We generalize the eigenstate thermalization hypothesis to systems with global symmetries. We present two versions, one with microscopic charge conservation and one with exponentially suppressed violations. They agree for correlation functions of simple operators, but differ in the variance of charged one-point functions at finite temperature. We then apply these ideas to holography and to gravitational low-energy effective theories with a global symmetry. We show that Euclidean wormholes predict a non-zero variance for charged one-point functions, which is incompatible with microscopic charge conservation. This implies that global symmetries in quantum gravity must either be gauged or explicitly broken by non-perturbative effects.
The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS$_2$, the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approx…
Eigenstate thermalization hypothesis (ETH) is discussed. We show that one common formulation of ETH does not necessarily imply thermalization of an observable of isolated many body quantum system. To get thermalization one has to postulate the canonical or microcanonical distribution in the ETH-ansatz. More generally, any other average can be postulated in the generalized ETH-ansatz which leads to a corresponding equilibration condition.
We investigate the eigenstate thermalization hypothesis (ETH) in d+1 dimensional conformal field theories by studying reduced density matrices in energy eigenstates. We show that if local probes of high energy primary eigenstates satisfy ETH, then any finite energy observable with support on a subsystem of finite size satisfies ETH. In two dimensions, we discover that if ETH holds locally, the finite size reduced density matrix of states created by heavy primary operators is well-approximated by a projection to the Virasoro identity block.
In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypot…
We discuss eigenstate correlations for ergodic, spatially extended many-body quantum systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the butterfly effect implies structure beyond ETH. We determine the universal form of this structure at long distances and small eigenvalue separations for Floquet systems. We use numerical studies of a Floquet quantum circuit to illustrate both the accuracy of ETH and the existence of our predicted additiona…
We study the characteristics of thermalizing and non-thermalizing operators in integrable theories as we turn on a non-integrable deformation. Specifically, we show that $σ^z$, an operator that thermalizes in the integrable transverse field Ising model, has mean matrix elements that resemble ETH, but with fluctuations around the mean that are sharply suppressed. This suppression rapidly dwindles as the Ising model becomes non-integrable by the turning on of a longitudinal field. We also construct a non-thermalizing operator in the integrable regime, which slowly approaches the ETH form as the …
We show that macroscopic thermalization and transport impose constraints on matrix elements entering the Eigenstate Thermalization Hypothesis (ETH) ansatz and require them to be correlated. It is often assumed that the ETH reduces to Random Matrix Theory (RMT) below the Thouless energy scale. We show this conventional picture is not self-consistent. We prove that energy scale at which the RMT behavior emerges has to be parametrically smaller than the inverse timescale of the slowest thermalization mode coupled to the operator of interest. We argue that the timescale marking the onset of the RM…
We show that the known bound on the growth rate of the out-of-time-order four-point correlator in chaotic many-body quantum systems follows directly from the general structure of operator matrix elements in systems that obey the eigenstate thermalization hypothesis. This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.
The Eigenstate Thermalization Hypothesis makes a prediction for the statistical distribution of matrix elements of simple operators in energy eigenstates of chaotic quantum systems. As a leading approximation, off-diagonal matrix elements are described by Gaussian random variables but higher-point correlation functions enforce non-Gaussian corrections which are further exponentially suppressed in the entropy. In this paper, we investigate non-Gaussian corrections to the statistical distribution of heavy-heavy-heavy OPE coefficients in chaotic two-dimensional conformal field theories. Using the…
In this work, we use quantum complexity theory to quantify the difficulty of distinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis (ETH). After identifying simple operators with an algebra of low-energy observables and tracing out the complementary high-energy Hilbert space, the ETH leads to an exponential suppression of trace distance between the coarse-grained eigenstates. Conversely, we show that an exponential hardness of distinguishing between states implies ETH-like matrix elements. The BBBV lower bound on the query complexity of Grover search then translates direct…
The Eigenstate Thermalization Hypothesis (ETH) explains emergence of the thermodynamic equilibrium by assuming a particular structure of observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by Random Matrix Theory (RMT). To what extent physical operators can be described by RMT, more precisely at which energy scale strict RMT description applies, is however not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matri…
Naively, a resolution of the black hole information paradox appears to involve microscopic details of a theory of quantum gravity. However, recent work has argued that a unitary Page curve can be recovered by including novel replica instantons in the gravitational path integral. Moreover, replica instantons seem to rely on disorder averaging the microscopic theory, without a definite connection to a single, underlying unitary quantum system. In this letter, we show that disorder averaging and replica instantons emerge naturally from a gravitational effective theory built out of typical microsc…
Having analytical instances of the Eigenstate Thermalization Hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generically a hard task, due to the belief that non-linear interactions are basic ingredients of the thermalization mechanism. In this article we proof that random gaussian free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian sys…
The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out-of-time-order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length, relevant for systems with slow dynamics. These correlations, as well as those between el…
Generic ETH: Eigenstate Thermalization beyond the Microcanonical
