Perturbative calculations of entanglement entropy

Published for SISSA by Springer Received: December 11, 2020 Accepted: February 5, 2021 Published: March 22, 2021 Pouria Dadras and Alexei Kitaev California Institute of Technology, Pasadena, CA 91125, U.S.A. E-mail: , Abstract: This paper is an attempt to extend the recent understanding of the Page curve for evaporating black holes to more general systems coupled to a heat bath. Although calculating the von Neumann entropy by the replica trick is usually a challenge, we have identified two solvable cases. For the initial section of the Page curve, we sum up the perturbation series in the system-bath coupling κ; the most interesting contribution is of order 2s, where s is the number of replicas. For the saturated regime, we consider the effect of an external impulse on the entropy at a later time and relate it to OTOCs. A significant simplification occurs in the maximal chaos case such that the effect may be interpreted in terms of an intermediate object, analogous to the branching surface of a replica wormhole. Keywords: AdS-CFT Correspondence, Black Holes, 2D Gravity ArXiv ePrint: 2011.09622 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)198 JHEP03(2021)198 Perturbative calculations of entanglement entropy Contents 1 2 Early phase of entanglement growth 2.1 The model and general formulas 2.2 Short initial period vs. linear growth 2 3 6 3 Perturbations to the saturated phase 3.1 Statement of the problem 3.2 Thermodynamic response theory for the replicated system 3.3 Branched two-point correlator 3.4 Branched correlator related to early-time OTOCs 3.5 Locking two operators in the same replica 7 8 9 11 11 13 4 Summary and discussion 15 1 Introduction While not an observable quantity, entropy is useful as an abstract measure of active degrees of freedom and correlations in the system. Quantum correlations can be elusive, particularly in black holes, where the classical space-time picture is incomplete. There has been a long but ultimately successful chase of correlations in the Hawking radiation. If a black hole forms from an object in a pure quantum state and then evaporates, the resulting radiation must also be in a pure state. Thus, it is strongly (albeit nonlocally) correlated. The general form of such correlations was predicted by Don Page [1], who considered a black hole as a generic quantum system. Still, it long remained unclear how such correlations could emerge in semiclassical gravity. Some important works that contributed to the solution include the Dray-’t Hooft mechanism of gravitational interaction between infalling matter and subsequent radiation [2, 3] and the calculation of out-of-time-order correlators (OTOCs) in the black hole setting [4]. However, the OTOC physics is relevant on short time scales and explains correlations that are present not in the radiation itself but relative to a purifying system [5] (that is, under the assumption that the black hole is part of a thermofield double and that we have unrestricted access to the other part). The recent breakthrough in understanding the correlations developing over the Page time [6, 7] required a careful formulation of the problem, which we will now summarize. The problem at hand is a semiclassical one. We do not have a complete theory of quantum gravity, nor should it be required. When working at the semiclassical level, it is not possible to derive long-term evolution from the short-term one. Rather, one should look for a global solution, which may depend on the quantity of interest. We consider the –1– JHEP03(2021)198 1 Introduction entanglement entropy between the black hole and the emitted radiation at a particular time t. So let ρ = ρ(t) be the black hole’s density matrix; we want to compute its von Neumann entropy, S(ρ) = − Tr(ρ ln ρ). The latter is expressed as the s → 1 limit of the s-Renyi entropy, 1 Ss (ρ) = ln Tr ρs . (1.1) 1−s For integer s, the expression Tr ρs may be interpreted as the partition functions of s replicas of the system. The von Neumann and Renyi entropies are nonlinear functions of the quantum state, which is why they are not observables. However, the logarithmic nonlinearity is mild, such that in the thermodynamic limit, S(ρ) is determined by typical microstates that contribute to the mixed state ρ. In contrast, Renyi entropies are often dominated by a fraction of microstates of tiny overall weight. This distinction is also evident from the replica wormhole picture. The s-Renyi entropy is related to an s-fold cover of space-time, whose metric is different from the physical one. But when we analytically continue the solution in s and take s to 1, we get the standard metric with an additional piece of data, the branching surface. Thus, the s → 1 limit is essential for compatibility with the usual (non-entropic) physics. Our main technical advance is how to take this limit in some specific cases. 2 Early phase of entanglement growth We adopt a simpler variant of the evaporation problem, where instead of radiating energy, the system comes into contact with a heat bath at the same temperature. Turning the system-bath interaction on represents a slight change in the Hamiltonian and results in a brief period of non-equilibrium dynamics. Then a steady state is achieved such that all simple correlation functions are thermal. However, if the system’s initial state was pure (though mimicking the thermal state), its von Neumann entropy will grow at a constant rate. We focus on this regime as well as the very beginning of quantum evolution. The entropy growth eventually saturates at the thermal (i.e. coarse-grained) entropy, but that is not captured by our method. Our calculation is perturbative in the system-bath coupling strength κ. Note that the von Neumann entropy has a logarithmic singularity at the unperturbed state, which is pure. This is reflected by the fact that in addition to terms of order κ2 (or any constant power of κ), terms of order κ2s (where s is the number of replicas) play an important role. –2– JHEP03(2021)198 Now, it turns out that the transition from the early phase of the black hole evaporation (when the radiation is uncorrelated as the naive theory predicts) to the later phase (when the entanglement entropy equals the black hole’s coarse-grained entropy) is first order. The later phase is described by a new type of space-time geometry, the replica wormhole [6, 7]. Although choosing the correct solution of the two is a global problem, each of them can be examined locally. We will study some properties of both solutions for general many-body systems, where the geometric description is not applicable. 2.1 The model and general formulas Let us consider a quantum system (meant to represent a black hole) with some Hilbert space HB and Hamiltonian HB . For an exact analogy with the evaporation problem, we would have to pick a pure state that looks like thermal to all simple measurements. (...truncated)