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 AdS$_2$ 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.
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 t…
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 w…
Motivated by recent studies of the information paradox in (1+1)-D anti-de Sitter spacetime with a bath described by a (1+1)-D conformal field theory, we study the dynamics of second Rényi entropy of the Sachdev-Ye-Kitaev (SYK) model ($χ$) coupled to a Majorana chain bath ($ψ$). The system is prepared in the thermofield double (TFD) state and then evolved by $H_L+H_R$. For small system-bath coupling, we find that the second Rényi entropy $S^{(2)}_{χ_L, χ_R}$ of the SYK model undergoes a first order transition during the evolution. In the sense of holographic duality, the long-time solution corr…
We study the dynamical decoherence of a qubit weakly coupled to a two-body random interaction model (TBRIM) describing a quantum dot of interacting fermions or the Sachdev-Ye-Kitaev (SYK) black hole model. We determine the rates of qubit relaxation and dephasing for regimes of dynamical thermalization of the quantum dot or of quantum chaos in the SYK model. These rates are found to correspond to the Fermi golden rule and quantum Zeno regimes depending on the qubit-fermion coupling strength. An unusual regime is found where these rates are practically independent of TBRIM parameters. We push fo…
As shown in [1], two copies of the large $N$ Majorana SYK model can produce spontaneous breaking of a $Z_2$ symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the $U(1) \times U(1)$ symmetry. We also present a tensor counterpart of this coupled model. When the coefficient $α$ of the quartic term lies in a certain range, the coupled large $N$ theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of $α$. We show that the operator…
Sachdev-Ye-Kitaev (SYK) model, which describes $N$ randomly interacting Majorana fermions in 0+1 dimension, is found to be an solvable UV-complete toy model for holographic duality in nearly AdS$_2$ dilaton gravity. Ref. [1] proposed a modified model by coupling two identical SYK models, which at low-energy limit is dual to a global AdS$_2$ geometry. This geometry is an "eternal wormhole" because the two boundaries are causally connected. Increasing the temperature drives a Hawking-Page like transition from the eternal wormhole geometry to two disconnected black holes with coupled matter field…
The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with $N$ Majorana fermion modes coupled to another SYK model with $N^2$ Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to tempe…
We study four coupled SYK models and nearly AdS$_2$ gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS$_2$ gravity side, we entangle matter fields in two copies of traversable wormholes. In both cases, the systems show first order phase transitions at zero temperature by changing couplings, which is understood as the exchange of traversable wormhole configurations. In nearly AdS$_2$ gravity cases, by exchanging the role of space and time the wormholes are interpreted as bra-ket wormholes. In Lorentzian signature, these bra-ke…
In this paper, we study the entanglement entropy between two SYK systems with bilinear coupling. We use the replica trick to calculate the entanglement entropy in the ground state. In parallel, we calculate the entanglement entropy through the Ryu-Takayanagi formula in gravity. For the ground state that is dual to an eternal traversable wormhole in AdS_2, the bulk quantum correction to the entanglement entropy is of the same order as the minimal surface area. The ground state of the coupled system is close to a thermofield double state with particular temperature and they have the same entangl…
We study a large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions, $ψ_1^{abc}$ and $ψ_2^{abc}$. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev models, each one containing $N_{\rm SYK}$ Majorana fermions. In these models we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter $α$. We find a duality relation between two Hamiltonians with different values of $α$, which allows us to restrict the model to the range of $-1\leq α\leq 1/3$. The scaling dimension of the fermion number operator $Q=iψ_1^{abc…
We investigate two sparse Sachdev-Ye-Kitaev (SYK) systems coupled by a bilinear term as a holographic quantum mechanical description of an eternal traversable wormhole in the low temperature limit. Each SYK system consists of $N$ Majorana fermions coupled by random $q$-body interactions. The degree of sparseness is captured by a regular hypergraph in such a way that the Hamiltonian contains exactly $k\,N$ independent terms. We improve on the theoretical understanding of the sparseness property by using known measures of hypergraph expansion. We show that the sparse version of the two coupled S…
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 $μ$ 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) st…
Recent work has shown that coupling two identical Sachdev-Ye-Kitaev (SYK) models can realize a phase of matter that is holographically dual to an eternal traversable wormhole. This phase supports revival oscillations between two quantum chaotic systems that can be interpreted as information traversing the wormhole. Here we generalize these ideas to a pair of coupled SYK models with complex fermions that respect a global U(1) charge symmetry. Such models show richer behavior than conventional SYK models with Majorana fermions and may be easier to realize experimentally. We consider two differen…