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Syracuse University（雪城大学）；Sarah Lawrence College（莎拉劳伦斯学院）
Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2d conformal field theories through an infinite tower of local conserved qKdV charges in involution. We discuss the role this integrable structure plays in equilibration of 2d CFTs. We show that in the thermodynamic limit large central charge 2d CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Our work settles the qu…
Quantum thermalization is well understood via the Eigenstate Thermalization Hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a `typicality' argument of invariance with respect to local rotations involving nearby energy levels. In this work, we uncover the close relation between this perspective on ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition g…
The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi-)energy eigenbasis. Here we study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits in dependence of the frequency associated with the corresponding eigenstates. We provide an exact asymptotic expression for the spectral function, i.e., the second moment of this frequency resolved distribution. The latter is obtained from the…
We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the `Eigenstate Thermalization Hypothesis' (ETH), and the resulting `single-eigenstate statistical mechanics'. We then focus on a class of systems which fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can locally remember for…
We study aspects of chaos and thermodynamics at strong coupling in a scalar model using LCT numerical methods. We find that our eigenstate spectrum satisfies Wigner-Dyson statistics and that the coefficients describing eigenstates in our basis satisfy Random Matrix Theory (RMT) statistics. At weak coupling, though the bulk of states satisfy RMT statistics, we find several scar states as well. We then use these chaotic states to compute the equation of state of the model, obtaining results consistent with Conformal Field Theory (CFT) expectations at temperatures above the scale of relevant inte…
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well a…
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with …
Bypassing eigenstate thermalization with experimentally accessible quantum dynamics
JILA and Center for Theory of Quantum Matter, Department of Physics, University of Colorado, Boulder（科罗拉多大学博尔德分校物理系量子物质理论中心及JILA）；University of Colorado, Boulder（科罗拉多大学博尔德分校）
We use field-theoretic methods to explore the statistics of eigenfunctions of the Floquet operator for a large family of Floquet random quantum circuits. The correlation function of the quasienergy eigenstates is calculated and shown to exhibit random matrix circular unitary ensemble statistics, which is consistent with the analogue of Berry's conjecture for quantum circuits. This quantity determines all key metrics of quantum chaos, such as the spectral form factor and thermalizing time-dependence of the expectation value of an arbitrary observable. It also allows us to explicitly show that t…
The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there ex…
We study the quantum Fisher information (QFI) and, thus, the multipartite entanglement structure of thermal pure states in the context of the Eigenstate Thermalization Hypothesis (ETH). In both the canonical ensemble and the ETH, the quantum Fisher information may be explicitly calculated from the response functions. In the case of ETH, we find that the expression of the QFI bounds the corresponding canonical expression from above. This implies that although average values and fluctuations of local observables are indistinguishable from their canonical counterpart, the entanglement structure o…
Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $ k $-ETH ($ k=1,2,\dots $), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, wh…
Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence
School of Quantum & Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学量子物质科学与量子技术研究院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）
We develop a class of matrix models which implement and formalize the `eigenstate thermalization hypothesis' (ETH) and point out that in general these models must contain non-Gaussian corrections, already in order to correctly capture thermal mean-field theory, or to capture non-trivial OTOCs as well as their higher-order generalizations. We develop the framework of these `ETH matrix models', and put it in the context of recent studies in statistical physics incorporating higher statistical moments into the ETH ansatz. We then use the `ETH matrix model' in order to develop a matrix-integral de…
We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Rényi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Rényi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central …
The eigenstate thermalization hypothesis as well as the quantum ergodic theorem are studied in the light of quantum Fisher information. We show how global bounds on quantum Fisher information set the ETH and ergodicity conditions. Complexity and operator growth are briefly discussed in this frame.
Numerical evidence for the non-Abelian eigenstate thermalization hypothesis
Joint Center for Quantum Information and Computer Science, NIST and University of Maryland（美国国家标准与技术研究院与马里兰大学联合量子信息与计算科学中心）；Department of Physics, University of Seoul（首尔大学物理系）；Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究所）
Eigenstate Thermalization in 1+1-Dimensional SU(2) Lattice Gauge Theory Coupled with Dynamical Fermions
Indian Institute of Technology - Kanpur（坎普尔印度理工学院）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington（华盛顿大学物理系量子模拟孵化器）；Department of Physics, Birla Institute of Technology and Science Pilani, K K Birla Goa Campus（比拉理工学院皮拉尼分校K.K.比尔拉果阿校区物理系）；Center for Research in Quantum Information and Technology, Birla Institute of Technology and Science Pilani（比拉理工学院皮拉尼分校量子信息技术研究中心）
We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the Rényi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of th…
Understanding the evolution towards thermal equilibrium of an isolated quantum system is at the foundation of statistical mechanics and a subject of interest in such diverse areas as cold atom physics or the quantum mechanics of black holes. Since a pure state can never evolve into a thermal density matrix, the Eigenstate Thermalization Hypothesis (ETH) has been put forward by Deutsch and Srednicki as a way to explain this apparent thermalization, similarly to what the ergodic theorem does in classical mechanics. In this paper this hypothesis is tested numerically. First, it is observed that t…
