid theory,one introduces the interaction fab via �"a =

Pb fab�nb, where

a, b label quasiparticle states. For a di↵usive disordered Fermiliquid, we take fab = F/g(0), where g(0) is the quasi-particleDOS, and F is the dimensionless Fermi liquid interaction pa-rameter. The standard result of Fermi liquid theory[32], isthat � is una↵ected by F but K is renormalized, leading to�/K = ⇡

2

3 (1 + F). We see that F ⇠ (U0/t0)2� 1, so that the

......

......

G(i!) = tG(i!)

strong similarities to DMFT equations

mathematical structure appeared in early study of doped t-J model with double large N and infinite dimension limits: O. Parcollet+A. Georges, 1999

UtLarge N equations





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3

d1


• There is a low coherence scale Ec ⇠ t20/U , with SYKcriticality at T > Ec and heavy Fermi liquid behaviorfor T < Ec.

• From the Kubo formula, we have the conductivity

Re[�(!)] / t20

Zd⌦

f(! + ⌦)� f(⌦)

!A(⌦)A(! + ⌦)

where A(!) = Im[GR(!)].At T > Ec, using A(!) ⇠ !�1/2F (!/T ), this yieldsthe bad metal behavior

� ⇠ e2

h

t20U

1

T; ⇢ ⇠

✓h

e2

◆T

Ec

Aavishkar Patel

1/12/18, 4(52 AMarXiv.org Search




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The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Sachdev/0/1/0/all/0/1

Showing results 1 through 25 (of 333 total) for au:Sachdev

1. arXiv:1801.01125 [pdf, other]Title: Topological order and Fermi surface reconstructionAuthors: Subir SachdevComments: 48 pages, 22 figures. Review article, comments welcome. Partly based on lectures at the34th Jerusalem Winter School in Theoretical Physics: New Horizons in Quantum Matter, December 27,2016 - January 5, 2017, this https URL ;(v2) added refs, figures, remarksSubjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th);Quantum Physics (quant-ph)

2. arXiv:1712.05026 [pdf, other]Title: Magnetotransport in a model of a disordered strange metalAuthors: Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir SachdevComments: 18+epsilon pages + Appendices + References, 4 figuresSubjects: Strongly Correlated Electrons (cond-mat.str-el); Disordered Systems and Neural Networks(cond-mat.dis-nn); High Energy Physics - Theory (hep-th)

3. arXiv:1711.09925 [pdf, other]Title: Topological order in the pseudogap metalAuthors: Mathias S. Scheurer, Shubhayu Chatterjee, Wei Wu, Michel Ferrero, Antoine Georges, SubirSachdevComments: 8+15 pages, 10 figures

A. A. Patel, J. McGreevy, D. P. Arovas and S. Sachdev, arXiv:1712.05026

Infecting a Fermi liquid and making it SYKMobile electrons (c, green) interacting with SYK quantum

dots (f, blue) with exchange interactions.This yields the first model agreeing with

magnetotransport in strange metals !

H = �t

MX

hrr0i; i=1

(c†ricr0i + h.c.)� µc

MX

r; i=1

c†ricri � µ

NX

r; i=1

f†rifri

+1

NM1/2

NX

r; i,j=1

MX

k,l=1

grijklf

†rifrjc

†rkcrl +

1

N3/2

NX

r; i,j,k,l=1

Jrijklf

†rif

†rjfrkfrl.

Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

⌃(⌧ � ⌧ 0) = �J2G2(⌧ � ⌧ 0)G(⌧ 0 � ⌧)� M

Ng2G(⌧ � ⌧ 0)Gc(⌧ � ⌧ 0)Gc(⌧ 0 � ⌧),

G(i!n) =1

i!n + µ� ⌃(i!n), (f electrons)

⌃c(⌧ � ⌧ 0) = �g2Gc(⌧ � ⌧ 0)G(⌧ � ⌧ 0)G(⌧ 0 � ⌧),

Gc(i!n) =X

k

1

i!n � ✏k + µc � ⌃c(i!n). (c electrons)

Exactly solvable in the large N,M limits! • Low-T phase: c electrons form a Marginal Fermi-liquid (MFL), f electrons are local SYK models

Infecting a Fermi liquid and making it SYK



Gc(⌧) = � Ccp1 + e�4⇡Ec

✓T

sin(⇡T ⌧)

◆1/2

e�2⇡EcT⌧ , G(⌧) = � Cp1 + e�4⇡E

✓T

sin(⇡T ⌧)

◆1/2

e�2⇡ET⌧ , 0 ⌧ < �.

Gc(⌧) = � Ccp1 + e�4⇡Ec

✓T

sin(⇡T ⌧)

◆1/2

e�2⇡EcT⌧ , G(⌧) = � Cp1 + e�4⇡E

✓T

sin(⇡T ⌧)

◆1/2

e�2⇡ET⌧ , 0 ⌧ < �.

