Sachdev-Ye-Kitaev Model Research Articles

Sign-problem-free variant of the complex Sachdev-Ye-Kitaev model

We construct a sign-problem free variant of the complex Sachdev-Ye-Kitaev (SYK) model which keeps all the essential properties of the SYK model, including the analytic solvability in the large-$N$ limit and being maximally chaotic. In addition to the number of complex fermions $N$, our model has an additional parameter $M$ controlling the number of terms in the Hamiltonian which we take $M \to \infty$ with keeping $M/N$ constant in the large-$N$ limit. While our model respects global $U(1)$ symmetry associated with the fermion number conservation, both the large-$N$ limit and the sign-problem free nature become explicit in the Majorana representation. We present a detailed analysis on our model, i.e., the random matrix classification based on the symmetry analysis, analytic approach, and the quantum Monte Carlo simulations. All these analysis show that our model exhibit a non-Fermi liquid (NFL) physics, a gapless fermionic system lying beyond the conventional Fermi liquid picture.

The Extended Diffusive Sachdev–Ye–Kitaev Model as a Sort of “Strange Metal”

The linear dependence on temperature of the resistivity in metals undergoing high‐temperature superconducting transition is a puzzle, which is generally attributed to non‐Fermi liquid behavior on a large range of temperature values. No particle and momentum conservation characterizes the 0 + 1 − d Sachdev–Ye–Kitaev model as a tractable non‐Fermi liquid starting paradigm, when extended to higher space dimensions. Due to the conformal symmetry breaking in the incoherent phase, Goldstone bosons arise, which provide energy diffusion in the lattice. The linear response to energy diffusion in the model is discussed, which is in principle charge neutral and non‐momentum conserving, and the temperature dependence of the thermal and electric conductivity in the incoherent prechaotic, intermediate temperature phase is obtained.

Q -Laguerre spectral density and quantum chaos in the Wishart-Sachdev-Ye-Kitaev model

We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\hat{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number $N$ of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara $Q$-Laguerre polynomials. For $\hat{q} = 3, 4$, the $Q$-Laguerre prediction, with $Q=Q(\hat{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30 < N \leq 34$ while we observe some deviations for $\hat q = 2$. The most salient feature of the spectral density is that, for odd $\hat{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\hat q = 4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on $N$, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chGUE universality class, which completes the tenfold-way classification in the SYK model.

Effects of the Symmetries and Related Orders to the Thermalization of Many-Body Localized System

In this paper, we discuss the effects of the symmetries and related topological orders to the thermalization of many-body localized system. We consider the one-dimensional fermion chain system with open (or periodic) boundary condition, whose boundaries are characterized by Sachdev-Ye-Kitaev (SYK) intercation. Just like in the SYK model and the tendor models, there are many-body quantum chaos in out-of-time-ordered correlation when the system is being thermalized by the interactions (usually nonuniform and being randomly distributed), and satisfies the eigenstate thermalization hypothesis (ETH). While the continuous or discrete symmetries usually protect the related topological orders against the thermalization, which may lead to the localization of quantum states and generate large degeneracy. We discuss these effects in terms of the fermionic or spin languages. In many-body localized phase, a large number of degenerate ETH-violated states can be found in both the frustration-free AKLT model or the integrable biquadratic (Wishart) SYK model. The quantum scars appear when such degenerate states are embeded into an ETH-satisfying spectrum of the Hilbert space enlarged by a much larger number of bosonic flavors (e.g., SU(M) multiplets degeneracy). Usually, the emergence of quantum chaos require large limits of boson flavor M and the number of coupled (undegenerate) states (N; which related to Z N symmetry). Further, the boson (or excitations) flavor number M can often to related to the fermion number N through the duality transformations, in which case the Z M symmetry is possible to generted by SU(M) model and leads to asymptotic degeneracy in large-M limit.

Thermalization of randomly coupled SYK models

We investigate the thermalization of Sachdev–Ye–Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the Rényi entropies do not saturate to the expected thermal values in the large-N limit. Using numerical large-N methods, we first show that the Rényi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large-N artifact by repeating the quench for finite N and finding that the saturation value of the Rényi entropy extrapolates to the expected thermal value in the N → ∞ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations.

