SYK Model Research Articles

Half-wormholes in SYK with one time point

In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a ``wormhole'' saddle plus a ``pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.

Universal ratio in random matrix theory and chaotic-to-integrable transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models

We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both $q=4$ SYK with interaction $J$ and $q=2$ SYK with an interaction $K$ in type-I or ${(q=2)}^{2}$ SYK with an interaction $\sqrt{K}$ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $N$ even and $N$ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $1/N$ expansion in the large $N$ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $N=\ensuremath{\infty}$, $1/N$ expansions at a finite $N$, and the CIT characterized by the RMT at a finite $N$: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.

Free energy landscape and kinetics of phase transition in two coupled SYK models and the corresponding wormhole-two black hole switching

We propose that the thermodynamics and the kinetics of the phase transition between wormhole and two black hole described by the two coupled SYK model can be investigated in terms of the stochastic dynamics on the underlying free energy landscape. We assume that the phase transition is a stochastic process under the thermal fluctuations. By quantifying the underlying free energy landscape, we study the phase diagram, the kinetic time and its fluctuations in details, which reveal the underlying thermodynamics and kinetics. It is shown that the first order phase transition between wormhole and two black hole described by two coupled SYK model is analogous to the Van der Waals phase transition. Therefore, the emergence of wormhole and two black hole phases, the phase transition and associated kinetics can be quantitatively addressed in our free energy landscape and kinetic framework through the dependence on the barrier height and the temperature.

A dynamical version of the SYK model and theq-Brownian motion

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.

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).

Multi-trace correlators in the SYK model and non-geometric wormholes

We consider multi-energy level distributions in the SYK model, and in particular, the role of global fluctuations in the density of states of the SYK model. The connected contributions to the moments of the density of states go to zero as N → ∞, however, they are much larger than the standard RMT correlations. We provide a diagrammatic description of the leading behavior of these connected moments, showing that the dominant diagrams are given by 1PI cactus graphs, and derive a vector model of the couplings which reproduces these results. We generalize these results to the first subleading corrections, and to fluctuations of correlation functions. In either case, the new set of correlations between traces (i.e. between boundaries) are not associated with, and are much larger than, the ones given by topological wormholes. The connected contributions that we discuss are the beginning of an infinite series of terms, associated with more and more information about the ensemble of couplings, which hints towards the dual of a single realization. In particular, we suggest that incorporating them in the gravity description requires the introduction of new, lighter and lighter, fields in the bulk with fluctuating boundary couplings.

A sharp transition in quantum chaos and thermodynamics of mass deformed SYK model

We review our recent work [1] where we studied the chaotic property of the two coupled Sachdev-Ye-Kitaev systems exhibiting a Hawking-Page like phase transition. By computing the out-of-time-ordered correlator in the large NN limit by using the bilocal field formalism, we found that the chaos exponent of this model shows a discontinuous fall-off at the phase transition temperature. Hence in this model the Hawking-Page like transition is correlated with a transition in chaoticity, as expected from the relation between a black hole geometry and the chaotic behavior in the dual field theory.

Quantum Ergodicity in the Many-Body Localization Problem.

We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long-range interactions. This extension leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "nonergodic extended states" in many-body systems discussed in the recent literature.

A traversable wormhole teleportation protocol in the SYK model

In this paper, we propose a concrete teleportation protocol in the SYK model based on a particle traversing a wormhole. The required operations for the communication, and insertion and extraction of the qubit, are all simple operators in terms of the basic qubits. We determine the effectiveness of this protocol, and find a version achieves almost perfect fidelity. Many features of semiclassical traversable wormholes are manifested in this setup.

On 1D, mathcal{N} = 4 supersymmetric SYK-type models. Part I

Proposals are made to describe 1D, mathcal{N} = 4 supersymmetrical systems that ex- tend SYK models by compactifying from 4D, mathcal{N} = 1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via in- tegrals over the whole superspace, while 2(q − 1)-point fermionic vertices are generated via superpotentials. The coupling constants in the superfield Lagrangians are arbitrary, and can be chosen to be Gaussian random. In that case, these 1D, mathcal{N} = 4 supersymmetric SYK models would exhibit Wishart-Laguerre randomness, which share the same feature among other 1D supersymmetric SYK models in literature. One difference with 1D, mathcal{N} = 1 and mathcal{N} = 2 models though, is our models contain dynamical bosons, but this is consistent with other 1D, mathcal{N} = 4 and 2D, mathcal{N} = 2 models in literature. Added conjectures on duality and possible mirror symmetry realizations in these models is noted.

