the disordered hubbard model: from si:p to the high ...qpt.physics.harvard.edu/talks/nist18.pdf ·...

The disordered Hubbard model: from Si:P to

the high temperature superconductors

HARVARD

Subir SachdevApril 25, 2018

Workshop on 2D Quantum MetamaterialsNIST, Gaithersburg, MD

1. Disordered Hubbard model for Si:P

2. Solvable disordered models:

(A) Random matrix model of a `quantum dot’ Metal with quasiparticles

(B) SYK model of a `quantum dot’ Metal without quasiparticles

3. High temperature superconductivity and strange metals.

Disordered Hubbard model for Si:P

H = �X

i,j

tijc†i↵cj↵

� µ

X

i

c†i↵ci↵

+ U

X

i

c†i"ci"c

†i#ci#

tij ⇠ exp(�|ri � rj |/a0)<latexit sha1_base64="Cx0KlFibznm9K5vrM8MRoWgC8NQ=">AAADFHicdVJNbxMxEPUuXyV8pXDkwIiIqogm7LaghANSBZcei0TaStmw8nq9iVN717K9LdF2e+Qn8Cu4wokb4sqdA/8Fb7KBtGpHsub5zTyPZ+xIcqaN5/123CtXr12/sXKzcev2nbv3mqv393SWK0L7JOOZOoiwppyltG+Y4fRAKopFxOl+dPi2iu8fUaVZlr43U0mHAo9SljCCjaXCVefR2g68hjYEOhdhwTYmJRjrwXpS+Q BzOcblhyDGoxFVFTlZkBAEjbXTZbPniHx+FrtUz5b0sKx/Bv2z2lxipbLj8+oFXW/j7Di9MO9/wFY6rS5btWY7CzQTENCPEtbbJ1AEUQKqtFXb//AETp7j0HvaCJstr/Nyq9d91QWv483MAt/b7G11wa+ZFqptN2z+sXVJLmhqCMdaD3xPmmGBlWGE07IR5JpKTA7xiA4sTLGgeiM+YlLP4LCYPWoJT2wwhiRTdqUGZuyyuMBC66mIbKbAZqzPxyryotggN0lvWLBU5oamZF4oyTmYDKofAjFTlBg+tQATxey1gYyxwsTYf1TNY9E0XA72Nu2EOv67F63tN/VkVtBD9BitIx910TbaQbuoj4jzyfnifHW+uZ/d7+4P9+c81XVqzQN0xtxffwHapvuT</latexit><latexit sha1_base64="Cx0KlFibznm9K5vrM8MRoWgC8NQ=">AAADFHicdVJNbxMxEPUuXyV8pXDkwIiIqogm7LaghANSBZcei0TaStmw8nq9iVN717K9LdF2e+Qn8Cu4wokb4sqdA/8Fb7KBtGpHsub5zTyPZ+xIcqaN5/123CtXr12/sXKzcev2nbv3mqv393SWK0L7JOOZOoiwppyltG+Y4fRAKopFxOl+dPi2iu8fUaVZlr43U0mHAo9SljCCjaXCVefR2g68hjYEOhdhwTYmJRjrwXpS+Q BzOcblhyDGoxFVFTlZkBAEjbXTZbPniHx+FrtUz5b0sKx/Bv2z2lxipbLj8+oFXW/j7Di9MO9/wFY6rS5btWY7CzQTENCPEtbbJ1AEUQKqtFXb//AETp7j0HvaCJstr/Nyq9d91QWv483MAt/b7G11wa+ZFqptN2z+sXVJLmhqCMdaD3xPmmGBlWGE07IR5JpKTA7xiA4sTLGgeiM+YlLP4LCYPWoJT2wwhiRTdqUGZuyyuMBC66mIbKbAZqzPxyryotggN0lvWLBU5oamZF4oyTmYDKofAjFTlBg+tQATxey1gYyxwsTYf1TNY9E0XA72Nu2EOv67F63tN/VkVtBD9BitIx910TbaQbuoj4jzyfnifHW+uZ/d7+4P9+c81XVqzQN0xtxffwHapvuT</latexit><latexit sha1_base64="Cx0KlFibznm9K5vrM8MRoWgC8NQ=">AAADFHicdVJNbxMxEPUuXyV8pXDkwIiIqogm7LaghANSBZcei0TaStmw8nq9iVN717K9LdF2e+Qn8Cu4wokb4sqdA/8Fb7KBtGpHsub5zTyPZ+xIcqaN5/123CtXr12/sXKzcev2nbv3mqv393SWK0L7JOOZOoiwppyltG+Y4fRAKopFxOl+dPi2iu8fUaVZlr43U0mHAo9SljCCjaXCVefR2g68hjYEOhdhwTYmJRjrwXpS+Q BzOcblhyDGoxFVFTlZkBAEjbXTZbPniHx+FrtUz5b0sKx/Bv2z2lxipbLj8+oFXW/j7Di9MO9/wFY6rS5btWY7CzQTENCPEtbbJ1AEUQKqtFXb//AETp7j0HvaCJstr/Nyq9d91QWv483MAt/b7G11wa+ZFqptN2z+sXVJLmhqCMdaD3xPmmGBlWGE07IR5JpKTA7xiA4sTLGgeiM+YlLP4LCYPWoJT2wwhiRTdqUGZuyyuMBC66mIbKbAZqzPxyryotggN0lvWLBU5oamZF4oyTmYDKofAjFTlBg+tQATxey1gYyxwsTYf1TNY9E0XA72Nu2EOv67F63tN/VkVtBD9BitIx910TbaQbuoj4jzyfnifHW+uZ/d7+4P9+c81XVqzQN0xtxffwHapvuT</latexit><latexit sha1_base64="Cx0KlFibznm9K5vrM8MRoWgC8NQ=">AAADFHicdVJNbxMxEPUuXyV8pXDkwIiIqogm7LaghANSBZcei0TaStmw8nq9iVN717K9LdF2e+Qn8Cu4wokb4sqdA/8Fb7KBtGpHsub5zTyPZ+xIcqaN5/123CtXr12/sXKzcev2nbv3mqv393SWK0L7JOOZOoiwppyltG+Y4fRAKopFxOl+dPi2iu8fUaVZlr43U0mHAo9SljCCjaXCVefR2g68hjYEOhdhwTYmJRjrwXpS+Q BzOcblhyDGoxFVFTlZkBAEjbXTZbPniHx+FrtUz5b0sKx/Bv2z2lxipbLj8+oFXW/j7Di9MO9/wFY6rS5btWY7CzQTENCPEtbbJ1AEUQKqtFXb//AETp7j0HvaCJstr/Nyq9d91QWv483MAt/b7G11wa+ZFqptN2z+sXVJLmhqCMdaD3xPmmGBlWGE07IR5JpKTA7xiA4sTLGgeiM+YlLP4LCYPWoJT2wwhiRTdqUGZuyyuMBC66mIbKbAZqzPxyryotggN0lvWLBU5oamZF4oyTmYDKofAjFTlBg+tQATxey1gYyxwsTYf1TNY9E0XA72Nu2EOv67F63tN/VkVtBD9BitIx910TbaQbuoj4jzyfnifHW+uZ/d7+4P9+c81XVqzQN0xtxffwHapvuT</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)

