Non-Fermi liquid behavior in the Sachdev-Ye-Kitaev model for a one dimensionalincoherent semimetal

Geo Jose1, Kangjun Seo2, Bruno Uchoa1

1Center for Quantum Research and Technology, Department of Physics and astronomy,University of Oklahoma, Norman, Oklahoma 73019, USA and

2School of Electrical and Computer Engineering,The University of Oklahoma, Tulsa, OK 74135, USA

(Dated: January 21, 2022)

We study a two-band dispersive Sachdev-Ye-Kitaev (SYK) model in 1 + 1 dimension. We suggesta model that describes a semimetal with quadratic dispersion at half-filling. We compute the Green’sfunction at the saddle point using a combination of analytical and numerical methods. Employinga scaling symmetry of the Schwinger-Dyson equations that becomes transparent in the stronglydispersive limit, we show that the exact solution of the problem yields a distinct type of non-Fermi liquid with sublinear ρ ∝ T 2/5 temperature dependence of the resistivity. A scaling analysisindicates that this state corresponds to the fixed point of the dispersive SYK model for a quadraticband touching semimetal.

I. INTRODUCTION

Sachdev-Ye-Kitaev (SYKq) models describe stronglyinteracting fermions with infinite range, q−body, randomall-to-all interactions. The 0 + 1 dimensional SYKq dotmodel [1, 2] exhibits an approximate conformal symme-try in the infrared, is exactly solvable in the limit of alarge number of fermion flavors, saturates the bound onquantum chaos and are dual to gravitational theories in1 + 1 dimensions. [3–5]. Useful connections to the blackhole information problem have also been established [6].

In these models, approximate conformal symmetry inthe strong coupling/low frequency regime leads to powerlaw behavior of correlation functions. In the SYK4

model, the zero temperature two-point correlation func-tion decays as G (ω) ∼ 1/

√ω, with ω the frequency. Fi-

nite temperature Green’s functions can be then obtainedby appealing to conformal symmetry [1]. In dispersiveversions of the SYK model, they result in the linear tem-perature dependence of the dc resistivity, ρ ∝ T , a char-acteristic feature of strange metal phases. It was origi-nally conjectured that the linear scaling of the scatteringrate in the strange metal phase was due to hyperscalingin the proximity of a quantum critical point buried insidethe superconducting phase. Recent momentum resolvedelectron energy spectroscopy experiments in the cupratesrevealed the emergence of a mysterious momentum inde-pendent energy scale nearly one order of magnitude largerthan the temperature range of the quantum critical fan[7, 8], at odds with conventional quantum critical the-ories. One may speculate [9–13] that the origin of thestrange metal phase could be related to aspects of thephysics of incoherent metals.

Various lattice generalizations [13–15, 17–19] of the dotmodel comprising of connected SYK dots have been re-cently proposed in the regime where the SYK coupling isthe dominant energy scale of the problem. The generalidea behind many of those extensions is to build on thesolution of the dot model including lattice effects pertur-batively. Weakly dispersive versions of the SYK model

were used to describe incoherent or ‘Planckian’ metalswhich lack well defined quasiparticles [12, 20, 21]. Theseincoherent metals typically have a crossover between theincoherent high temperature regime and a low temper-ature Fermi liquid behavior[15, 17]. The crossover en-ergy scale between the two regimes is set by t2/J , witht proportional to the band width and J � t the SYKcoupling. In the low temperature regime, the coherenceof the quasiparticles is restored by the presence of a largeFermi surface. Semimetals, on the other hand, abridgea large class of gapless multiband systems that lack aFermi surface. One could ask what is the nature of thenormal state in a disordered semimetal with random localcouplings.

Motivated by these ideas, we study a 1D ladder withlocal random couplings at every unit cell, as shown inFig. 1. The hopping amplitudes between lattice sites isfinely tuned such that this system describes a half-filledsemimetal with quadratic dispersion and local SYK cou-plings. The weakly dispersive limit t2/J � T � J hasapproximate conformal invariance and recovers the usualSYK transport behavior, as expected. In the strongly

t1

t2

1

t1

t2

1

t

t?

a

b

a

b

1

t

t?

a

b

a

b

1

Figure 1. Dispersive SYK ladder model: The unit cell con-tains two sites, one for each chain (color). Each color site hostsN complex fermions, which interact locally through randomcouplings. We assume that hopping is only allowed betweendifferent color sites, with t1 the NN hopping and t2 the NNNone. The two-band quadratic dispersion in Eq. (3) can beobtained by tuning t1 = −2t2 = t, with m = 2/(ta2) theeffective mass of the fermions, where a is the lattice constant.