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…
Josephson wormhole in coupled superconducting Yukawa-SYK metals
Leiden University（莱顿大学）；The Abdus Salam International Center for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）；Silicon Quantum Computing Pty Ltd（硅量子计算有限公司）；UNSW Sydney（新南威尔士大学悉尼分校）
We continue the investigation of coupled Sachdev-Ye-Kitaev(SYK) models without Schwarzian action dominance. Like the original SYK, at large N and low energies these models have an approximate reparametrization symmetry. However, the dominant action for reparametrizations is non-local due to the presence of irrelevant local operator with small conformal dimension. We semi-analytically study different thermodynamic properties and the 4-point function and demonstrate that they significantly differ from the Schwarzian prediction. However, the residual entropy and maximal chaos exponent are the sam…
We show that a non-Hermitian two coupled Sachdev-Ye-Kitaev (SYK) model can provide thermodynamic structure equivalent to Hermitian two coupled SYK model. The energy spectrum, the entanglement degree of the ground states and the low energy effective action of this model are not influenced by the non-Hermiticity. The novel biorthogonal ground states demonstrates that two SYK sites, one of which can be in the ground state and the other in the Schwarzian excited state by tuning the non-Hermiticity. We find evidence that the free energy is independent of the non-Hermiticity.
We argue that in certain class of coupled Sachdev-Ye-Kitaev(SYK) models the low energy physics at large N is governed by a non-local action rather than the Schwartzian action. We present a partial analytic and extensive numerical evidence for this. We find that these models are maximally chaotic and have the same residual entropy as Majorana SYK. However, thermodynamic quantities, such as heat capacity and diffusion constant are different.
We study the dynamics of a quantum system in thermal equilibrium that is suddenly coupled to a bath at a different temperature, a situation inspired by a particular black hole evaporation protocol. We prove a universal positivity bound on the integrated rate of change of the system energy which holds perturbatively in the system-bath coupling. Applied to holographic systems, this bound implies a particular instance of the averaged null energy condition. We also study in detail the particular case of two coupled SYK models in the limit of many fermions using the Schwinger-Keldysh non-equilibriu…
We show that the spectrum of the SYK model can be interpreted as that of a 3D scalar coupled to gravity. The scalar has a mass which is at the Breitenholer-Freedman bound of AdS$_2$, and subject to a delta function potential at the center of the interval along the third direction. This, through Kaluza-Klein procedure on AdS$_2 \times (S^1)/Z_2$, generates the spectrum reproducing the bi-local propagator at strong coupling. Furthermore, the leading $1/J$ correction calculated in this picture reproduces the known correction to the poles of the SYK propagator, providing credence to a conjecture t…
We study the large $N$ saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We start from analytic treatment of the free case with $q=2$ and perform the numerical study of the interacting case $q=4$. We show that in both cases there is a nontrivial phase structure with infinite number of phases. Every phase correspond to a saddle point in the non-interacting two-replica SYK. The nontrivial saddle points have non-zero value of the replica-nondiagonal correlator in the sense of quasi-averaging, when the coupling between replicas is turned off. Thus, the n…
We study some general properties of coupled quantum systems. We consider simple interactions between two copies of identical Hamiltonians such as the SYK model, Pauli spin chains with random magnetic field and harmonic oscillators. Such couplings make the ground states close to the thermofield double states of the uncoupled Hamiltonians. For the coupled SYK model, we push the numerical computation further towards the thermodynamic limit so that an extrapolation in the size of the system is possible. We find good agreement between the extrapolated numerical result and the analytic result in the…
In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound $2π/β$ at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. I…
We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a coupling $λ$ in which all observables are governed by a non-trivial saddle point. The Liouville description of double-scaled SYK suggests that the correlation functions define a geometry in a two-dimensional bulk, with the 2-point function describing the metric. For small coupling, the fluctuations are highly suppressed, and the bulk describes a rigid (A)dS spacetime. As the coupling increases, the fluctuations become stronger. We study the correction to the curvature of the bulk geometry indu…
The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a Fermi liquid non-chaotic phase sits at the critical point with equal m…
Leinweber Institute for Theoretical Physics, University of California, Berkeley（莱因韦伯理论物理研究所，加州大学伯克利分校）；Martin Fisher School of Physics, Brandeis University（马丁·费舍尔物理学院，布兰迪斯大学）；Department of Physics and Astronomy, University of British Columbia（不列颠哥伦比亚大学物理与天文学系）