Theory of local moment formation in Si:P Choose an e↵ective Hamiltonian of the form

He↵ = �X

i,j

tijc†i↵cj↵ �

X

i

�ic†i↵ci↵

+1

2

X

i

~hi · c†i↵~�↵�ci�

where ~hi and �i are variational parameters. Taking the density matrix

defined by He↵ as the trial density matrix, and minimizing the free energy,

we obtain an expansion in power of ~hi

Fe↵ [~hi] = F0 +

1

4

X

i,j,k

�ij(�jk � U�jk)(~hi · ~hk) + . . .

where

�ij ⌘ �X

�,⇢

�(i) ⇤⇢(i)

⇤�(j) ⇢(j)

f(✏�)� f(✏⇢)

✏� � ✏⇢

where � and ✏� are the eigenfunctions and eigenvalues of He↵ at ~hi = 0.<latexit sha1_base64="oRdtzT4FeNvVf3OcIbHpU6OPsRY=">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</latexit><latexit sha1_base64="oRdtzT4FeNvVf3OcIbHpU6OPsRY=">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</latexit><latexit sha1_base64="oRdtzT4FeNvVf3OcIbHpU6OPsRY=">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</latexit><latexit sha1_base64="oRdtzT4FeNvVf3OcIbHpU6OPsRY=">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</latexit>

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

Theory of local moment formation in Si:P

Fe↵ [~hi] = F0 +1

4

X

i,j,k

�ij(�jk � U�jk)(~hi · ~hk) + . . .