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dispersive regime, T � t2/J , the scaling symmetry ofthe problem becomes transparent, albeit the absence ofconformal symmetry. In this limit, the incoherent regimeextends down to zero frequency and temperature, unlikein the more conventional metallic case. We show thatthe resistivity of this model has a sublinear scaling withtemperature,

ρ ∝ T 2/5, (1)

whereas the Lorentz ratio L = κ/(σT ) ≈ 3.2 is fairlyclose to the value expected for a Fermi liquid, L = π2/3.We find through a scaling argument that when the sys-tem starts from the SYK fixed point at high tempera-ture, it flows towards a distinct non-Fermi liquid (NFL)fixed point at zero temperature. At intermediate energyscales, away from the low temperature fixed point, thesystem crosses over from a “Planckian” semimetal to anincoherent NFL with sublinear temperature scaling of theresistivity.

This paper is organized in the following way: in sec-tion II we introduce the lattice model of a 1D ladderof SYK quantum dots that behaves as a 1D semimetalwith quadratic dispersion. In section III we address theGreen’s function of this system at zero and finite tem-perature. In the strongly dispersive regime (T � t2/J),where conformal symmetry is not present, we numeri-cally extract the finite temperature scaling functions ofthe Green’s function and of the self-energy. In sectionIV, we discuss a scaling analysis of the problem andthe crossover between the high temperature incoherentPlanckian regime and the low temperature NFL one. Insection V we address the temperature scaling of the dcresistivity of the model. Finally, in section VI we presentour conclusions.

II. MODEL

We consider N−flavors of complex fermions hopping onan 1D lattice. Each lattice site hosts an SYK dot withrandom, site dependent interactions Jxijk` between them.The indexes i, j, k, ` = 1, .., N label the N−flavors/colorsper site. We start from a 1D ladder shown in Fig. 1, withtwo sites per unit cell, shown in blue and red. Allowinghopping processes between blue and red sites only, theHamiltonian of the kinetic term can be written as

H0 =

ˆk

∑

ν

f(k)ψ†k,i,ν (σx)ν,ν′ ψk,i,ν′ , (2)

where

f(k) = t1 + 2t2 cos(ka), (3)

with t1 and t2 the hopping between nearest neighbors(NN) and second nearest neighbors (NNN) respectivelyamong different color sites, and

´k≡ a(2π)−1

´ Λ

−Λdk

with Λ = π/a the ultraviolet cutoff. ψi,ν is a two-component spinor in the site basis of the unit cell ν = a, b

Figure 2. Finely tuned energy dispersion for the ladder modelillustrated in Fig. 1. The half-filled band describes a 1Dsemimetal with parabolic band touching at k = 0.

and σx is the real off-diagonal Pauli matrix in that basis.Fine tuning the hopping constants to t1 = −2t2 = t, then

H0 =

ˆk

∑

iν

k2

2mψ†k,i,ν (σx)νν′ ψk,i,ν′ , (4)

in the continuum limit (k � 1/a), with m−1 = ta2/2.The dispersion of this model has two quadratic bandstouching at a single point k = 0, as shown in Fig. 2.At half filling, the band lacks a Fermi momentum andbehaves as a 1D semimetal. In the following, we assumethe band to be half filled and set a→ 1.

These fermions interact via a local, instantaneous twobody SYK interaction,

HSYK =1

(N)32

∑

νν′,ijk`

ˆx

Jxijk`ψ†x,i,νψ

†x,j,ν′ψx,k,ν′ψx,`,ν

(5)with random, color site independent matrix elementsJxijk` that are properly antisymmetrized with Jxijk` =−Jxjik` = −Jxij`k. As in the other SYK models, we takethese to be complex Gaussian distributed coupling with azero mean value 〈〈Jxijk`〉〉 = 0 and variance 〈〈|Jxijk`|2〉〉 =

J2/8.The standard method to study the current problem is

the imaginary time path integral formalism, where thepartition function is given by Z =

´ [DψDψ

]e−S , where

S = S0 + SSYK, with

S0 =

ˆτ,x

∑

`,ν

ψ`,ν (τ, x)

[∂τ − (σx)νν′

∂2x

2m

]ψ`,ν′ (τ, x) .