• High-T phase: c electrons form an “incoherent metal” (IM), with local Green’s function, and no notion of momentum; f electrons remain local SYK models



• Low-T phase: c electrons form a Marginal Fermi-liquid (MFL), f electrons are local SYK models

⌃c(i!n) =ig2⌫(0)T

2J cosh1/2(2⇡E)⇡3/2

✓!n

Tln

✓2⇡Te�E�1

J

◆+!n

T ⇣ !n

2⇡T

⌘+ ⇡

◆,

⌃c(i!n) !ig2⌫(0)

2J cosh1/2(2⇡E)⇡3/2!n ln

✓|!n|e�E�1

J

◆, |!n| � T (⌫(0) ⇠ 1/t)


Linear-in-T resistivity

Both the MFL and the IM are not translationally-invariant and have linear-in-T resistivities!

�IM0 = (⇡1/2/8)⇥MT�1J ⇥

✓⇤

⌫(0)g2

◆cosh1/2(2⇡E)cosh(2⇡Ec)

.

[Can be obtained straightforwardly from Kubo formula in the large-N,M limits]

The IM is also a “Bad metal” with �IM0 ⌧ 1

�MFL0 = 0.120251⇥MT�1J ⇥

✓v2Fg2

◆cosh1/2(2⇡E). (vF ⇠ t)


Magnetotransport: Marginal-Fermi liquid• Thanks to large N,M, we can also exactly derive the linear-

response Boltzmann equation for non-quantizing magnetic fields…

�MFLL

= Mv2F⌫(0)

16T

Z 1

�1

dE1

2⇡sech2

✓E1

2T

◆�Im[⌃c

R(E1)]

Im[⌃c

R(E1)]2 + (vF /(2kF ))2B2

,

�MFLH

= �Mv2F⌫(0)

16T

Z 1

�1

dE1

2⇡sech2

✓E1

2T

◆(vF /(2kF ))B

Im[⌃c

R(E1)]2 + (vF /(2kF ))2B2

.

�MFLL

⇠ T�1sL((vF /kF )(B/T )), �MFLH

⇠ �BT�2sH((vF /kF )(B/T )).

sL,H(x ! 1) / 1/x2, sL,H(x ! 0) / x0.

Scaling between magnetic field and temperature in orbital magnetotransport!

(1� @!Re[⌃cR(!)])@t�n(t, k,!) + vF k ·E(t) n0

f (!) + vF (k ⇥ Bz) ·rk�n(t, k,!) = 2�n(t, k,!)Im[⌃cR(!)],

(B = eBa2/~) (i.e. flux per unit cell)

Magnetotransport with mesoscopic homogeneity• No macroscopic momentum, so equations describing charge transport are

justr · I(x) = 0, I(x) = �(x) ·E(x), E(x) = �r�(x).

22

[8] A. Y. Kitaev, “Talks at KITP, University of California, Santa Barbara,” Entanglement in Strongly-Correlated Quantum

Matter (2015).

[9] O. Parcollet and A. Georges, “Non-Fermi-liquid regime of a doped Mott insulator,” Phys. Rev. B 59, 5341 (1999),

cond-mat/9806119 .

[10] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, “Emergent quantum criticality, Fermi surfaces, and AdS2,” Phys. Rev.

D 83, 125002 (2011), arXiv:0907.2694 [hep-th] .

[11] S. Sachdev, “Bekenstein-Hawking Entropy and Strange Metals,” Phys. Rev. X 5, 041025 (2015), arXiv:1506.05111 [hep-

th] .

[12] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, “Phenomenology of the normal

state of Cu-O high-temperature superconductors,” Phys. Rev. Lett. 63, 1996 (1989).

[13] M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong,

G. D. Gu, and P. Abbamonte, “Singular density fluctuations in the strange metal phase of a copper-oxide supercon-

ductor,” ArXiv e-prints (2017), arXiv:1708.01929 [cond-mat.str-el] .

[14] S. Sachdev, “Holographic Metals and the Fractionalized Fermi Liquid,” Phys. Rev. Lett. 105, 151602 (2010),

arXiv:1006.3794 [hep-th] .

[15] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, “Charge transport by holographic Fermi surfaces,” Phys.

Rev. D 88, 045016 (2013), arXiv:1306.6396 [hep-th] .

[16] X.-Y. Song, C.-M. Jian, and L. Balents, “Strongly Correlated Metal Built from Sachdev-Ye-Kitaev Models,” Phys. Rev.

Lett. 119, 216601 (2017), arXiv:1705.00117 [cond-mat.str-el] .

[17] Y. Gu, X.-L. Qi, and D. Stanford, “Local criticality, di↵usion and chaos in generalized Sachdev-Ye-Kitaev models,”

JHEP 11, 014 (2017), arXiv:1609.07832 [hep-th] .