We numerically diagonalize �ij , and choose eigenmodes with eigen-values ⇤i > 1/U : these are the eigenmodes corresponding to localmoment instabilities.

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Magnetic excitations are localized

(eigenmodes of the spin susceptibility)

Charged fermionic excitations are

extended(eigenmodes of the

electron Hamiltonian)

Theory of local moment formation in Si:P

A simple model of a metal with quasiparticles

Pick a set of random positions

Place electrons randomly on some sites


Electrons move one-by-one randomly


H =1

(N)1/2

NX

i,j=1

tijc†i cj + . . .

cicj + cjci = 0 , cic†j + c

†jci = �ij

1

N

X

i

c†i ci = Q

Fermions occupying the eigenstates of a N x N random matrix

tij are independent random variables with tij = 0 and |tij |2 = t2


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

G(i!) =1

i! + µ� ⌃(i!), ⌃(⌧) = t2G(⌧)

G(⌧ = 0�) = Q.

G(!) can be determined by solving a quadratic equation.

!

�ImG(!)

µ

Infinite-range model with quasiparticles

Infinite-range model with quasiparticles

Fermi liquid state: Two-body interactions lead to a scattering timeof quasiparticle excitations from in (random) single-particle eigen-states which diverges as ⇠ T�2 at the Fermi level.

Now add weak interactions

H =1

(N)1/2

NX

i,j=1

tijc†i cj +

1

(2N)3/2

NX

i,j,k,`=1

Uij;k` c†i c

†jckc`

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

compute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the

free particle Hamitonian with energy E↵. By Fermi’s Golden rule, for E↵ at the

Fermi energy

1

⌧↵= ⇡U

2⇢20

ZdE�dE�dE�f(E�)(1� f(E�))(1� f(E�))�(E↵ + E� � E� � E�)

=⇡3U

2⇢20

4T

2

where ⇢0 is the density of states at the Fermi energy.<latexit sha1_base64="cmzYtW7HgBFDnMmx0Ij330jBM9E=">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</latexit><latexit sha1_base64="cmzYtW7HgBFDnMmx0Ij330jBM9E=">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</latexit><latexit sha1_base64="cmzYtW7HgBFDnMmx0Ij330jBM9E=">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</latexit><latexit sha1_base64="cmzYtW7HgBFDnMmx0Ij330jBM9E=">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</latexit>

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.

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

p2(E � E0)N�) , 2s0 = 0.464848...

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

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

the coherent: G(⌧) ⇠ 1/⌧)

• T > 0 Green’s function has conformal invarianceG ⇠ (T/ sin(⇡kBT ⌧/~))1/2

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The SYK model

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


p2(E � E0)N�) , 2s0 = 0.464848...







The SYK model


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


p2(E � E0)N�) , 2s0 = 0.464848...







The SYK model


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

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, PRB 96, 205123 (2017)

• If there are no quasiparticles, then

E 6=X

↵

n↵"↵ +X

↵,�

F↵�n↵n� + . . .


⌧eq = #~

kBT

• Systems without quasiparticles are the fastest possible in reaching localequilibrium, and all many-body quantum systems obey, as T ! 0

⌧eq > C~

kBT.

– In Fermi liquids ⌧eq ⇠ 1/T 2, and so the bound is obeyed as T ! 0.

– This bound rules out quantum systems with e.g. ⌧eq ⇠ ~/(JkBT )1/2.– There is no bound in classical mechanics (~ ! 0). By cranking up

frequencies, we can attain equilibrium as quickly as we desire.

S. Sachdev, Quantum Phase Transitions,

Cambridge (1999)

Quantum matter without quasiparticles:


E 6=X

↵

n↵"↵ +X

↵,�

F↵�n↵n� + . . .