(6)We define

´τ≡´ β

0dτ , with β = 1/T , and SSYK the

corresponding two body action of (5) with with same timeGrassmann fields ψ(τ, x) and ψ(τ, x). In order to dealwith the disorder, we use the replica trick and averageover disorder realizations. This procedure amounts to anannealed approximation[16]. Using

〈〈e−∑ijk` Jijk`Aijk`〉〉 = eJ

2 ∑ijk` Aijk`Aijk` (7)

and defining the Green’s function

Gνν′ (τ, x) = − 1

N

N∑

`=1

〈T [ψν,` (0, 0) ψν′,` (x, τ)]〉, (8)

3

the integration over the fermionic fields results in the saddle point action,

Seff = − log det[δ (τ − τ ′) δ (x− x′)

(∂τ + σx (i∂x)

2)

+ Σ (x− x′, τ − τ ′)]

− J2

8

ˆτ,τ ′

trG2 (0, τ − τ ′) trG2 (0, τ ′ − τ)−ˆx,x′

ˆτ,τ ′

tr[Σ (x− x′, τ − τ ′) G (x′ − x, τ ′ − τ)

], (9)

where Σ (x− x′, τ − τ ′) is the self-energy. The action canbe minimized exactly in G and Σ in the large-N limit.Following the minimization, the solutions form a set ofSchwinger-Dyson equations

G−1 (iωn, k) = iωn −k2

2mσx − Σ (iωn, k) , (10)

and

Σ (τ, x) = −J2

2δ(x)G (−τ, 0) tr

[G (τ, 0) G (τ, 0)

], (11)

The self-energy Σ (iωn, k) ≡ Σ (iωn) is therefore momen-tum independent, reflecting the x−dependence of thecouplings Jxijk`. The disorder averaged SYK term is un-correlated and purely local. We denote the Fourier trans-form of the momentum independent self-energy as Σ (τ).We also denote

G(τ) ≡ G(τ, 0) =

ˆk

G(τ, k) (12)

for the momentum integrated Green’s function.

III. GREEN’S FUNCTION

If one takes the ansatz for the Green’s function, G(τ) =G(τ)σx, the self energy then has to be of the formΣ(τ) = Σ(τ)σx. As in usual SYK models, we make theusual infrared assumption iωn � Σ(iωn). The SchwingerDyson equations (10) and (11) can be written as

G(iωn) = −ˆk

1k2

2m + Σ(iωn)

= −√

2m

π√

Σ(iω)tan−1

(Λ√

2m√

Σ(iω)

), (13)

and

Σ(τ) = −J2G2(τ)G(−τ), (14)

There are two limits of particular interest. As it willbe clear in the next section, one is the intermediate fre-quency limit t2/J � ω � J , where the argument of thetan−1(y) function,

y ≡ Λ√2m√

Σ(iω)(15)

is small. This regime corresponds to the weakly disper-sive limit, which recovers the physics of the 0 + 1 di-mensional SYK dot. The other is the strongly dispersiveregime, ω � t2/J , where y � 1. We show that this limitis exactly solvable and leads to a different NFL regime.

A. Weakly dispersive limit

In this weakly dispersive limit (y � 1), the SYKphysics dominates and typically one obtains a fully in-coherent system with a linear in T dc resistivity. In thatregime, Eq. (13) becomes

G(iωn) ≈ − 1

π

Λ

Σ (iωn). (16)

The Schwinger-Dyson Eq. (14) and (16) have the sameform as in the SYK dot model [1]. They have confor-mal/reparametrization invariance, indicating a power lawsolution at T = 0.

