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Kudler-Flam J. Rényi Mutual Information in Quantum Field Theory. PHYSICAL REVIEW LETTERS 2023; 130:021603. [PMID: 36706421 DOI: 10.1103/physrevlett.130.021603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2022] [Accepted: 12/23/2022] [Indexed: 06/18/2023]
Abstract
We study a proper definition of Rényi mutual information (RMI) in quantum field theory as defined via the Petz Rényi relative entropy. Unlike the standard definition, the RMI we compute is a genuine measure of correlations between subsystems, as evidenced by its non-negativity and monotonicity under local operations. Furthermore, the RMI is UV finite and well defined in the continuum limit. We develop a replica path integral approach for the RMI in quantum field theories and evaluate it explicitly in 1+1D conformal field theory using twist fields. We prove that it bounds connected correlation functions and check our results against exact numerics in the massless free fermion theory.
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Affiliation(s)
- Jonah Kudler-Flam
- School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA and Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
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Dai YW, Chen XH, Cho SY, Zhou HQ. Critical exponents of block-block mutual information in one-dimensional infinite lattice systems. Phys Rev E 2021; 104:044137. [PMID: 34781450 DOI: 10.1103/physreve.104.044137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2020] [Accepted: 10/04/2021] [Indexed: 11/07/2022]
Abstract
We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2 XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. As the lattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.
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Affiliation(s)
- Yan-Wei Dai
- Centre for Modern Physics, Chongqing University, Chongqing 400044, China
| | - Xi-Hao Chen
- Research Institute for New Materials and Technology, Chongqing University of Arts and Sciences, Chongqing 402160, China
| | - Sam Young Cho
- Centre for Modern Physics, Chongqing University, Chongqing 400044, China.,Department of Physics, Chongqing University, Chongqing 400044, China
| | - Huan-Qiang Zhou
- Centre for Modern Physics, Chongqing University, Chongqing 400044, China.,Department of Physics, Chongqing University, Chongqing 400044, China
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3
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Walsh C, Sémon P, Poulin D, Sordi G, Tremblay AMS. Local Entanglement Entropy and Mutual Information across the Mott Transition in the Two-Dimensional Hubbard Model. PHYSICAL REVIEW LETTERS 2019; 122:067203. [PMID: 30822052 DOI: 10.1103/physrevlett.122.067203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2018] [Indexed: 06/09/2023]
Abstract
Entanglement and information are powerful lenses to probe phases transitions in many-body systems. Motivated by recent cold atom experiments, which are now able to measure the corresponding information-theoretic quantities, we study the Mott transition in the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory, and focus on two key measures of quantum correlations: entanglement entropy and a measure of total mutual information. We show that they detect the first-order nature of the transition, the universality class of the end point, and the crossover emanating from the end point.
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Affiliation(s)
- C Walsh
- Department of Physics, Royal Holloway, University of London, Egham, Surrey, United Kingdom, TW20 0EX
| | - P Sémon
- Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
| | - D Poulin
- Département de physique & Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
- Canadian Institute for Advanced Research, Toronto, Ontario, Canada, M5G 1Z8
| | - G Sordi
- Department of Physics, Royal Holloway, University of London, Egham, Surrey, United Kingdom, TW20 0EX
| | - A-M S Tremblay
- Département de physique & Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
- Canadian Institute for Advanced Research, Toronto, Ontario, Canada, M5G 1Z8
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De Chiara G, Sanpera A. Genuine quantum correlations in quantum many-body systems: a review of recent progress. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:074002. [PMID: 29671752 DOI: 10.1088/1361-6633/aabf61] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Quantum information theory has considerably helped in the understanding of quantum many-body systems. The role of quantum correlations and in particular, bipartite entanglement, has become crucial to characterise, classify and simulate quantum many body systems. Furthermore, the scaling of entanglement has inspired modifications to numerical techniques for the simulation of many-body systems leading to the, now established, area of tensor networks. However, the notions and methods brought by quantum information do not end with bipartite entanglement. There are other forms of correlations embedded in the ground, excited and thermal states of quantum many-body systems that also need to be explored and might be utilised as potential resources for quantum technologies. The aim of this work is to review the most recent developments regarding correlations in quantum many-body systems focussing on multipartite entanglement, quantum nonlocality, quantum discord, mutual information but also other non classical measures of correlations based on quantum coherence. Moreover, we also discuss applications of quantum metrology in quantum many-body systems.
