We analytically study low-temperature universal properties of a class of SYK-type models in the large N limit from the ${\mathrm{AdS}}_{2}$ gravity dual side in terms of $\mathrm{SL}(2,R)$-invariant Schwarzian action. The quantum correction to the conformal field theory ${\mathrm{CFT}}_{1}$ two-point correlation function due to the Schwarzian action produces a transfer of degree of freedom from the quasiparticle peak to the Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher-point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having a Lyapunov exponent. Higher-order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as the spin liquid in disordered metals, marginal Fermi liquid, non-Fermi liquid, etc.
We analytically study low-temperature universal properties of a class of SYK-type models in the large N limit from the AdS2 gravity dual side in terms of SL(2,R)-invariant Schwarzian action. The quantum correction to the conformal field theory CFT1 two-point correlation function due to the Schwarzian action produces a transfer of degree of freedom from the quasiparticle peak to the Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher-point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having a Lyapunov exponent. Higher-order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as the spin liquid in disordered metals, marginal Fermi liquid, non-Fermi liquid, etc.
We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=\infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=\infty $ at any finite $ M $, nor at $ M=\infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.
The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows peculiar non-Fermi-liquid properties without quasiparticles. It exhibits a maximally chaotic behavior characterized by out-of-time-ordered correlators (OTOCs), and is expected to be a holographic dual to black holes. While a faithful realization of the SYK model in condensed-matter systems may be involved, a striking similarity between the SYK model and the Hund-coupling-induced spin-freezing crossover in multiorbital Hubbard models has recently been pointed out. To further explore this connection, we study OTOCs for fermionic single-orbital and multiorbital Hubbard models, which are prototypical models for strongly correlated electrons in solids. We introduce an imaginary-time four-point correlation function with an appropriate time ordering, which by means of the spectral representation and the out-of-time-order fluctuation-dissipation theorem can be analytically continued to real-time OTOCs. Based on this approach, we numerically evaluate real-time OTOCs for Hubbard models in the thermodynamic limit, using the dynamical mean-field theory in combination with a numerically exact continuous-time Monte Carlo impurity solver. The results for the single-orbital model show that a certain spin-related OTOC captures local moment formation in the vicinity of the metal-insulator transition, while the self-energy does not show SYK-type non-Fermi-liquid behavior. On the other hand, for the two- and three-orbital models with nonzero Hund coupling we find that the OTOC exhibits a rapid damping at short times and an approximate power-law decay at longer times in the spin-freezing crossover regime characterized by fluctuating local moments and a non-Fermi-liquid self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})\ensuremath{\sim}\sqrt{\ensuremath{\omega}}$. These results are in a good agreement with the behavior of the SYK model, providing firm evidence for the close relation between the spin-freezing crossover physics of multiorbital Hubbard models and the SYK strange metal.
This dissertation presents a comprehensive investigation of fermionic quantum matter into the intricate relationship between quantum chaos, thermalization, and their connection to gravitational duals, with particular emphasis on the Sachdev-Ye-Kitaev (SYK) model and its extensions. Through an integrated approach combining analytical solutions in the large-N limit with advanced numerical techniques, we explore fundamental questions in quantum many-body physics, including the interplay of chaos and ergodicity, universal transport in non-Fermi liquids, and the role of gravitational analogs in strongly correlated systems. We first demonstrate that coupled SYK lattices evade instantaneous thermalization despite weak charge fluctuations, governed by a discrete wave equation dependent solely on hopping strength, independent of on-site interactions. Thermodynamically, coupled SYK models exhibit critical exponents universal to van der Waals fluids and AdS black holes, yet diverge from the Hawking-Page transition at ultralow temperatures, revealing a continuum of first-order phase transitions. They simultaneously retain the quantum chaotic properties of both a single-dot SYK model and black holes. In extended SYK chains, we identify a universal DC conductivity bound across interaction regimes, identifying crossovers between insulating, Fermi liquid, strange metal, and bad metal phases with signatures of holographic insulator behavior. Quench dynamics in mixed SYK systems reveal rapid thermalization without prethermal plateaus, enabling direct observation of thermalization in closed quantum systems in the thermodynamic limit. To diagnose ergodicity-breaking, we introduce the Krylov variance, a probe of operator localization in Krylov space, validated through mass-deformed SYK and quantum East models. Finally, we uncover Page curve entanglement dynamics and temporal quantum phase transitions in entanglement Hamiltonians of interacting fermionic chains, where symmetry-sector crossings highlight limitations of perturbative calculations performed around initial times, such as in Hawking’s analysis of black hole evaporation. We provide a condensed matter analog of the holographic “island formula” — a mechanism invoking quantum extremal surfaces to resolve the black hole information paradox. Methodologically, this work integrates large-N-type expansions, Keldysh contour techniques, and numerical methods to bridge condensed matter physics, quantum information, and gravitational duality. The results establish SYK-type systems as a universal framework for studying non-Fermi liquids, quantum criticality, and black hole analogs, while paving the way for exploring bosonic extensions and generalizations to open quantum systems.
Extreme correlations arise as the limit of strong correlations, when the local interaction constant U goes to infinity. This singular limit transforms canonical fermions to noncanonical Hubbard-type operators with a specific graded Lie algebra replacing the standard anticommutators. We are forced to deal with a fundamentally different and more complex lattice field theory. We study the $t\text{\ensuremath{-}}J$ model, embodying such extreme correlations. We formulate the picture of an extremely correlated electron liquid, generalizing the standard Fermi liquid. This quantum liquid breaks no symmetries, and has specific signatures in various physical properties, such as the Fermi-surface volume and the narrowing of electronic bands by spin- and density-correlation functions. We use Schwinger's source field idea to generate equations for the Green's function for the Hubbard operators. A local (matrix) scale transformation in the time domain to a quasiparticle Green's function is found to be optimal. This transformation allows us to generate vertex functions that are guaranteed to reduce to the bare values for high frequencies, i.e., are ``asymptotically free.'' The quasiparticles are fractionally charged objects, and we find an exact Schwinger-Dyson equation for their Green's function, i.e., the self-energy is given explicitly in terms of the singlet and triplet particle-hole vertex functions. We find a hierarchy of equations for the vertex functions, and further we obtain Ward identities so that systematic approximations are feasible. An expansion in terms of the density of holes measured from the Mott Hubbard insulating state follows from the nature of the theory. A systematic presentation of the formalism is followed by some preliminary explicit calculations. We find a d-wave superconducting instability at low T that formally resembles that found in the resonating valence-bond theory, but with a much reduced Tc.
The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical many body system with random all-to-all interactions on fermionic N sites (N>>1). This model is shown to saturate the known maximal chaos bound of many body system and then based on this observation it is conjectured to be dual to a quantum black hole in the sense of the AdS/CFT correspondence. In this dissertation, we show that the large N physics of the SYK model is systematically described by a single bi-local field. In particular, we emphasize the appearance of the emergent conformal reparametrization symmetry at the critical IR fixed point and the corresponding divergent contribution of the symmetry modes in the propagator of the bi-local field. We discuss non-linear-level derivation of the zero modes effective action, which is given by the Schwarzian derivative for finite reparametrization symmetry. Besides the symmetry modes, which correspond to the dilaton-gravity sector in the dual AdS theory, the SYK model also predicts an infinite tower of matter fields in AdS_2. We demonstrate that this infinite spectrum can be nicely packaged into a single field in 3-dimensional space-time. Finally, we consider the question of identifying the dual space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of n-point Green's functions. This results in a dual theory with Euclidean AdS signature with additional leg-factors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2D string theory.
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.