At zero temperature, Eq. (14) and (16) can be solvedby the ansatz G (iω) = c1e−i

π4 (iω)

− 12 for the time or-

dered Green’s function [1]. Using this result in Eq. (14)and taking a Fourier transform one finds

Σ (iω) = −J2c31π

eiπ4

√iω, (17)

where the constant c1 = Λ14 /√J. The zero temperature

dispersive Green’s function is

G(iω, k) =1

Σ (iω)− k2/2m

Σ (iω)2 + . . . . (18)

This solution introduces a perturbative correction to theSYK Green’s function, in the same spirit as in the metal-lic case [17].

To get the finite temperature solutions, one can thenuse the conformal map, τ → f (τ) = tan πτ

β . Applyingthis to the Fourier transform of G (iω) gives

G (τ) = sgn(τ) c1

√1/β

2 sinπτ/β. (19)

4

Figure 3. Scaling functions for the Green’s function (FG) and the self-energy (FΣ) versus Matsubara frequency ωn normalizedby temperature T . Top row: numerical solution of Eq. (14) and (16) for FG and FΣ in the SYK limit for various temperatures,namely T = 10, 20 and 30 (green line, blue and red, respectively). β = T−1and J = 100 are set in units of 2m with a → 1(Λ = π). In this case, the Green’s function and self energy are purely imaginary and admit an analytical solution [see Eq.(17)]. Bottom row: numerical solution of Eqs. (14) and (25), in the strongly dispersive regime. The real and imaginaryparts of the scaling functions were computed at various temperatures, namely β = 1/T = 256, 64 and 4 (green line, blue andred, respectively). All curves nearly coincide at low frequencies, where the scaling functions are expected to be temperatureindependent.

The finite temperature self-energy Σ (iωn) can then beobtained from a Fourier transform of (14),

Σ (iωn) = iJ2c31

(2π)3/2√

2Γ(

34 + ωnβ

2π

)Γ(− 1

2

)

√βπΓ

(14 + ωnβ

2π

) , (20)

with ωn a Matsubara frequency. The dispersive Green’sfunction at finite temperature follows from Eq. (16) and(20),

G(iωn, k) =1

Σ (iωn)− k2/2m

Σ2 (iωn)+ . . . . (21)

It is well known that the Greens function and self en-ergies in this regime have some convenient scaling prop-erties. Eqs. (14) and (16) admit solutions of the form:

G(iωn) = (JT )−12 Λ−

34FG

(ωnT

)(22)

Σ(iωn) = (JT )12 Λ

34FΣ

(ωnT

)(23)

where FG,Σ are scaling functions which are independentof all parameters. The scaling functions are plotted in thetop row panels of Fig. 3. They were obtained by numer-ically solving Eq. (14) and (16) for various temperaturesand then stripping away the power law dependence in Tand J from the above equations. The results show goodagreement with the scaling arguments.

B. Strongly dispersive regime

Next we consider the regime where y � 1. As we willshow below, this inequality corresponds to the regimewhere

ω � t2

J, (24)

and leads a different kind of NFL behavior compared toweakly dispersive SYK models.

In the y � 1 regime, the Schwinger-Dyson equation

5

(13) reads

G(iω) = −1

2

√2m√

Σ(iω). (25)

Eq. (14) and (25) admit a power law solution at T = 0,given by the ansatz

G(τ) = C1

|τ |2∆, (26)

where 2∆ = 3/5, as found in a related model [18], with

C =

[Γ( 1

5 )−1Γ( 25 )−2

20 sin2(

3π10

)sin(

9π10

)] 1

5

≈ 0.40. (27)

That solution corresponds to a self-energy

Σ(iω) = C ′J45m

35 |ω|6∆−1 � |ω|, (28)

from equation (14). Explicit verification of this solutionfollows by Fourier transforming (25),

G(iω) = 2C sinπ∆ Γ(1− 2∆)J−25m

15 |ω|2∆−1

, (29)

and calculating Σ (iω) from (14). The zero-temperatureGreen’s function is hence

G(iω, k) =−σx

k2

2m + C ′J45m

35 |ω|4/5

, (30)

where

C ′ = −2C3 sin

(9π

10

)Γ

(−4

5

)≈ 0.22. (31)

The Green’s function above describes a distinct type ofincoherent semimetal, and is valid all the way down tozero frequency. It contrasts with the result in the coher-ent J → 0 limit of the ladder problem model (J/|ω| � 1),where the Green’s function has a pole with well definedquasiparticles. One needs to analytically continue theabove solution and impose physical restrictions to obtainthe exact Green’s function.

