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 corresponds to a "replica wormhole". The transition time corresponds to the Page time of a black hole coupled to a thermal bath. We further study the information scrambling and retrieval by introducing a classical control bit, which controls whether or not we add a perturbation in the SYK system. The mutual information between the bath and the control bit shows a positive jump at the Page time, indicating that the entanglement wedge of the bath includes an island in the holographic bulk.
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 …
We study "half-wormhole-like" saddle point contributions to spectral correlators in a variety of ensemble average models, including various statistical models, generalized 0d SYK models, 1d Brownian SYK models and an extension of it. In statistical ensemble models, where more general distributions of the random variables could be studied in great details, we find the accuracy of the previously proposed approximation for the half-wormholes could be improved when the distribution of the random variables deviate significantly from Gaussian distributions. We propose a modified approximation scheme…
We consider the factorization problem in toy models of holography, in SYK and in Matrix Models. In a theory with fixed couplings, we introduce a fictitious ensemble averaging by inserting a projector onto fixed couplings. We compute the squared partition function and find that at large $N$ for a typical choice of the fixed couplings it can be approximated by two terms: a "wormhole" plus a "pair of linked half-wormholes". This resolves the factorization problem. We find that the second, half-wormhole, term can be thought of as averaging over the imaginary part of the couplings. In SYK, this rep…
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…
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.
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…
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 introduce a "partially disorder-averaged" SYK model. This model has a real parameter that smoothly interpolates between the ordinary totally disorder-averaged SYK model and the totally fixed-coupling model. For the large $N$ effective description, in addition to the usual bi-local collective fields, we also introduce a new additional set of local collective fields. These local fields can be understood as "half" of the bi-local collective fields, and in the totally fixed-coupling limit, they represent the "half-wormholes" which were found in recent studies. We find that the la…
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…
We study a two-site Sachdev-Ye-Kitaev (SYK) model with complex couplings, and identify a low temperature transition to a gapped phase characterized by a constant in temperature free energy. This transition is observed without introducing a coupling between the two sites, and only appears after ensemble average over the complex couplings. We propose a gravity interpretation of these results by constructing an explicit solution of Jackiw-Teitelboim (JT) gravity with matter: a two-dimensional Euclidean wormhole whose geometry is the double trumpet. This solution is sustained by imaginary sources …
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…
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）
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…
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…
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 …
Replica geometries are not rigid when gravity is dynamical. We numerically construct $1<n\leq 2$ replica saddles in $2d$ gravity coupled to a CFT and compare the resulting Renyi entropies with the field theory result.
The information paradox can be realized in anti-de Sitter spacetime joined to a Minkowski region. In this setting, we show that the large discrepancy between the von Neumann entropy as calculated by Hawking and the requirements of unitarity is fixed by including new saddles in the gravitational path integral. These saddles arise in the replica method as complexified wormholes connecting different copies of the black hole. As the replica number $n \to 1$, the presence of these wormholes leads to the island rule for the computation of the fine-grained gravitational entropy. We discuss these repl…
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…
The replica wormholes are a key to the existence of the islands that play a central role in a recent proposal for the resolution of the black hole information paradox. In this paper, we study the replica wormholes in the JT gravity, a model of two-dimensional quantum gravity coupled to a non-dynamical dilaton, by making use of the 2$d$ conformal field theory (CFT) description, namely, the Liouville theory coupled to the $(2,p)$ minimal matter in the $p\to\infty$ limit. In the Liouville CFT description, the replica wormholes are created by the twist operators and the gravitational part of the b…
In the Sachdev-Ye-Kitaev model with generic order $q \ge 4$ random couplings, we compute the critical temperature relating the Majorana fermions high temperature perturbative vacuum to the vacuum where the replica diagonal collective field $G(τ, τ')$ condenses. We study, by a finite temperature diagrammatic analysis, the effective action of an auxiliary Hubbard-Stratonovich bilocal field related to $G(τ, τ')$ in the large $N$ limit. Subtelties that arise in switching from the operatorial to the functional integral representation of the SYK thermal partition function are also discussed.