⌧eq = #~

kBT


⌧eq > C~

kBT.





Cambridge (1999)

Quantum matter without quasiparticles:


Cambridge (1999)

Quantum matter without quasiparticles:• If there are no quasiparticles, then

E 6=X

↵

n↵"↵ +X

↵,�

F↵�n↵n� + . . .


⌧eq = #~

kBT


⌧eq > C~

kBT.





Cambridge (1999)

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

• Black holes relax to thermal equilibrium in a time⇠ ~/(kBTH) = 8⇡GM/c3.

• The entropy of black holes is proportional to theirsurface area: the SYK model (in 0+1 dimensions)maps to a black hole in 1+1 dimensions.

<latexit sha1_base64="C/XOVl74ROyNoXDuL6VugiJR6gk=">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</latexit><latexit sha1_base64="C/XOVl74ROyNoXDuL6VugiJR6gk=">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</latexit><latexit sha1_base64="C/XOVl74ROyNoXDuL6VugiJR6gk=">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</latexit><latexit sha1_base64="C/XOVl74ROyNoXDuL6VugiJR6gk=">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</latexit>

Black holes

SYK models and black holes

⇣ ~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

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

Described by a Hubbard model on the Cu sites

SM

FL

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

YBa2Cu3O6+x

SM

FL

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

YBa2Cu3O6+x

Strange Metal

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.

<latexit sha1_base64="K7jx8EFBma2lRUjXY27Hncmbd18=">AAADI3icdVJNb9NAELXNVzFfKRy5jEhQilRSJwKl4VSVC9yKaGilbEjW67G9qr3r7q4bRVb4L1z4K1w4QMWFA/+FTeKigmBOz++9mZ1967DIuDZB8MP1rly9dv3Gxk3/1u07d+81Nu+/07JUDIdMZlIdh1RjxgUODTcZHhcKaR5meBSevFzqR2eoNJfi0MwLHOc0ETzmjBpLTTbdFyTEhIuKoTCoFv4w5KclN7LUhPgVWZ1QKYwWMJ2+NYqKBNvtbUI+WHk63afRpa/XgskUlZ30m5TK9g0z2/j0tKTClHm7vbB8joZmkFINFBRqe1F+xs18G1pEpbK1DbOUsxRkiHO9nrQSgGi ew2Grnk5FtAZcgJY5ArNRaKitSQLpDr7vtQiBLeswMwkRz1Es09BP7Ax/tlwXWmsbcA0mRagXhVJwAzKu96OCYccnKKKLrCaNZtAJukFvtwsWrMqC/mDQf/4MujXTdOo6mDTOSSRZaTcwLKNaj7pBYcYVVYazDBc+KTUWlJ3QBEcWCpqjHlerJ1jAY8tEENtAYykMrNjLHRXNtZ7noXXm1KT6b21J/ksblSbeHVdcFKVBwdYHxWUGRsLyj7GRKWQmm1tAmeJ2V2ApVZTZDLRvQ7i4KfwfDHudQSd402vu7ddpbDgPnUfOltN1+s6e88o5cIYOcz+6n92v7jfvk/fFO/e+r62eW/c8cP4o7+cvfKr9BQ==</latexit><latexit sha1_base64="K7jx8EFBma2lRUjXY27Hncmbd18=">AAADI3icdVJNb9NAELXNVzFfKRy5jEhQilRSJwKl4VSVC9yKaGilbEjW67G9qr3r7q4bRVb4L1z4K1w4QMWFA/+FTeKigmBOz++9mZ1967DIuDZB8MP1rly9dv3Gxk3/1u07d+81Nu+/07JUDIdMZlIdh1RjxgUODTcZHhcKaR5meBSevFzqR2eoNJfi0MwLHOc0ETzmjBpLTTbdFyTEhIuKoTCoFv4w5KclN7LUhPgVWZ1QKYwWMJ2+NYqKBNvtbUI+WHk63afRpa/XgskUlZ30m5TK9g0z2/j0tKTClHm7vbB8joZmkFINFBRqe1F+xs18G1pEpbK1DbOUsxRkiHO9nrQSgGi ew2Grnk5FtAZcgJY5ArNRaKitSQLpDr7vtQiBLeswMwkRz1Es09BP7Ax/tlwXWmsbcA0mRagXhVJwAzKu96OCYccnKKKLrCaNZtAJukFvtwsWrMqC/mDQf/4MujXTdOo6mDTOSSRZaTcwLKNaj7pBYcYVVYazDBc+KTUWlJ3QBEcWCpqjHlerJ1jAY8tEENtAYykMrNjLHRXNtZ7noXXm1KT6b21J/ksblSbeHVdcFKVBwdYHxWUGRsLyj7GRKWQmm1tAmeJ2V2ApVZTZDLRvQ7i4KfwfDHudQSd402vu7ddpbDgPnUfOltN1+s6e88o5cIYOcz+6n92v7jfvk/fFO/e+r62eW/c8cP4o7+cvfKr9BQ==</latexit>