Note that the linear in T resistivity in SYK modelsstems from ∆ = 1

4 . In strongly dispersive semimetals,with y � 1, finite temperature solutions cannot be ob-tained using a conformal map on the T = 0 solutionbecause (14) and (25) do not have the requisite con-formal/reparametrization symmetry. Finite temperaturesolutions to these equations then have to be obtainednumerically. However, we still have a scaling symmetrywhich dictates a certain scaling form for the solutions.

Rewriting (25) in τ−space,ˆτ1,τ2

G(τ)G(τ − τ2)Σ(τ − τ1) =m

2δ(τ). (32)

It is easy to see that these equations are invariant under

τ → bτ, G → b−35G, Σ→ b−

95 Σ. (33)

Under this scaling, T → T/b leaving Tτ invariant. Withthis information, one can see that (14) and (25) admit asolution of the form

G(τ) = m15 J−

25T

35 G(Tτ) (34)

and

Σ(τ) = m35 J

45T

95 Σ(Tτ). (35)

Equivalently in Fourier space, we find

G(iωn) = m15 (JT )−

25FG

(ωnT

)(36)

Σ(iωn) = m35 (JT )

45FΣ

(ωnT

)(37)

where FG,Σ are scaling functions that are independent oftemperature T and the coupling J , with ωn a Matsub-ara frequency. The dispersive finite temperature Green’sfunction of the problem in this regime is

G−1(iωn, k) = −(k2

2m+m

35 (JT )

45FΣ(ωn/T )

)σx.

(38)As shown in the next section, this will suffice to deter-mine the temperature dependence of various transportcoefficients. Strictly speaking, the scaling symmetry isonly present in the infrared limit of the theory. Thismeans that the exact numerical solutions may violatethese scaling forms at very high frequencies. The realand imaginary parts of the numerically obtained scalingfunctions FG and FΣ are plotted in the bottom row ofFig. 3. The plots show good agreement with Eq. (37)even outside the infrared limit.

IV. SCALING ANALYSIS

After averaging over the disorder, which is spatiallyuncorrelated, the SYK term in the action has eightfermionic fields, which we symbolically write as

SSYK = J2

ˆτ1,τ2,x

[ψ(τ1, x)ψ(τ1, x)]2[ψ(τ2, x)ψ(τ2, x)]2.

(39)Rescaling time as τ ′ = τ/s and imposing the SYKcoupling J to be marginal, then the fields rescale asψ′ = ψ/s

14 . As pointed out before [15, 17], analyzing

the problem in the vicinity of the fixed point of the 0 + 1dimensional SYK model (t = 0), the kinetic term is arelevant perturbation and the hopping parameter growsas t′ = t

√s. If one starts from temperature T in the

weakly dispersive regime t � J , the hopping will growuntil t′(s∗) ∼ J for a rescaling parameter no larger thans∗ = J/T . Hence, the scaling stops at T = t2/J . The va-lidity of the incoherent “Planckian” regime requires that

T � T∗ = t2/J, (40)

6t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

NFL SYK

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

NFL SYK

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

NFL SYK

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

NFL SYK

1

t ⌧ T ⌧ J

T ⌧ t ⌧ J

T ⌧ J ⌧ t

⇢ ⇠ T

⇢ ⇠ T25

8><>:

NFL SYKT

1

t2

J⌧ T ⌧ J

T ⌧ t2

J

1

t2

J⌧ T ⌧ J

T ⌧ t2

J

1

Figure 4. Different temperature regimes in the scaling. Attemperature T > T∗ = t2/J , the system is close to the 0 + 1dimensional SYK fixed point and shows Planckian behavior,with linear dependence of the resistivity in temperature. Be-low T∗ = t2/J the system crosses over towards a distinct typeof incoherent NFL, with ρ ∼ T

25 .

as in the case of incoherent metals with a finite Fermisurface [17].