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Affiliation(s)
- Gabriele De Chiara
- Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
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Alba V. Out-of-equilibrium protocol for Rényi entropies via the Jarzynski equality. Phys Rev E 2017; 95:062132. [PMID: 28709283 DOI: 10.1103/physreve.95.062132] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2016] [Indexed: 11/07/2022]
Abstract
In recent years entanglement measures, such as the von Neumann and the Rényi entropies, provided a unique opportunity to access elusive features of quantum many-body systems. However, extracting entanglement properties analytically, experimentally, or in numerical simulations can be a formidable task. Here, by combining the replica trick and the Jarzynski equality we devise an alternative effective out-of-equilibrium protocol for measuring the equilibrium Rényi entropies. The key idea is to perform a quench in the geometry of the replicas. The Rényi entropies are obtained as the exponential average of the work performed during the quench. We illustrate an application of the method in classical Monte Carlo simulations, although it could be useful in different contexts, such as in quantum Monte Carlo, or experimentally in cold-atom systems. The method is most effective in the quasistatic regime, i.e., for a slow quench. As a benchmark, we compute the Rényi entropies in the Ising universality class in 1+1 dimensions. We find perfect agreement with the well-known conformal field theory predictions.
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Affiliation(s)
- Vincenzo Alba
- International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di Trieste, Italy
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De Tomasi G, Bera S, Bardarson JH, Pollmann F. Quantum Mutual Information as a Probe for Many-Body Localization. PHYSICAL REVIEW LETTERS 2017; 118:016804. [PMID: 28106445 DOI: 10.1103/physrevlett.118.016804] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2016] [Indexed: 06/06/2023]
Abstract
We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal-insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from a MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.
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Affiliation(s)
- Giuseppe De Tomasi
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
| | - Soumya Bera
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
| | - Jens H Bardarson
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
| | - Frank Pollmann
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
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Drut JE, Porter WJ. Entanglement, noise, and the cumulant expansion. Phys Rev E 2016; 93:043301. [PMID: 27176422 DOI: 10.1103/physreve.93.043301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2015] [Indexed: 06/05/2023]
Abstract
We put forward a simpler and improved variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions. The present method takes advantage of the approximate log-normal distributions that characterize the signal-to-noise properties of other approaches. In addition, we show that a simple rewriting of the formalism allows circumvention of the inversion of the restricted one-body density matrix in the calculation of the nth Rényi entanglement entropy for n>2. We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the n=2,3,4,⋯,10 Rényi entropies of the one-dimensional attractive Hubbard model. We use that data to extrapolate to the von Neumann (n=1) and n→∞ cases.
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Affiliation(s)
- Joaquín E Drut
- Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599-3255, USA
| | - William J Porter
- Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599-3255, USA
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Sherman NE, Devakul T, Hastings MB, Singh RRP. Nonzero-temperature entanglement negativity of quantum spin models: Area law, linked cluster expansions, and sudden death. Phys Rev E 2016; 93:022128. [PMID: 26986309 DOI: 10.1103/physreve.93.022128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2015] [Indexed: 06/05/2023]
Abstract
We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the "area-law" logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional XXZ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at T=0, negativity shows critical scaling in size and temperature.
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Affiliation(s)
- Nicholas E Sherman
- Department of Physics, University of California Davis, California 95616, USA
| | - Trithep Devakul
- Department of Physics, Princeton University, New Jersey 08544, USA
| | - Matthew B Hastings
- Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, USA
| | - Rajiv R P Singh
- Department of Physics, University of California Davis, California 95616, USA
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Deng Z, Wu J, Guo W. Rényi information flow in the Ising model with single-spin dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:063308. [PMID: 25615223 DOI: 10.1103/physreve.90.063308] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2014] [Indexed: 06/04/2023]
Abstract
The n-index Rényi mutual information and transfer entropies for the two-dimensional kinetic Ising model with arbitrary single-spin dynamics in the thermodynamic limit are derived as functions of ensemble averages of observables and spin-flip probabilities. Cluster Monte Carlo algorithms with different dynamics from the single-spin dynamics are thus applicable to estimate the transfer entropies. By means of Monte Carlo simulations with the Wolff algorithm, we calculate the information flows in the Ising model with the Metropolis dynamics and the Glauber dynamics, respectively. We find that not only the global Rényi transfer entropy, but also the pairwise Rényi transfer entropy, peaks in the disorder phase.