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

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⌧, c̄ix⌧. Owing to theself-averaging established for the SYK model at large N, it issu�cient to study Z̄ =

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

indices are summed over)

S c =X

x

Z �

0d⌧ c̄ix⌧(@⌧ � µ)cix⌧ �

Z �

0d⌧1d⌧2

hX

x

U20

4N3 c̄ix⌧1 c̄ jx⌧1 ckx⌧1 clx⌧1 c̄lx⌧2 c̄kx⌧2 c jx⌧2 cix⌧2 +X

hxx0i

t20

Nc̄ix⌧1 c jx0⌧1 c̄ 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

SYK quantum dots of electrons with random hopping between them.

Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, 216601 (2017)

SYK building blocks for a strange metal

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






(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⌧ c̄ix⌧(@⌧ � 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 M

ay 2

017

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⌧ c̄ix⌧(@⌧ � µ)cix⌧ �

Z �

0d⌧1d⌧2

hX

x

U20


hxx0i

t20


i. (2)


Ut

t

U

4

The density-density correlator is expressed as

DRn(x,t; x0,t0) ⌘ i✓(t � t

0)h[N(x,t),N(x0,t0)]i

=i

2hNc(x,t)Nq(x

0,t0)i, (10)

where Ns ⌘N�S '�'̇s

, Nc/q = N+ ± N�(keeping momentum-independent components- See Sec.B). Adding a contact termto ensure that limp!0 DRn(p,! , 0) = 0[31], the action (9)yields the di↵usive form [32]

DRn(p,!) =�iNK!

i! � D'p2 + NK =�NKD'p

2

i! � D'p2 . (11)

From this we identify NK and D' as the compressibility andcharge di↵usion constant, respectively. The electric conduc-tivity is given by Einstein relation � ⌘ 1/⇢ = NKD', or,restoring all units,� = NKD'

e2

~ a2�d(a is lattice spacing).

Note the proportionality to N: in the standard non-linearsigma model formulation, the dimensionless conductance islarge, suppressing localization e↵ects. This occurs becauseboth U and t interactions scatter between all orbitals, destroy-ing interference from closed loops.

The analysis of energy transport proceeds similarly. Sinceenergy is the generator of time translations, one considers thetime-reparametrization (TRP) modes induced by ts ! ts+✏s(t)and defines ✏c/q = 1

2 (✏+ ± ✏�). The e↵ective action for TRPmodes to the lowest-order in p,! reads (Sec. B)

iS ✏ =X

p

Z +1

�1

d! ✏c,!(2i�!2T

2� p

2⇤3(!))✏q,�! + · · · , (12)

where the ellipses has the same meaning as in (9). At lowfrequency, the correlation function integral, given in Sec. B,behaves as ⇤3(!) ⇡ 2�D✏T 2!, which defines the energy dif-fusion constant D✏ . This identification is seen from the corre-lator for energy density modes "c/q ⌘

iN�S ✏�✏̇c/q

,

DR"(p,!) =i

2h"c"qip,! =

�NT2�D✏ p2

i! � D✏ p2 , (13)

where we add a contact term to ensure conservation of energyat p = 0. The thermal conductivity reads = NT�D✏ (kB = 1)–like �, is O(N).

