Entanglement and symmetry resolution in two dimensional free quantum field theories
TL;DR: In this paper, a thorough analysis of entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry is presented.
Abstract: We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux α, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painleve V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.
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TL;DR: In this article, the authors provided an in-depth numerical study of the disordered Heisenberg chain and showed that the behavior is not transient and persists even for very strong disorder.
Abstract: We have recently shown that the logarithmic growth of the entanglement entropy following a quantum quench in a many-body localized phase is accompanied by a slow growth of the number entropy ${S}_{N}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}ln\phantom{\rule{0.16em}{0ex}}t$. Here we provide an in-depth numerical study of ${S}_{N}(t)$ for the disordered Heisenberg chain and show that this behavior is not transient and persists even for very strong disorder. Calculating the truncated R\'enyi number entropy ${S}_{N}^{(\ensuremath{\alpha})}(t)={(1\ensuremath{-}\ensuremath{\alpha})}^{\ensuremath{-}1}ln{\ensuremath{\sum}}_{n}{p}^{\ensuremath{\alpha}}(n)$ for $\ensuremath{\alpha}\ensuremath{\ll}1$ and $p(n)g{p}_{c}$---which is sensitive to large number fluctuations occurring with low probability---we demonstrate that the particle number distribution $p(n)$ in one half of the system has a continuously growing tail. This indicates a slow but steady increase in the number of particles crossing between the partitions in the interacting case and is in sharp contrast to Anderson localization for which we show that ${S}_{N}^{(\ensuremath{\alpha}\ensuremath{\rightarrow}0)}(t)$ saturates for any cutoff ${p}_{c}g0$. We show, furthermore, that the growth of ${S}_{N}$ is not the consequence of rare states or rare regions but rather represents typical behavior. These findings indicate that the interacting system is never fully localized even for very strong but finite disorder.
109 citations
TL;DR: In this paper, the Fourier transform of the entanglement entropy in a quantum many-body system has been studied in terms of a semiclassical picture of moving quasiparticles spreading the entagglement throughout the system.
Abstract: The time evolution of the entanglement entropy is a key concept to understand the structure of a nonequilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a general conjecture for the charged entropies whose Fourier transform gives the desired symmetry-resolved entanglement ${S}_{n}(q)$. We point out two physically relevant effects that should be easily observed in atomic experiments: a delay time for the onset of ${S}_{n}(q)$ which grows linearly with $|\mathrm{\ensuremath{\Delta}}q|$ (the difference between the charge $q$ and its mean value) and an effective equipartition when $|\mathrm{\ensuremath{\Delta}}q|$ is much smaller than the subsystem size.
86 citations
TL;DR: In this paper, the authors derived the symmetry-resolved entanglement entropy for Poincare patch and global AdS3, as well as for the conical defect geometries, by relating the generating function for the charged moments to the amount of charge in the entangling subregion.
Abstract: We consider symmetry-resolved entanglement entropy in AdS3/CFT2 coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincare patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons level k, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of $$ \hat{\mathfrak{u}}{(1)}_k $$
Kac-Moody type. Employing the $$ \hat{\mathfrak{u}}{(1)}_k $$
Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.
70 citations
TL;DR: In this paper, the authors developed a general approach to compute entanglement measures of SPTs in any dimension via a discrete path integral on multisheet Riemann surfaces with generalized defects.
Abstract: Symmetry-protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension. Here we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multisheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows us to identify universal entanglement properties. Specifically, we demonstrate an equiblock decomposition of the reduced density matrix into symmetry sectors, for all one-dimensional topological phases protected by finite Abelian unitary symmetries.
64 citations
TL;DR: In this paper, the authors considered the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies.
Abstract: We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a ℤ2 symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. The issue of symmetry resolution for discrete symmetries is also discussed. We show that entanglement equipartition is generically expected and we identify the first subleading term (in the UV cutoff) breaking it. We also present the complete computation of the symmetry resolved von Neumann entropy for an interval in the ground state of the paramagnetic phase of the Ising model. In particular, we compute the universal functions entering in the charged and symmetry resolved entanglement.
56 citations
References
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01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.
14,825 citations
Book•
01 Jan 1982
TL;DR: In this article, exactly solved models of statistical mechanics are discussed. But they do not consider exactly solvable models in statistical mechanics, which is a special issue in the statistical mechanics of the classical two-dimensional faculty of science.
Abstract: exactly solved models in statistical mechanics exactly solved models in statistical mechanics rodney j baxter exactly solved models in statistical mechanics exactly solved models in statistical mechanics flae exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books hatsutori in size 15 gvg7bzbookyo.qhigh literature cited r. j. baxter, exactly solved models in exactly solvable models in statistical mechanics exactly solved models in statistical mechanics dover books okazaki in size 24 vk19j3book.buncivy exactly solved models of statistical mechanics valerio nishizawa in size 11 b4zntdbookntey fukuda in size 13 33oloxbooknhuy yamada in size 19 x6g84ybook.zolay in honour of r j baxter’s 75th birthday arxiv:1608.04899v2 statistical mechanics, threedimensionality and np beautiful models: 70 years of exactly solved quantum many exactly solved models in statistical mechanics (dover solved lattice models: 1944 2010 university of melbourne exactly solved models and beyond: a special issue in the statistical mechanics of the classical two-dimensional faculty of science, p. j. saf ́arik university in ko?sice? a one-dimensional statistical mechanics model with exact statistical mechanics department of physics and astronomy statistical mechanics principles and selected applications graph theory and statistical physics yaroslavvb chapter 4 methods of statistical mechanics ijs thermodynamics and an introduction to thermostatistics potts models and related problems in statistical mechanics methods of quantum field theory in statistical physics statistical mechanics: theory and molecular simulation exactly solvable su(n) mixed spin ladders springer statistical field theory : an introduction to exactly
7,761 citations
TL;DR: It is argued that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy.
Abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.
4,395 citations
Book•
01 Jan 2004
TL;DR: In this paper, the Sine-Gordon F.1. Peculiarities of d = 1 2. Bosonization 3. Luttinger liquids 4. Refinements 5. Microscopic methods 6. Spin 1/2 chains 7. Interacting fermions on a lattice 8. Coupled fermionic chains 9. Disordered systems 10. Boundaries and isolated impurities 11.
Abstract: 1. Peculiarities of d=1 2. Bosonization 3. Luttinger liquids 4. Refinements 5. Microscopic methods 6. Spin 1/2 chains 7. Interacting fermions on a lattice 8. Coupled fermionic chains 9. Disordered systems 10. Boundaries and isolated impurities 11. Significant others A. Basics of many body B. Not so important fine technical points C. Correlation functions D. Bosonization directory E. Sine-Gordon F. Numerical solution
3,131 citations
TL;DR: In this article, the properties of entanglement in many-body systems are reviewed and both bipartite and multipartite entanglements are considered, and the zero and finite temperature properties of entangled states in interacting spin, fermion and boson model systems are discussed.
Abstract: Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.
3,096 citations