If one continues to lower the temperature further belowT∗, we claim that the system crosses over to a differenttype of incoherent NFL regime with sublinear temper-ature scaling of the resistivity, as illustrated in Fig. 4.From the perspective of the Schwinger-Dyson equations(13) and (14), the parameter that controls the crossoverbetween the weakly and the strongly dispersive regimesis

y(T ) ∼√

t

Σ(T ). (41)

In the strongly dispersive regime y(T )� 1, setting m ∼t−1, one can write the solution of the finite temperatureself-energy (37) as

Σ(T ) ∝ t− 35 (JT )

45 � t. (42)

This inequality leads to T � T∗ = t2/J . In the sameway, in the weakly dispersive regime (y(T )� 1),

Σ(T ) ∝√JT � t, (43)

as seen from Eq. (20), implying that T � T∗.We note that in the case of metals, the T < T ∗ regime

was found to realize a Fermi liquid. In the case ofa 1D half-filled semimetal with parabolic touching, weshowed that the low temperature regime does not leadto a semimetal, but to another type of incoherent NFL,whose transport properties will be addressed in the nextsection.

V. TRANSPORT

In this section, we look at the electric and thermal con-ductivities using the above finite temperature solutions.

These can be computed using the Kubo formula. Thecharge current operator for this model is

je =e

2m

ˆk

∑

i

kψ†kiσxψki. (44)

The zero frequency conductivity is given by

σdc = limω→0ImKret(ω)

ω, (45)

where Kret(ω) is the retarded current-current correlationfunction, K(τ) = 〈T [je(0, x)je(τ, x)〉, given by the seriesof diagrams in Fig. 5. Each diagram displayed in thatfigure is of order N . For instance, in diagram (b) eachSYK vertex contributes a factor of N−

32 , while the three

independent flavor sums contribute N4, making it a totalof order N. Diagrams in (b) and (c) vanish because thecurrent vertex is an odd function of momenta. This canbe readly seen by noticing that because of the disorderaveraging, the summation over momenta through eachcurrent vertex can be performed independently, result-ing in a zero contribution of those diagrams [22]. Theremaining diagram is shown in Fig. 5a. It can be writtenin terms of the Green’s functions derived before as

K(iωn) =Ne2

(2m)2T tr

∑

νn

ˆk

k2G(iνn, k)G(iνn + iωn, k).

(46)The transport properties of the weakly dispersive regimerecovers the expected behavior of incoherent metalsfound in Ref. [13, 17], σdc ∝ 1/T , and we will focusinstead in the strongly dispersive case.

It is usually challenging to sum over Matsubara fre-quencies in the absence of poles in the Green’s functions.One can circumvent that difficulty by using the spectralrepresentation of the Green’s function

G(iωn, k) =

ˆω′

A(k, ω′)iωn − ω′

, (47)

with

A(k, ω) =1

πσxIm

1k2

2m + Σ(ω + i0+)(48)

the spectral function. One arrives at

σdc =Ne2√

2m

π2T

ˆω

[ImΣ(ωT )

]2

cosh2( ω2T )

ˆk

k2

∣∣ k22m + Σ

(ω2T

)∣∣4 .

(49)Equivalently, casting Eq. (49) in terms of the scalingfunctions (37), the dc conductivity is

σdc(T ) =Ne2

√2m(JT )

25

I1, (50)

7

Figure 5. Diagrams that contribute to the current-currentcorrelation function to leading order in N (see text). Redrectangles represent the current vertex. Black lines representthe fermion Green’s function and dotted blue line representsdisorder average. Since the current vertex is an odd functionof momenta, diagrams (b) and (c) and so on vanish, leaving(a) as the sole contribution in the large-N limit.

where

I1 =1

π2

ˆ ∞0

dz[ImFΣ(z)]

2

cosh2( z2 )

ˆ ∞0

dyy2

|y2 + FΣ (z)|4(51)

is a dimensionless integral, and FΣ(z) the analyticallycontinued scaling function of the self energy.

A signature property of Fermi liquids is captured bythe Lorentz ratio, L = κ/(σT ), which is the ratio of ther-mal (κ) and electric (σ) conductivities. In the particle-hole symmetric model, the thermoelectric contribution tothe thermal conductivity, which is proportional to T/σ,vanishes by symmetry and can be ignored in the com-putation of κ [13, 20]. The energy current, whose cor-relation function determines the remaining piece of thethermal conductivity, is given by

jE =

ˆk

∑

i

k

2mψ†k,iσx∂τψk,i. (52)

In the present particle-hole symmetric case, the thermalconductivity is then given by the same diagrams as in thecase of σ. Following the same prescription as above,

κ = limω→0KretE (ω)

ωT, (53)

where KE(τ) = 〈T [jE(0, x)jE(τ, x)〉 is a thermal current-current correlation function. This leads to

κ =Ne2

√2mJ

25

T35 I2, (54)

where

I2 =1

π2

ˆ ∞0

dz z2 Im [FΣ(z)]2

cosh2( z2 )

ˆ ∞0

dyy2

|y2 + FΣ (z)|4.