Quantum tunneling dynamics in a complex-valued Sachdev-Ye-Kitaev model quench-coupled to a cool bath

The Sachdev-Ye-Kitaev (SYK) model describes interacting fermionic zero modes in zero spatial dimensions, e.g. quantum dot, with interactions strong enough to completely washout quasiparticle excitations in the infrared. In this paper, we consider the complex-valued SYK model at initial temperature $T$ and chemical potential $\mu$ coupled to a large reservoir by a quench at time $t=0$. The reservoir is kept at zero temperature and charge neutrality. We find that the dynamics of the discharging process of the SYK quantum dot reveals a distinctive characteristic of the SYK non-Fermi liquid (nFl) state. In particular, we focus on the tunneling current induced by the quench. We show that the temperature dependent contribution to the current's half-life scales linearly in $T$ at low temperatures for the SYK nFl state, while for the Fermi liquid it scales as $T^2$.

Correlation functions of $U(N)$-tensor models and their Schwinger–Dyson equations

We analyze the correlation functions of U(N) -tensor models (or complex tensor models) and use the Ward–Takahashi identity in order to derive the full tower of exact, analytic Schwinger–Dyson equations. We write them explicitly for ranks D=3 and D=4 . Throughout, we follow a non-perturbative approach. We propose the extension of this program to the Gurău–Witten model, a holographic tensor model based on the Sachdev–Ye–Kitaev model (SYK model).

Variational preparation of the thermofield double state of the Sachdev-Ye-Kitaev model

We provide an algorithm for preparing the thermofield double (TFD) state of the Sachdev-Ye-Kitaev (SYK) model without the need for an auxiliary bath. Following previous work, the TFD can be cast as the approximate ground state of a Hamiltonian, ${H}_{\text{TFD}}$. Using variational quantum circuits, we propose and implement a gradient-based algorithm for learning parameters that find this ground state, an application of the variational quantum eigensolver. Concretely, we find shallow quantum circuits that prepare the ground state of ${H}_{\text{TFD}}$ for the $q=4$ SYK model for $N=8$ Majoranas per side. For $N=12$, we achieve a variational energy within 1% of the true ground-state energy.

Traversable wormhole in coupled Sachdev-Ye-Kitaev models with imbalanced interactions

A pair of identical Sachdev-Ye-Kitaev (SYK) models with bilinear coupling forms a quantum dual of a traversable wormhole, with a ground state close to the so called thermofield double state. We use adiabaticity arguments and numerical simulations to show that the character of the ground state remains unchanged when interaction strengths in the two SYK models become imbalanced. Analysis of thermodynamic and dynamical quantities highlights the key signatures of the wormhole phase in the imbalanced case. Further adiabatic evolution naturally leads to the ``maximally imbalanced'' limit where fermions in a single SYK model are each coupled to a free Majorana zero mode. This limit is interesting because it could be more easily realized using various setups proposed to implement the SYK model in a laboratory. We find, based on numerical studies of the quantum mechanical model, that this limiting case is marginal in that it retains some of the characteristic signatures of the wormhole physics, such as the spectral gap and revival dynamics, but lacks others so that it likely does not represent the full-fledged wormhole dual. We discuss how this scenario could be implemented in the proposed realization of the SYK physics in quantum wires of finite length coupled to a disordered quantum dot.