We study the time evolution of Rényi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Rényi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary p…
We study a model of $N$ fermions in a quantum dot, coupled to $M$ bosons by a disorder-induced complex Yukawa coupling (Yukawa-SYK model), in order to explore the interplay between non-Fermi liquid and superconductivity in a strongly coupled, (quantum-)critical environment. We analyze the phase diagram of the model for an arbitrary complex interaction and arbitrary ratio of $N/M$, with special focus on the two regimes of non-Fermi-liquid behavior: an SYK-like behavior with a power-law frequency dependence of the fermionic self-energy and an impurity-like behavior with frequency independent sel…
We construct a nearly-$AdS_2$ solution describing an eternal traversable wormhole. The solution contains negative null energy generated by quantum fields under the influence of an external coupling between the two boundaries. In parallel, we discuss two SYK systems coupled by a relevant interaction. The physics of the two cases is very similar. They both share a "gravitational" subsector which is identical. The solution within this subsector sets the stage for dynamics which is almost conformal invariant. We study this system in detail, both in gravity and in the SYK model. The coupled SYK mod…
We introduce a family of tensor quantum-mechanical models based on irreducible rank-$3$ representations of $\mathrm{Sp}(N)$. In contrast to irreducible tensor models with $\mathrm{O}(N)$ symmetry, the fermionic tetrahedral interaction does not vanish and can therefore support a melonic large $N$ limit. The strongly-coupled regime has a very analogous structure as in the complex SYK model or in $\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ tensor quantum mechanics, the main difference being that the states are now singlets under $\mathrm{Sp}(N)$. We introduce character formulas that enu…
It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two dimensional Anti de Sitter space (${\rm AdS}_2$) is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). In this paper, we study spectral and eigenstate properties of this coupled SYK model. We have found that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, shows substantial deviations from random matrix theory which suggests that traversable wormholes are n…
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 …
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…
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 hi…
The shear viscosity is an important characterization of how a many-body system behaves like a fluid. We study the shear viscosity in a strongly interacting solvable model, consisting of coupled Sachdev-Ye-Kitaev (SYK) islands. As temperature is lowered, the model exhibits a crossover from an incoherent metal with local criticality to a marginal fermi liquid. We find that while the ratio of shear viscosity to entropy density in the marginal Fermi liquid regime satisfies a Kovtun-Son-Starinets (KSS) like bound, it can strongly violate the KSS bound in a robust temperature range of the incoherent…
We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exp…
We construct three models to describe the scenario where two eternal black holes are separated by a flat space, and can eventually be entangled by exchanging radiations. In the doubly holographic setup, we compute the entanglement entropy and the mutual information among the subsystems and obtain the dynamic phase structure of the entanglement. The formation of entanglement between the two black holes is delayed by the space where the radiations must travel through. Finally, if the two black holes exchange sufficient Hawking modes, the final state is characterized by a connected entanglement w…
Recently it has been shown by Almheiri and Lin [1] that the reconstruction of black-hole interior is sensitive to knowing the exact coupling of the boundary theory even if the coupling is irrelevant. This motivates us to enlarge the set of the one-time and two-time toy models inspired from the SYK by deforming the same with \textit{irrelevant} coupling. We find that both half-wormholes as well as the wormholes persist in presence of the deformation, leading to a similar mechanism for curing the factorization problem. While for the one time case, the deformed partition function and its moments …
We compute the energy diffusion constant $D$, Lyapunov time $τ_{\text{L}}$ and butterfly velocity $v_{\text{B}}$ in an inhomogeneous chain of coupled Majorana Sachdev-Ye-Kitaev (SYK) models in the large $N$ and strong coupling limit. We find $D\le v_{\text{B}}^2 τ_{\text{L}}$ from a combination of analytical and numerical approaches. Our example necessitates the sharpening of postulated transport bounds based on quantum chaos.
We consider a black hole with a stretched horizon as a toy model for a fuzzball microstate. The stretched horizon provides a cut-off, and therefore one can determine the normal (as opposed to quasi-normal) modes of a probe scalar in this geometry. For the BTZ black hole, we compute these as a function of the level $n$ and the angular quantum number $J$. Conventional level repulsion is absent in this system, and yet we find that the Spectral Form Factor (SFF) shows clear evidence for a dip-ramp-plateau structure with a linear ramp of slope $\sim 1$ on a log-log plot, with or without ensemble av…