Time dependent entanglement entropy in SYK models and page curve

In this work new interaction terms between two SYK models are proposed, which allow to define an interaction picture such that it is possible to calculate the vacuum time dependent entanglement entropy at the conformal and large N limit. It is shown that the vacuum evolves as a time dependent SU(2) squeezed state and the time dependent entanglement entropy has the same form of the Page curve of Black Hole formation and evaporation.

On systems of maximal quantum chaos

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the ‘hydrodynamic’ origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.

Exact four point function for large q SYK from Regge theory

Motivated by the goal of understanding quantum systems away from maximal chaos, in this note we derive a simple closed form expression for the fermion four point function of the large q SYK model valid at arbitrary temperatures and to leading order in 1/N. The result captures both the large temperature, weakly coupled regime, and the low temperature, nearly conformal, maximally chaotic regime of the model. The derivation proceeds by the Sommerfeld-Watson resummation of an infinite series that recasts the four point function as a sum of three Regge poles. The location of these poles determines the Lyapunov exponent that interpolates between zero and the maximal value as the temperature is decreased. Our results are in complete agreement with the ones by Streicher [1] obtained using a different method.

AdS3 gravity and the complex SYK models

We provide a non-conformal generalization of the Compère-Song-Strominger (CSS) boundary conditions for AdS3 gravity that breaks the hat{u}(1) Kac-Moody-Virasoro symmetry to two u(1)s. The holographic dual specified by the new boundary conditions can be understood as an irrelevant deformation of a warped conformal field theory (WCFT). Upon consistent reduction to two dimensions, AdS3 gravity results in a deformed Jackiw-Teitelboim dilaton gravity model coupled to a Maxwell field. We show that near extremality the boundary conditions inherited from generalized CSS boundary conditions in three dimensions give rise to an effective action exhibiting the same symmetry breaking pattern as the complex Sachdev-Ye-Kitaev models. Besides the Schwarzian term reflecting the breaking of conformal symmetry, the effective action contains an additional term that captures the breaking of the hat{u}(1) Kac-Moody symmetry.

SYK wormhole formation in real time

We study the real time formation of the ground state of two coupled SYK models. This is a highly entangled state which is close to the thermofield double state and can be viewed as a wormhole. We start from a high temperature state, we let it cool by coupling to a cold bath. We numerically solve for the large N dynamics. Our main result is that the system forms a wormhole by going through a region with negative specific heat, taking time that is independent of N. The dynamics is smooth everywhere and it seems to follow the equilibrium thermodynamic configurations of the microcanonical ensemble. Also we comment on the relation between this coupled SYK model and Jackiw-Teitelboim gravity theory with bulk fields.

Thermalization in different phases of charged SYK model

We study thermalization of charged SYK model in two different phases. We show that both the highly chaotic liquid phase and the dilute gas phase thermalize. Surprisingly the dilute gas state thermalizes instantaneously. We argue that this phenomenon arises because the system in this phase consists of only long-lived quasi-particles at very low density. The liquid state thermalizes exponentially fast. We also show that the additional introduction of random mass deformation (q = 2 SYK term) slows down thermalization but the system thermalizes exponentially fast. This is observed despite the fact that the addition of large q = 2 SYK interaction forces spectral statistics to obey Poisson statistics. An interesting new observation is that the effective temperature is non-monotonic during thermalization in the liquid state. It has a bump at relatively long time before settling down to the final value. With non-zero chemical potential, the effective temperature oscillates noticeably before settling down to the final value.

Entanglement Dynamics of Random GUE Hamiltonians

In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential n-point correlation functions in the gas of GUE eigenvalues.

García-García etal. Reply.

In a recent comment to the paper Chaotic Integrable transition in the SYK model, it was claimed that, in a certain region of parameters, the Lyapunov exponent of the N Majoranas Sachdev-Ye-Kitaev model with a quadratic perturbation, is always positive. This implies that the model is quantum chaotic. In this reply, we show that the employed perturbative formalism breaks down precisely in the range of parameters investigated in the comment due to a lack of separation of time scales. Moreover, based on recent analytical results, we show that for any large and fixed N, the model has indeed a chaotic-integrable transition that invalidate the results of the comment.

Complexity and momentum

Previous work has explored the connections between three concepts — operator size, complexity, and the bulk radial momentum of an infalling object — in the context of JT gravity and the SYK model. In this paper we investigate the higher dimensional generalizations of these connections. We use a toy model to study the growth of an operator when perturbing the vacuum of a CFT. From circuit analysis we relate the operator growth to the rate of increase of complexity and check it by complexity-volume duality. We further give an empirical formula relating complexity and the bulk radial momentum that works from the time that the perturbation just comes in from the cutoff boundary, to after the scrambling time.

Spectral form factor in the double-scaled SYK model

In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.