(55)In order to calculate integrals I1 and I2, one needs to

perform a numerical analytical continuation of the scal-ing functions obtained in section III. In the spirit of SYKmodels, we compute these integrals assuming the scal-ing form to be valid over the entire range of frequencies.Numerical analytical continuation is a challenging prob-lem. The numerical integrals were done with the Padeapproximation method in the TRIQS library [23, 24].

We numerically find that slight variations in the Mat-subara scaling forms can significantly affect the integralsI1 and I2, but their ratio is insensitive to numerical issueswith the analytical continuation process in the regime ofinterest. The Lorentz ratio is

L =κ

Tσ=I2I1≈ 3.2, (56)

which is rather close to the Fermi liquid value of LFL =π2/3.

VI. DISCUSSION

In this work, we studied a simple semimetallic ver-sion of a dispersive SYK model in one dimension. Con-trary to most studies of dispersive models in the litera-ture [13–15, 17, 18], we focus on the strongly dispersivelimit, which corresponds to the stable fixed point of thisproblem from the scaling point of view. In this limit,we find that the Schwinger-Dyson equations do not ad-mit an exact analytic finite temperature solution evenin the infrared approximation, where it is assumed thatΣ(iω) � iω. In particular, the model does not exhibitconformal symmetry, which makes it difficult to solve theSchwinger-Dyson equations analytically. We solve thoseequations exactly exploiting the scaling symmetry of themodel, combined with numerical calculations. We findthat the Greens function and self-energy scale with tem-perature with a power law of T−

25 and T

45 respectively.

Using this solution to study transport properties, weshow that dc resistivity scales with a sublinear power lawdependence on temperature, ρ ∼ T

25 . We compute the

Lorentz ratio L = κ/(σT ) with the analytically continuedscaling functions and find that L ≈ 3.2, rather close tothat of Fermi liquids. The scaling analysis of this problemindicates that if one starts in the high temperature SYKfixed point of the problem, where t2/J � T � J , thehopping parameter will grow as one scales the tempera-ture down, while the SYK coupling is marginal. The scal-ing flows towards the strongly dispersive regime, wherethe Schwinger-Dyson equations indicate the presence ofa distinct incoherent NFL regime at T � t2/J . Thatcontrasts with the behavior of incoherent metals, whichhave a finite Fermi surface. In the latter, the system flowstowards a Fermi liquid at low temperature [17].

Those results should be compared with the several lat-tice models of SYK dots that have been studied recently[13, 17, 18]. Ref. [13, 17] studied a lattice of coupleddots in the limit where the SYK coupling J is the high-est energy scale. In those cases, the physics of a singledot dominates, with the effects of hopping being pertur-bative. The Σ ∝ √ω scaling of the self energy in thislimit ultimately leads to a linear in T dc resistivity.

Among the dispersive SYK models, the one studied inRef. [18] is the closest to ours. They examined a two-band model for arbitrary dimension and dispersion, witha color site dependent SYK interaction, which forces the

8

saddle point solution of the Green’s function to be di-agonal but still color site dependent. Their solution forthe self-energy is purely imaginary and color site inde-pendent, differently from our results. That leads to anapproximate conformal symmetry in the problem in theNFL regime, in contrast with our work, where we findthat conformal symmetry is absent. In this paper, we fo-cused in the crossover between the regime dominated by

the 0 + 1 dimensional SYK fixed point and the low tem-perature NFL regime for a 1D semimetal with parabolicband touching, and addressed the transport properties ofthis novel state.

Acknowledgements.− We thank A. Haldar for severalhelpful discussions. BU and GJ acknowledge Carl T.Bush fellowship and NSF grant DMR-2024864 for partialsupport.

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