Sachdev-Ye-Kitaev models and beyond: Window into non-Fermi liquids

This is a review of the Sachdev-Ye-Kitaev (SYK) model of compressible quantum many-body systems without quasiparticle excitations, and its connections to various theoretical studies of non-Fermi liquids in condensed matter physics. The review is placed in the context of numerous experimental observations on correlated electron materials. Strong correlations in metals are often associated with their proximity to a Mott transition to an insulator created by the local Coulomb repulsion between the electrons. The phase diagrams of a number of models of such a local electronic correlation are explored, employing a dynamical mean-field theory in the presence of random spin exchange interactions. Numerical analyses and analytical solutions, using renormalization group methods and expansions in large spin degeneracy, lead to critical regions that display SYK physics. The models studied include the single-band Hubbard model, the $t\text{−}J$ model, and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. Also examined are non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multiorbital atom, with the SYK atoms arranged in an infinite lattice. Connections are made to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large-$N$ theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, an overview of the links between the SYK model and quantum gravity is presented, and the review ends with an outlook on open questions.

Figure 1

(a) NFL obtained by coupling a critical boson (nematic order with $\mathit{Q}=0$) to an electronic Fermi surface. (b) Bandwidth-tuned metal to paramagnetic Mott insulator transition. The Mott insulator hosts a neutral Fermi surface (dashed circle) of fractionalized degrees of freedom coupled to an emergent gauge field. (c) A Fermi volume changing transition between two distinct metals across a “Kondo breakdown” quantum-critical point. The quantum-critical point hosts a critical Fermi surface of electrons in all the examples. The Mott insulator and the FL* phases host a critical Fermi surface of “spinons” in (b) and (c), respectively.

Figure 2

Measurement of (a) the diffusion constant and (inset) compressibility for a gas of ultracold ${}^6\mathrm{Li}$ atoms in an optical lattice, with a two-dimensional Fermi-Hubbard model realized with $U/t\simeq 7.5$ at a density $n\simeq 0.825$. (b) Reconstructed resistivity using Einstein-Sutherland relation. The gray horizontal dashed line represents the estimated MIR value. Theoretical calculations using DMFT (in green) and the finite-$T$ Lanczos method (in blue) are shown; the band representation indicates the estimated error bars. Adapted from [53].

Figure 3

Examples of $T$-linear resistivity extending over a wide range of temperature scales in (a) hole-doped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO) near optimal doping and (b) magic-angle twisted bilayer graphene (MATBG) near $\nu \approx -2$ relative to the charge neutrality $\nu =0$. In LSCO, $T_{\mathrm{coh}}$ can be inferred to be much lower than any characteristic energy scales by turning on a magnetic field and accounting for the finite magnetoresistance [(a)-top inset]; the variation of the slope ($A$) on hole-doping is shown in the bottom inset of (a). In MATBG, the linearity for a range of dopings near $\nu \approx -2$ [inset of (b)] persists down to $\sim 40\mathrm{mK}$. Both families of materials also display a Planckian form of ${\mathrm{\Gamma }}_{\mathrm{dc}}$ [Eq. (3.5)]. (a) Adapted from [147]. (b) Adapted from [201].

Figure 4

Graph of the electron self-energy $\mathrm{\Delta }(\tau )$ in Eq. (4.6b). Solid lines denote fully dressed electron Green’s functions. The dashed line represents the disorder averaging associated with $\overline{|t_{ij}{|}^2}$.

Figure 5

Display of 65 536 many-body eigenvalues of an $N=32$ Majorana matrix model with random $q=2$ fermion terms. $\mathcal{N}(E)$ is plotted in (a) and (b) in 200 and 100 bins, while (b) and (c) are enlargements of the bottom of the band. Individual energy levels are shown in (c) and are expected to have the spacing $1/N\rho (\mu )$ at the bottom of the band as $N\to \infty$.

Figure 6

Display of 65 536 many-body eigenvalues of an $N=32$ Majorana SYK Hamiltonian with random $q=4$ fermion terms. $\mathcal{N}(E)$ is plotted in (a) and (b) in 200 and 100 bins, while (b) and (c) are enlargements of the bottom of the band. Individual energy levels are shown in (c) and are expected to have the spacing $e^{-N\mathcal{S}}$ at the bottom of the band as $N\to \infty$. Compare this to Fig. 5 for the random-matrix model, which has a much sparser spacing $\sim 1/N$ at the bottom of the band.

Figure 7

The “melon graph” for the electron self-energy $\mathrm{\Sigma }(\tau )$ in Eq. (5.2b). Solid lines denote fully dressed electron Green’s functions. The dashed line represents the disorder averaging associated with the interaction vertices (denoted as solid circles) $\overline{|U_{ij;k\ell }{|}^2}$.

Figure 8

Electron spectral density in the SYK model, obtained from imaginary part of Eq. (5.26).

Figure 9

$T=0$ entropy density $\mathcal{S}$ vs $\mathcal{Q}$. From [142].

Figure 10

Large-$N$ equation satisfied by the three-point correlator in Eq. (5.46). The red circles represent the operator $O_h$.

Figure 11

Local spin response function for the spin-$1/2$ doped random-exchange $t\text{−}J$ model, as obtained by exact diagonalization of an $N=18$ site cluster averaged over 100 disorder realizations, for $t=J=1$. $n=1-p$ is the particle density. From [374].