Quantum extremal islands reproduce the unitary Page curve of an evaporating black hole. This has been derived by including replica wormholes in the gravitational path integral, but for the transient, evaporating black holes most relevant to Hawking's paradox, these wormholes have not been analyzed in any detail. In this paper we study replica wormholes for black holes formed by gravitational collapse in Jackiw-Teitelboim gravity, and confirm that they lead to the island rule for the entropy. The main technical challenge is that replica wormholes rely on a Euclidean path integral, while the qua…
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 $N \to \infty$, 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 sublea…
We present an explicit reconstruction of the interior of an AdS$_2$ black hole in Jackiw-Teitelboim gravity, that is entirely formulated in the dual SYK model and makes no direct reference to the gravitational bulk. We do this by introducing a probe "observer" in the right wormhole exterior and using the prescription of [arXiv:2009.04476] to transport SYK operators along the probe's infalling worldline and into the black hole interior, using an appropriate SYK modular Hamiltonian. Our SYK computation recovers the precise proper time at which signals sent from the left boundary are registered b…
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 discuss generalizations of the TFD to a density matrix on the doubled Hilbert space. We suggest that a semiclassical wormhole corresponds to a certain class of such density matrices, and specify how they are constructed. Different semi-classical profiles correspond to different non-overlapping density matrices. We show that this language allows for a finer criteria for when the wormhole is semiclassical, which goes beyond entanglement. Our main tool is the SYK model. We focus on the simplest class of such density matrices, in a scaling limit where the ER bridge is captured by chords going f…
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 …
We argue that the `island conjecture' and the replica wormhole derivation of the Page curve break monogamy of entanglement through allowing black hole interior states to be non-classically correlated while also pairwise entangled with radiation states. The reason is that quantum degrees of freedom (present in any half of a Hawking pair) cannot all be identified with the environment at semi-classical pair production, and can only be fixed relative to a subsystem, as required for the Page curve, by correlations equivalent to entanglement - regardless of what those correlations are attributed to.…
A recent article [Nature 612, 51-55 (2022)] claims to observe traversable wormhole dynamics in an experiment. This claim is based upon performing a teleportation protocol using a Hamiltonian that consists of seven Majorana fermions with five fully-commuting terms. The Hamiltonian is generated via a machine-learning procedure designed to replicate the teleportation behavior of the Sachdev-Ye-Kitaev (SYK) model. The authors claim that the learned Hamiltonian reproduces gravitational dynamics of the SYK model and demonstrates gravitational teleportation through an emergent wormhole. We find: (i) …
We study a series of powerful correspondences among new multi-gravity extensions of the Jackiw-Teitelboim model, multi-SYK models and multi-Schwarzian quantum mechanics, in the $\rm{(A)dS_{2}/CFT}$ arena. Deploying a $BF$-like formulation of the model, we discuss the counting of the degrees of freedom for some specific classes of multi-gravity potentials, and unveil connections among a variety of apparently different models. Quantization of multi-gravity models can be then achieved from both the Hartle-Hawking no-boundary proposal, the SYK partition function and the spin-foam approaches. We co…
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 the SYK model in the large $N$ limit beyond the replica-diagonal approximation. First we show that there are exact replica-nondiagonal solutions of the saddle point equations for $q=2$ for any finite replica number $M$. In the interacting $q=4$ case we are able to construct the numerical solutions, which are in one-to-one correspondence to the analytic solutions of the quadratic model. These solutions are singular in the $M \to 0$ limit in both quadratic and quartic interaction cases. The calculations of the on-shell action at finite integer $M$ show that the nondiagonal replica-symme…
Through tracing back to EA/SK models, we disentangle the construction logic of SYK model. During the construction, we recall the technical essence of replica method. Then we point out the redundance of the flavor group and the slave indices as well as the color group via the generalization from 2-index SY model to its 4-index version and the simplification of the latter in terms of Majorana fermions. Spectacularly, such a simplification reserves the system's self-consistency condition. Getting rid of the redundance, the model itself and its effective action have the same $O(N_s)$ structural sy…
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…
We study the replica symmetry breaking (RSB) in the Gaussian Unitary Ensemble (GUE) random matrix model. We find that the RSB occurs at the transition temperature $T_{\text{RSB}}\sim N^{-2/3}$ in the large $N$ limit. We argue that this transition originates from a landscape of almost degenerate local minima of free energy labeled by Young diagrams. We also discuss a possible implication of our findings for the multiply-coupled Sachdev-Ye-Kitaev models and their holographic dual Jackiw-Teitelboim gravity from the viewpoint of ER=EPR conjecture.