Scaling collapse, Kadowaki-Woods and Lorentz ra-

tios – Electric/thermal conductivities are obtained fromlim!!0 ⇤2/3(!)/!, expressed as integrals of real-time corre-lation functions, and can be evaluated numerically for anyT, t0,U0. Introducing generalized resistivities, ⇢' = ⇢, ⇢" =T/, we find remarkably that for t0,T ⌧ U0, they collapse touniversal functions of one variable,

⇢⇣(t0,T ⌧ U0) =1N

R⇣( T

Ec

) ⇣ 2 {', "}, (14)

where R'(T ), R"(T ) are dimensionless universal functions.This scaling collapse is verified by direct numerical calcula-tions shown in Fig. 3a. From the scaling form (B2), we see thelow temperature resistivity obeys the usual Fermi liquid form

⇢⇣(T ⌧ Ec) ⇡ ⇢⇣(0) + A⇣T2, (15)

(a)

(b)

FIG. 3. (a): For t0,T ⌧ U0, ⇢'/" “collapse” to R'/"( T

Ec)/N. (b): The

Lorentz ratio ⇢T

reaches two constants ⇡2

3 ,⇡2

8 , in the two regimes.The solid curves are guides to the eyes.

where the temperature coe�cient of resistivity A⇣ =R00

⇣ (0)2NE

2c

islarge due to small coherence scale in denominator, charac-teristic of a strongly correlated Fermi liquid. Famously, theKadowaki-Woods ratio, A'/(N�)2, is approximately system-independent for a wide range of correlated materials[33, 34].We find here A'

(N�)2 =R00' (0)

2[S0(0)]2N3 is independent of t0 and U0!Turning now to the incoherent metal regime, in limit of

large arguments, T � 1, the generalized resistivities vary lin-early with temperature: R⇣(T ) ⇠ c⇣ T . We analytically obtainc' =

2p⇡

and c" =16⇡5/2 (Supplementary Information), implying

that the Lorenz number, characterizing the Wiedemann-Franzlaw, takes the unusual value L =

�T!⇡2

8 for Ec ⌧ T ⌧ U0.More generally, the scaling form (B2) implies that L is a uni-versal function of T/Ec, verified numerically as shown inFig. 3b. The Lorenz number increases with lower tempera-ture, saturating at T ⌧ Ec to the Fermi liquid value ⇡2/3.

Conclusion – We have shown that the SYK model pro-vides a soluble source of strong local interactions which,when coupled into a higher-dimensional lattice by ordinarybut random electron hopping, reproduces a remarkable num-


Low ‘coherence’ scale

Ec ⇠t20U

4


DRn(x,t; x0,t0) ⌘ i✓(t � t

0)h[N(x,t),N(x0,t0)]i

=i

2hNc(x,t)Nq(x

0,t0)i, (10)



DRn(p,!) =�iNK!


2

i! � D'p2 . (11)


e2





iS ✏ =X

p

Z +1

�1

d! ✏c,!(2i�!2T

2� p

2⇤3(!))✏q,�! + · · · , (12)



,

DR"(p,!) =i

2h"c"qip,! =

�NT2�D✏ p2

i! � D✏ p2 , (13)




⇢⇣(t0,T ⌧ U0) =1N

R⇣( T

Ec

) ⇣ 2 {', "}, (14)


⇢⇣(T ⌧ Ec) ⇡ ⇢⇣(0) + A⇣T2, (15)

(a)

(b)


Ec)/N. (b): The

Lorentz ratio ⇢T


3 ,⇡2



⇣ (0)2NE

2c


(N�)2 =R00' (0)



2p⇡



�T!⇡2




For Ec < T < U , theresistivity, ⇢, and

entropy density, s, are

⇢ ⇠ h

e2

✓T

Ec

◆, s = s0


Ec ⇠t20U

4


DRn(x,t; x0,t0) ⌘ i✓(t � t

0)h[N(x,t),N(x0,t0)]i

=i

2hNc(x,t)Nq(x

0,t0)i, (10)



DRn(p,!) =�iNK!