Figure 12

Phase diagram of the spin-$1/2$, half-filled, random-exchange $t\text{−}J\text{−}U$ model. At low temperatures, a quantum-critical point separates the spin-glass (SG) phase from a Fermi-liquid (FL) phase. The background color corresponds to the fitted power-law exponent of the local spin correlation function $\chi (\tau )\sim 1/{\tau }^{2\mathrm{\Delta }}$, with $2\mathrm{\Delta }\simeq 1$ in the quantum-critical metal (QCM) (red) and $\mathrm{\Delta }=1$ in the Fermi liquid (blue). At high temperatures and $U$, one obtains a Mott insulator. From [62].

Figure 13

Phase diagram ([107]) of the spin-$1/2$ doped random-exchange $t\text{−}U\text{−}J$ model as obtained using a quantum Monte Carlo solution of the EDMFT equations. FL, Fermi liquid; SG, metallic spin glass for $p\ne 0$. The background color corresponds to the fitted power-law exponent of the local spin correlation function $\chi (\tau )\sim 1/{\tau }^{2\mathrm{\Delta }}$ (color scale on the right). Along the dashed gray line, SYK behavior $2\mathrm{\Delta }\simeq 1$ is found. A linear-in-$T$ resistivity is obtained in the quantum-critical region with a resistivity that becomes lower than the MIR resistivity. Inset: Enlargement close to the quantum-critical point.

Figure 14

Inverse single-electron excitations lifetime $1/{\tau }^{*}$ as a function of temperature $T$ in the spin-$1/2$ doped random-exchange $t\text{−}U\text{−}J$ model for different doping $p$. A Planckian behavior [Eq. (7.12)] is observed close to the quantum-critical point.

Figure 15

Phase diagram ([303]) of the doped $t\text{−}J$ model in the large-$M$ limit with a condensed bosonic holon $b$. The SY spin liquid (incoherent metal) displays linear-in-$T$ resistivity with a large bad metal resistivity.

Figure 16

Schematic phase diagram of the $t\text{−}J$ model in the non-Fermi-liquid $M$ limit of Sec. 7d2 ([76]). In the critical metal phase, the exponents obey $0<{\mathrm{\Delta }}_b<1/4$ and ${\mathrm{\Delta }}_f=1/2-{\mathrm{\Delta }}_b$, and ${\mathrm{\Delta }}_b$ decreases monotonically toward 0 (the Fermi-liquid value) with increasing $p$.

Figure 17

Schematic phase diagram of the RG analysis of the random, fully connected $t-J$ model. The spinon and holon states are nearly degenerate in the critical spin-liquid theory, while the holon (spinon) states have lower energy for $p>p_c$ ($p<p_c$). From [205].

Figure 18

RG flow of Eq. (7.35) in the $\gamma \text{−}g$ plane plotted for $\epsilon =1$ and $\overline{r}=0.5$. The red circle is the stable fixed point in this plane, which is unstable only to flows predominantly in the $s$ direction out of the plane; this fixed point describes the $p=p_c$ critical state in Fig. 17, and $p-p_c$ tunes the coefficient of the relevant perturbation (not shown), which presumably drives the system into the $p>p_c$ and $p<p_c$ phases shown in Fig. 17. From [205].

Figure 19

Phase diagram of the large-$M$ Kondo lattice with random exchange. The dashed lines are crossovers, but the $T=0$ filled circle marks a quantum phase transition. The FL* phase and the quantum-critical point exhibit linear-in-$T$ resistivity with the small carrier density of the conduction electrons. The HFL exhibits Fermi-liquid $T^2$ resistivity with a large carrier density of both the conduction electrons and the local moments. The critical theory of the HFL-FL* transition at $T=0$ has also been discussed for models with full translational symmetry ([370]; [371]). From [57].

Figure 20

RG flow of Eq. (8.21) for $\epsilon =1$. The $\gamma =0$ axis corresponds to the usual Kondo RG flow ([179]). The Kondo-breakdown fixed point has one relevant direction and describes a phase transition between a small Fermi surface phase [likely with magnetic order for $\mathrm{SU}(2)$] associated with the random quantum magnet fixed point of Sec. 6c, and a large Fermi surface HFL at large $J_K$. Compare this to Fig. 18 for the random $t\text{−}J$ model. From [366].

Figure 21

Basic building block for studying translationally invariant lattice models constructed out of SYK atoms with $N$ orbitals per site. The different sites are coupled together by single-electron hopping terms.

Figure 22

Melon graphs for the model in Eq. (10.7) for the electron self-energies for (a) $c$ and (b) $d$ electrons, respectively. Solid black (red) lines denote fully dressed $c$ ($d$) Green’s functions. The dashed (dotted) line represents the disorder averaging associated with the interaction vertex $\overline{|U_{ij;k\ell }{|}^2}$ ($\overline{|V_{ij;k\ell }{|}^2}$).

Figure 23

Bethe-Salpeter equation for the intraorbital pairing vertex $\mathrm{\Phi }$ in the large-$N$ limit.

Figure 24

Extended patch of the Fermi surface with momenta expanded about the point ${\mathit{k}}_0$ on the Fermi surface. This yields a theory of two-dimensional fermions $\psi$ in Eq. (11.3).

Figure 25

Spatial crossover boundaries outside a black hole of charge $\mathcal{Q}$. The value of $R_h$ is determined from $\mathcal{Q}$ via Eq. (12.13), and we describe $T\ll 1/R_h$ at a fixed $\mathcal{Q}$ and $R_2\sim R_h$. We indicate contributions to the entropy $\mathrm{\Delta }S$ from regions (A) and (B).