Phase diagram of a two-site coupled complex SYK model

We study the thermodynamic properties of a two-site coupled complex Sachdev-Ye-Kitaev (SYK) model in the large $N$ limit by solving the saddle-point Schwinger-Dyson (SD) equations. We find that its phase diagram is richer than in the Majorana case. In the grand canonical ensemble, we identify a region of small chemical potential, and weak coupling between the two SYKs, for which two first order thermodynamic phase transitions occur as a function of temperature. First, we observe a transition from a cold wormhole phase to an intermediate phase that may correspond to a charged wormhole. For a higher temperature, there is another first order transition to the black hole phase. As in the Majorana case, the low temperature wormhole phase is gapped and, for sufficiently large coupling between the two complex SYK, or chemical potential, the first order transitions become crossovers. The total charge is good indicator to study the phase diagram of the model: it is zero in the cold wormhole phase and jumps discontinuously at the temperatures at which the transitions take place. Based on the approximate conformal symmetry of the ground state, expected to be close to a thermofield double state, we identify the effective low energy action of the model. It is a generalized Schwarzian action with $SL(2,R)\times U(1)$ symmetry with an additional potential and a extra degree of freedom related to the charge. In the large $N$ limit, results from this low energy action are consistent with those from the solution of the SD equations. Our findings are a preliminary step towards the characterization of traversable wormholes by its field theory dual, a strongly interacting fermionic system with charger, that is easier to model experimentally.

An SYK-inspired model with density–density interactions: Spectral & wave function statistics, Green’s function and phase diagram

The Sachdev–Ye–Kitaev (SYK) model is a rare example of a strongly-interacting system that is analytically tractable. Tractability arises because the model is largely structureless by design and therefore artificial: while the interaction is restricted to two-body terms, interaction matrix elements are “randomized” and therefore the corresponding interaction operator does not commute with the local density. Unlike conventional density–density-type interactions, the SYK-interaction is, in this sense, not integrable. We here investigate a variant of the (complex) SYK model, which restores this integrability. It features a randomized single-body term and a density–density-type interaction. We present numerical investigations suggesting that the model exhibits two integrable phases separated by several intermediate phases including a chaotic one. The chaotic phase carries several characteristic SYK-signatures including in the spectral statistics and the frequency scaling of the Green’s function and therefore should be adiabatically connected to the non-Fermi liquid phase of the original SYK model. Thus, our model Hamiltonian provides a bridge from the SYK-model towards microscopic realism.

Quantum thermal transport in the charged Sachdev-Ye-Kitaev model: Thermoelectric Coulomb blockade

We present a microscopic theory for quantum thermoelectric and heat transport in the Schwarzian regime of the Sachdev-Ye-Kitaev (SYK) model. As a charged fermion realization of the SYK model in nanostructures we assume a setup based on a quantum dot connected to the charge reservoirs through weak tunnel barriers. We analyze particle-hole symmetry breaking effects crucial for both Seebeck and Peltier coefficients. We show that the quantum charge and heat transport at low temperatures are defined by the interplay between elastic and inelastic processes such that the inelastic processes provide a leading contribution to the transport coefficients at the temperatures that are smaller compared to the charging energy. We demonstrate that both electric and thermal conductance obey a power law in temperature behavior, while thermoelectric, Seebeck, and Peltier coefficients are exponentially suppressed. This represents selective suppression of only nondiagonal transport coefficients. We discuss the validity of the Kelvin formula in the presence of a strong Coulomb blockade.

Variational wave functions for Sachdev-Ye-Kitaev models

Given a class of $q$-local Hamiltonians, is it possible to find a simple variational state whose energy is a finite fraction of the ground state energy in the thermodynamic limit? Whereas product states often provide an affirmative answer in the case of bosonic (or qubit) models, we show that Gaussian states fail dramatically in the fermionic case, like for the Sachdev-Ye-Kitaev (SYK) models. This prompts us to propose a new class of wavefunctions for SYK models inspired by the variational coupled cluster algorithm. We introduce a static ("0+0D") large-$N$ field theory to study the energy, two-point correlators, and entanglement properties of these states. Most importantly, we demonstrate a finite disorder-averaged approximation ratio of $r \approx 0.62$ between the variational and ground state energy of SYK for $q=4$. Moreover, the variational states provide an exact description of spontaneous symmetry breaking in a related two-flavor SYK model.

Remarks on a melonic field theory with cubic interaction

We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of N scalar fields transforming in an N-dimensional irreducible representation of SO(3). The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner 3jm symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-N limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with n fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For 5.74 < d < 6 the fixed-point theory defines a real CFT, while for smaller d complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large N , and we find a real and unitary CFT for any d < 6, both for real and imaginary coupling constant, up to some critical coupling.