[18] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen, and S. Sachdev, “Thermoelectric transport in disordered

metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,” Phys. Rev. B 95, 155131 (2017),

arXiv:1612.00849 [cond-mat.str-el] .

[19] M. Milovanovic, S. Sachdev, and R. N. Bhatt, “E↵ective-field theory of local-moment formation in disordered metals,”

Phys. Rev. Lett. 63, 82 (1989).

[20] R. N. Bhatt and D. S. Fisher, “Absence of spin di↵usion in most random lattices,” Phys. Rev. Lett. 68, 3072 (1992).

[21] A. C. Potter, M. Barkeshli, J. McGreevy, and T. Senthil, “Quantum Spin Liquids and the Metal-Insulator Transition

in Doped Semiconductors,” Phys. Rev. Lett. 109, 077205 (2012), arXiv:1204.1342 [cond-mat.str-el] .

[22] S. Burdin, D. R. Grempel, and A. Georges, “Heavy-fermion and spin-liquid behavior in a Kondo lattice with magnetic

frustration,” Phys. Rev. B 66, 045111 (2002), cond-mat/0107288 .

[23] D. Ben-Zion and J. McGreevy, “Strange metal from local quantum chaos,” ArXiv e-prints (2017), arXiv:1711.02686

[cond-mat.str-el] .

[24] A. M. Dykhne, “Anomalous plasma resistance in a strong magnetic field,” Journal of Experimental and Theoretical

Physics 32, 348 (1971).

[25] D. Stroud, “Generalized e↵ective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B

12, 3368 (1975).

[26] M. M. Parish and P. B. Littlewood, “Non-saturating magnetoresistance in heavily disordered semiconductors,” Nature

426, 162 (2003), cond-mat/0312020 .

[27] M. M. Parish and P. B. Littlewood, “Classical magnetotransport of inhomogeneous conductors,” Phys. Rev. B 72,

094417 (2005), cond-mat/0508229 .

[28] V. Guttal and D. Stroud, “Model for a macroscopically disordered conductor with an exactly linear high-field magne-

toresistance,” Phys. Rev. B 71, 201304 (2005), cond-mat/0502162 .

[29] J. C. W. Song, G. Refael, and P. A. Lee, “Linear magnetoresistance in metals: Guiding center di↵usion in a smooth

random potential,” Phys. Rev. B 92, 180204 (2015), arXiv:1507.04730 [cond-mat.mes-hall] .

[30] N. Ramakrishnan, Y. T. Lai, S. Lara, M. M. Parish, and S. Adam, “Equivalence of E↵ective Medium and Ran-

dom Resistor Network models for disorder-induced unsaturating linear magnetoresistance,” ArXiv e-prints (2017),

arXiv:1703.05478 [cond-mat.mes-hall] .

• 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.

Magnetotransport with mesoscopic homogeneity• No macroscopic momentum, so equations describing charge transport are

justr · I(x) = 0, I(x) = �(x) ·E(x), E(x) = �r�(x).

M.M. Parish and P.B. Littlewood, Nature 426, 162 (2003)

Solvable toy model: two-component disorder• Two types of domains a,b with different carrier

densities and lifetimes randomly distributed in approximately equal fractions over sample.

• Effective medium equations can be solved exactly✓I+ �a � �e

2�eL

◆�1

· (�a � �e) +

✓I+ �b � �e

2�eL

◆�1

· (�b � �e) = 0.

⇢eL⌘ �e

L

�e2L

+ �e2H

=

q(B/m)2

��a�MFL

0a � �b�MFL0b

�2+ �2

a�2b

��MFL0a + �MFL

0b

�2

�a�b(�MFL0a �MFL

0b )1/2��MFL0a + �MFL

0b

� ,

⇢eH

⌘ � �e

H/B

�e2L

+ �e2H

=�a + �b

m�a�b��MFL0a + �MFL

0b

� .

�a,b ⇠ T (i.e. effective transport scattering rates)

(m = kF /vF ⇠ 1/t)

⇢eL ⇠p

c1T 2 + c2B2

Scaling between B and T at microscopic orbital level has been transferred to global MR!

Magnetotransport in strange metals

• Engineered a model of a Fermi surface coupled

to SYK quantum dots which leads to a marginal

Fermi liquid with a linear-in-T resistance, with

a magnetoresistance which scales as B ⇠ T .

• Mesoscopic disorder then leads to linear-in-Bmagnetoresistance, and a combined dependence

which scales as ⇠pB2 + T 2

• Higher temperatures lead to an incoherent metal

with a local Green’s function and a linear-in-Tresistance, but negligible magnetoresistance.

• 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 disorder, and end up with such

two-band fixed point, with emergent local

conservation laws?

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. Paalanen, PRL 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)

quantum matter without quasiparticlesqpt.physics.harvard.edu/talks/nhmfl18c.pdf · low a small...

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