It is widely believed that exact global symmetries do not exist in theories that admit quantum black holes. Here we propose a way to quantify the degree of global symmetry violation in the Hawking radiation of a black hole by using certain relative entropies. While the violations of global symmetry that we consider are non-perturbative effects, they nevertheless give $\mathcal{O}(1)$ contributions to the relative entropy after the Page time. Furthermore, using "island" formulas, these relative entropies can be computed within semi-classical gravity, which we demonstrate with explicit examples.…
We find a late times approximation for the SYK spectral form factor from a large $N$ steepest descent version of the path integral over two replica collective fields. Main ingredients are a suitable uv regularization of the two replica kinetic operator, the property of its Fourier transform and some spectral analysis of the four point function two replica ladder kernel.
We define bit models of evaporating black holes which incorporate the effects of replica wormholes. These enter as non-perturbative corrections to the gravitational inner product, arising from spacetime wormholes that connect replica black holes. The resulting models have exactly unitary evolution, predict measurements on Hawking radiation in accord with a Page curve for entropy, and are compatible with a smooth horizon for infalling observers.
In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Renyi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half …
We study the behavior of Rényi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system. We compute the exact long-time averages of Rényi entropies and show that the quantum noise around these values is exponentially suppressed in the microcanonical entropy. For delocalized states over the microcanonical band, the long-time average approximately reproduces the equilibration proposal of H. Liu and S. Vardhan, with extra structure arising at the order of non-planar permutations. We analyze the equilibrium approxi…
We show that the low temperature phase of a conjugate pair of uncoupled, quantum chaotic, nonhermitian systems such as the Sachdev-Ye-Kitaev (SYK) model or the Ginibre ensemble of random matrices are dominated by replica symmetry breaking configurations with a nearly flat free energy that terminates in a first order phase transition. In the case of the SYK model, we show explicitly that the spectrum of the effective replica theory has a gap. These features are strikingly similar to those induced by wormholes in the gravity path integral which suggests a close relation between both configuratio…
In a close look at the replica wormhole derivation of the island conjecture, we note a discrepancy in the derived matrix eigenvalues in the limit where the Hawking radiation entropy goes to zero. The small corrections to the density matrix for the radiation infer that ${\rm tr}(ρ_\textsf{R}^2)$, equally a sum over the eigenvalues squared, must be small ($\ll1$) at all times. This is in contradiction with an ensemble dominated by a density matrix with an eigenvalue close to unity. Also, the radiation entropy has been approximated by the ensemble averaged entropy, as in a final state analysis. F…
We study the conventional holographic recipes and its real-time extensions in the context of the correspondence between SYK quantum mechanics and JT gravity. We first observe that only closed contours are allowed to have a 2d space-time holographic dual. Thus, in any real-time formulation of the duality, the boundaries of a classical connected geometry are a set of closed curves, parameterized by a complex \emph{closed} time contour as in the Schwinger-Keldysh framework. Thereby, a consistent extension of the standard holographic formulas is proposed, in order to describe the correspondence be…