Constructive expansion for vector field theories I. Quartic models in low dimensions

This paper is the first of a series aiming to use the loop vertex expansion (LVE) to recover or prove analyticity and Borel summability for generic vector models with bosonic or fermionic statistics in various dimensions. We consider both non-relativistic and relativistic bosons and fermions coupled with a constant quartic tensor in zero-, one-, and two-dimensional space by limiting our investigations to the super-renormalizable models. This offers a unified perspective on classical constructive results, highlighting the usefulness of the LVE as a modern tool to address these questions and to tackle more challenging models in higher dimensions. Finally, we investigate the large N and massless limits along with quenching for fermions in one dimension. In particular, this work establishes the Borel summability of the Sachdev–Ye–Kitaev model.

Rescuing a black hole in the large-q coupled SYK model

In this paper, we develop a general effective theory for two copies of the Sachdev-Ye-Kitaev (SYK) model with a time-dependent bilinear coupling. For a quantum quench problem with an initial state of the thermofield double state, we show how the evolution of the system is described by a complex reparametrization field with a classical Hamiltonian. We study correlation functions in this system and compare the large-q theory with the bulk low energy effective theory. In particular, we study the special case of a “rescued black hole”, which describes how a time-evolved thermofield double state can evolve to the ground state of a coupled SYK model by a carefully tuned time-dependent coupling. In the low energy region, there is a holographic dual interpretation, which is a geometry that crosses over from an eternal black hole to a global AdS2 vacuum. This family of geometries allow us to access the bulk region that would be the black hole interior without the rescue process. By comparing the large-q and low energy theory, we find that even in the low energy region the deviation from the low energy theory cannot be neglected if the rescue process starts late. This provides evidence that the low energy effective theory of the bulk fails near the inner horizon of the black hole. We note the possibility of a connection to a two-dimensional analog of the higher-dimensional black hole singularity.

Complexity growth of operators in the SYK model and in JT gravity

The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.

More on complex Sachdev-Ye-Kitaev eternal wormholes

In this work, we study a generalization of the coupled Sachdev-Ye-Kitaev (SYK) model with U(1) charge conservations. The model contains two copies of the complex SYK model at different chemical potentials, coupled by a direct hopping term. In the zero-temperature and small coupling limit with small averaged chemical potential, the ground state is an eternal wormhole connecting two sides, with a specific charge Q = 0, which is equivalent to a thermofield double state. We derive the conformal Green’s functions and determine corresponding IR parameters. At higher chemical potential, the system transit into the black hole phase. We further derive the Schwarzian effective action and study its quench dynamics. Finally, we compare numerical results with the analytical predictions.

Optical lattice platform for the Sachdev-Ye-Kitaev model

The tractability of the Sachdev-Ye-Kitaev (SYK) model at large $N$ limit makes it ideal to theoretically study its chaotic non-Fermi liquid behavior and holographic duality properties. We show that the complex SYK Hamiltonian emerges from a system of spinless itinerant fermions in an optical Kagome lattice with a strong disorder. We discuss the regimes supporting flat band spectra in a Kagome lattice, where the system can be non-dispersive. Random interaction between non-dispersive fermions is induced due to randomly distributed immobile impurities in the optical lattice, that exclude the presence of itinerant fermions at their locations. We show that the proposed setup is a reliable experimental platform to realize the SYK model and study its exotic behavior. We show that the velocity distribution of the released fermions is a sensitive probe of the many-body Wigner-Dyson spectral density of states while the averaged many-body Loschmidt echo scheme can measure two-point out-of-time-ordered correlation functions of the SYK system.

Superconductivity from Luttinger surfaces: Emergent Sachdev-Ye-Kitaev physics with infinite-body interactions

The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature ($T=0$). The equivalent question of pairing instability on a Luttinger surface (LS) -- a contour of zeros of the propagator -- instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid (NFL) and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the $q\rightarrow \infty$ SYK model with $q$-body interactions \textcolor{black}{close to the LS}, and (ii) the enforcement of gauge invariance results in a $\frac{1}{\sqrt{\tau}}$ ($\tau\sim T^{-1}$) behavior of the fluctuation propagator near the QCP, as is a feature of the fundamental SYK fermion.