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arXiv:1709.09776 · Physical review. D/Physical review. D.

Entanglement entropy with a time-dependent Hamiltonian

Allic Sivaramakrishnan

Published Mar 5, 2018• 3 citations
arXivHTMLPDF

Abstract

The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in ${\mathrm{CFT}}_{2}$ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher-order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in ${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that interactions entangle unentangled excitations. Entanglement propagates according to ``entanglement diagrams,'' proposed structures that are motivated by accessory spacetime diagrams for real-time perturbation theory. To illustrate the mechanisms involved, we compute higher-order corrections to free fermion entanglement entropy. We identify an unentangled operator, one which does not change the entanglement entropy to any order. Then, we introduce an interaction and find it changes entanglement entropy by entangling the unentangled excitations. The entanglement propagates in line with our conjecture. We compute several entanglement diagrams. We provide tools to simplify the computation of loop entanglement diagrams, which probe UV effects in entanglement propagation in CFT and holography.

DOI
10.1103/physrevd.97.066003

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The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in CFT2{\mathrm{CFT}}_{2}CFT2​ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher-order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that interactions entangle unentangled excitations. Entanglement propagates according to ``entanglement diagrams,'' proposed structures that are motivated by accessory spacetime diagrams for real-time perturbation theory. To illustrate the mechanisms involved, we compute higher-order corrections to free fermion entanglement entropy. We identify an unentangled operator, one which does not change the entanglement entropy to any order. Then, we introduce an interaction and find it changes entanglement entropy by entangling the unentangled excitations. The entanglement propagates in line with our conjecture. We compute several entanglement diagrams. We provide tools to simplify the computation of loop entanglement diagrams, which probe UV effects in entanglement propagation in CFT and holography.
We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality. In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures. Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators. In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.
Entanglement entropy (EE) is a concept of paramount importance in modern theoretical physics. Nevertheless, calculating EE in generic interacting theories with non-trivial time dependence is notoriously difficult. In this thesis, we address the real-time dynamics of EE in certain strongly-coupled quantum field theories (QFT) using the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. We first study the time evolution of EE in an exactly solvable model of two-dimensional CFT with an inhomogeneous sine-square deformation (SSD). We argue that the non-trivial time evolutions driven by the SSD Hamiltonian can be regarded as changing the foliations of spacetime. Remarkably, this leads to the presence of new horizons in the dual three-dimensional bulk. We then compute the EE in SSD CFT using holography and found agreement with CFT results in all comparable cases. Afterwards, we proceed to entanglement dynamics in generic chaotic quantum many-body systems. In the limit of large subsystems and late times, entanglement dynamics is believed to be described by an entanglement membrane effective theory. For 2d CFTs, a degenerate tension function reproduces several results regarding the dynamics of the entropy; this stands in contrast to higher dimensions where the tension is non-degenerate. We use holography to show that, in order to correctly capture the reflected entropy in 2d CFT, one needs to add an additional degree of freedom to the membrane description. Furthermore, we show that the conventional non-degenerate membrane tension function emerges upon introducing a relevant deformation to the CFT, dual to a planar Banados-Teitelboim-Zanelli (BTZ) black hole with scalar hair and with an interior Kasner universe. Subsequently, we establish a membrane effective theory for the dynamics of reflected entropy in higher dimensions. Lastly, we apply the membrane theories for reflected entropy to study the time evolution of multiparty entanglement signals. We found that scrambling dynamics generate multiparty entanglement with rich structures.
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.
We study the evolution of the universal area law of entanglement entropy when the Hamiltonian of the system undergoes a time dependent perturbation. In particular, we derive a general formula for the time dependent first order correction to the area law under the assumption that the field theory resides in a vacuum state when a small time-dependent perturbation of a relevant coupling constant is turned on. Using this formula, we carry out explicit calculations in free field theories deformed by a time dependent mass, whereas for a generic QFT we show that the time dependent first order correction is governed by the spectral function defining the two-point correlation function of the trace of the energy-momentum tensor. We also carry out holographic calculations based on the HRT proposal and find qualitative and, in certain cases, quantitative agreement with the field theory calculations.
Abstract In this work, we perturbatively calculate the modular Hamiltonian to obtain the entanglement entropy in a free fermion theory on a torus with three typical deformations, e.g., TT‾ T\overline{T} TT T T ¯ deformation, local bilinear operator deformation, and mass deformation. For TT‾ T\overline{T} TT T T ¯ deformation, we find that the leading order correction of entanglement entropy is proportional to the expectation value of the undeformed modular Hamiltonian. As a check, in the high/low-temperature limit, the entanglement entropy coincides with that obtained by the replica trick in the literature. Following the same perturbative strategy, we obtain the entanglement entropy of the free fermion vacuum state up to second-order by inserting a local bilinear operator deformation in a moving mirror setting. In the uniformly accelerated mirror, the first-order and second-order correction of entanglement entropy vanishes in the late time limit. For mass deformation, we derive the entanglement entropy up to first-order deformation and comment on the second-order correction.
For a perturbation of the state of a conformal field theory (CFT), the response of the entanglement entropy is governed by the so-called ``first law'' of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time ttt of entanglement entropy for a CFT driven by a ttt-linear source for a conserved U(1)U(1)U(1) current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically (d+1d+1d+1)-dimensional anti--de Sitter metric perturbation whose ttt dependence first appears at order zd{z}^{d}zd in the Fefferman-Graham expansion about the boundary at z=0z=0z=0.