Sachdev-Ye-Kitaev Models and Beyond: A Window into Non-Fermi Liquids

TL;DR

The article surveys a broad class of non-Fermi liquids built around the Sachdev-Ye-Kitaev framework, emphasizing solvable, locally critical states without quasiparticles. It connects SYK physics to random-matrix models, lattice SYK atoms, and Fermi surfaces coupled to critical bosons or gauge fields, and it derives universal low-energy scalings, entropy structure, and transport properties including Planckian dynamics. A major thrust is establishing deep links to quantum gravity through Schwarzian dynamics and AdS2 holography, with OTOCs and chaos as diagnostic tools. The review also presents extensive analyses of random exchange magnets, Hubbard-type models, Kondo-Heisenberg lattices, and lattice generalizations, highlighting potential relevance to real materials such as cuprates and moiré systems, and outlining key open questions in the field.

Abstract

We present 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. We explore the phase diagrams of a number of models of such local electronic correlation, 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 which display SYK physics. The models studied include the single-band Hubbard model, the $t$-$J$ model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect 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, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.

Figures (24)

• Figure 1: (a) A nFL obtained by coupling a critical boson (e.g. nematic order with $\boldsymbol{Q}=0$) to an electronic Fermi surface. (b) A 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 the diffusion constant (a) and compressibility ((a)-inset) for a gas of ultra-cold $^6$Li atoms in an optical lattice, realizing a two-dimensional Fermi-Hubbard model with $U/t\simeq 7.5$ at a density $n\simeq 0.825$. (b) Reconstructed 'resistivity' using Einstein-Sutherland relation. Grey 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 estimated error bars. Adapted from Bakr.
• Figure 3: Examples of $T-$linear resistivity extending over a wide range of temperature scales in (a) hole-doped La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) near optimal doping (adapted from boebinger), and (b) magic-angle twisted bilayer graphene (MATBG) near $\nu\approx -2$, relative to charge neutrality, $\nu=0$ (adapted from efetov21). In LSCO, $T_{\textnormal{coh}}$ 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 (a)-bottom inset. In MATBG, the linearity for a range of dopings near $\nu\approx-2$ ((b)-inset) persists down to $\sim 40$ mK. Both family of materials also display a Planckian form of $\Gamma_\textnormal{dc}$ (Eq. \ref{['eq:Planckian']}).
• Figure 4: The graph for the electron self-energy, $\Delta(\tau)$, in Eq. (\ref{['rm2']}). Solid lines denote fully dressed electron Green's functions. The dashed line represents the disorder averaging associated with $\overline{|t_{ij}|^2}$.
• Figure 5: 65536 many-body eigenvalues of a $N=32$ Majorana matrix model with random $q=2$ fermion terms. $\mathcal{N}(E)$ is plotted in (a) and (b) in 200 and 100 bins, (b) and (c) zoom into the bottom of the band. Individual energy levels are shown in (c), and these are expected to have spacing $1/(N \rho(\mu))$ at the bottom of the band as $N \rightarrow \infty$.