2

i! � D'p2 . (11)


e2





iS ✏ =X

p

Z +1

�1

d! ✏c,!(2i�!2T

2� p

2⇤3(!))✏q,�! + · · · , (12)



,

DR"(p,!) =i

2h"c"qip,! =

�NT2�D✏ p2

i! � D✏ p2 , (13)




⇢⇣(t0,T ⌧ U0) =1N

R⇣( T

Ec

) ⇣ 2 {', "}, (14)


⇢⇣(T ⌧ Ec) ⇡ ⇢⇣(0) + A⇣T2, (15)

(a)

(b)


Ec)/N. (b): The

Lorentz ratio ⇢T


3 ,⇡2



⇣ (0)2NE

2c


(N�)2 =R00' (0)



2p⇡



�T!⇡2




Ec ⇠t20U

For T < Ec, theresistivity, ⇢, and

entropy density, s, are

⇢ =h

e2

"c1 + c2

✓T

Ec

◆2#

s ⇠ s0

✓T

Ec

◆


4


DRn(x,t; x0,t0) ⌘ i✓(t � t

0)h[N(x,t),N(x0,t0)]i

=i

2hNc(x,t)Nq(x

0,t0)i, (10)



DRn(p,!) =�iNK!


2

i! � D'p2 . (11)


e2





iS ✏ =X

p

Z +1

�1

d! ✏c,!(2i�!2T

2� p

2⇤3(!))✏q,�! + · · · , (12)



,

DR"(p,!) =i

2h"c"qip,! =

�NT2�D✏ p2

i! � D✏ p2 , (13)




⇢⇣(t0,T ⌧ U0) =1N

R⇣( T

Ec

) ⇣ 2 {', "}, (14)


⇢⇣(T ⌧ Ec) ⇡ ⇢⇣(0) + A⇣T2, (15)

(a)

(b)


Ec)/N. (b): The

Lorentz ratio ⇢T


3 ,⇡2



⇣ (0)2NE

2c


(N�)2 =R00' (0)



2p⇡



�T!⇡2





Ec ⇠t20U

But this model exhibits negligible magnetoresistivity !

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

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

fc

Infecting a Fermi liquid and making it SYK

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


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

fc

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



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 yxiZot2FCqBIHAv4sPAZpoeBXtAWBouuKkh+hQBZqZ4YC0uikkkw3xIQOTAqcI6TGFq86UYw+vMZrLawVk2kncpQVYlSPx1PYOt6B4TZsvtmEKLVC1v1pPZxCdJbT6BQihSlvLTzHc4+lLOdt8C3qqojRQhTdI06eQGzi3g0WmCf/C7MToU4eVHybo8VZS04KP5YM0krLhk0HpBltaZRgBOHuuvU8lkpMCsG5b7jJXJ/unPjq4kU2j2xuIfRJHRUQFbEZ155Fx0Kzr+i6Esnr+QWfDo9IpzxZYPR343eDeeWPsg9zcvfTx3FOMbGD7vHusHvXzw7EXghNg42WkI1FR00t0osnsrmZTct/F1xZ36lrZushIMoyGHZ7o7WNsNcfHO693ANvhId7YWPsvxwc/BRCvxfOzkawOGejtb+ixMiq8BqVSjh32Q9LvqqFZZIKPWWVw1LI9yLDS29qUaC7qme7MYUf/EsyqyA1mmH2+jCiFoVzkyL2PxsK3H99M16e8F1WnB5c1aTLilHLeaK0UsAGmkXz22FRspp4Q0jrtS5B5sILyUvAIzn0u6ozzuuIccy3lPg89c+kp56fOxLg08b5oHfYC38fbBwdL4haDZ4H3wdbQT/YD46Ck+AsOA9k60Prn+WV5dXlv9sv2t325vxra2kR813w6LR3/wU613d1</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? Experiments

on Si:P indicate an a�rmative answer.<latexit sha1_base64="8DDJRrJnv7Czdq/LBZN5ISOzyP8=">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</latexit><latexit sha1_base64="8DDJRrJnv7Czdq/LBZN5ISOzyP8=">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</latexit><latexit sha1_base64="8DDJRrJnv7Czdq/LBZN5ISOzyP8=">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</latexit><latexit sha1_base64="8DDJRrJnv7Czdq/LBZN5ISOzyP8=">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</latexit>

the disordered hubbard model: from si:p to the high ...qpt.physics.harvard.edu/talks/nist18.pdf ·...

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