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Affiliation(s)
- Zehui Deng
- Physics Department, Beijing Normal University, Beijing 100875, China
| | - Jinshan Wu
- School of Systems Science, Beijing Normal University, Beijing 100875, China
| | - Wenan Guo
- Physics Department, Beijing Normal University, Beijing 100875, China and State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China
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Wang L, Troyer M. Renyi entanglement entropy of interacting fermions calculated using the continuous-time quantum Monte Carlo method. PHYSICAL REVIEW LETTERS 2014; 113:110401. [PMID: 25259962 DOI: 10.1103/physrevlett.113.110401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2014] [Indexed: 06/03/2023]
Abstract
We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples the interaction correction of the entanglement entropy, which by design ensures the efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.
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Affiliation(s)
- Lei Wang
- Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
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Herdman CM, Inglis S, Roy PN, Melko RG, Del Maestro A. Path-integral Monte Carlo method for Rényi entanglement entropies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:013308. [PMID: 25122411 DOI: 10.1103/physreve.90.013308] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2014] [Indexed: 06/03/2023]
Abstract
We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path-integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability, and interactions. We present proof-of-principle calculations and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large-scale many-body systems of interacting bosons.
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Affiliation(s)
- C M Herdman
- Department of Physics, University of Vermont, Burlington, Vermont 05405, USA
| | - Stephen Inglis
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, and Center for Nanoscience, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 Munich, Germany and Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1
| | - P-N Roy
- Department of Chemistry, University of Waterloo, Ontario, Canada N2L 3G1
| | - R G Melko
- Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1 and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
| | - A Del Maestro
- Department of Physics, University of Vermont, Burlington, Vermont 05405, USA and Vermont Complex Systems Center, University of Vermont, Burlington, Vermont 05405, USA
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12
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Stéphan JM, Inglis S, Fendley P, Melko RG. Geometric mutual information at classical critical points. PHYSICAL REVIEW LETTERS 2014; 112:127204. [PMID: 24724678 DOI: 10.1103/physrevlett.112.127204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2013] [Indexed: 06/03/2023]
Abstract
A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.
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Affiliation(s)
- Jean-Marie Stéphan
- Physics Department, University of Virginia, Charlottesville, Virginia 22904-4714, USA
| | - Stephen Inglis
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
| | - Paul Fendley
- Physics Department, University of Virginia, Charlottesville, Virginia 22904-4714, USA
| | - Roger G Melko
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
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Storms M, Singh RRP. Entanglement in ground and excited states of gapped free-fermion systems and their relationship with Fermi surface and thermodynamic equilibrium properties. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012125. [PMID: 24580190 DOI: 10.1103/physreve.89.012125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2013] [Indexed: 06/03/2023]
Abstract
We study bipartite entanglement entropies in the ground and excited states of free-fermion models, where a staggered potential, μs, induces a gap in the spectrum. Ground-state entanglement entropies satisfy the "area law", and the "area-law" coefficient is found to diverge as a logarithm of the staggered potential, when the system has an extended Fermi surface at μs=0. On the square lattice, we show that the coefficient of the logarithmic divergence depends on the Fermi surface geometry and its orientation with respect to the real-space interface between subsystems and is related to the Widom conjecture as enunciated by Gioev and Klich [ Phys. Rev. Lett. 96 100503 (2006)]. For point Fermi surfaces in two-dimension, the "area-law" coefficient stays finite as μs→0. The von Neumann entanglement entropy associated with the excited states follows a "volume law" and allows us to calculate an entropy density function sV(e), which is substantially different from the thermodynamic entropy density function sT(e), when the lattice is bipartitioned into two equal subsystems but approaches the thermodynamic entropy density as the fraction of sites in the larger subsystem, that is integrated out, approaches unity.
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Affiliation(s)
- Michelle Storms
- Department of Physics, University of California Davis, California 95616, USA and Department of Physics, Ohio Wesleyan University, Delaware, Ohio 43015, USA
| | - Rajiv R P Singh
- Department of Physics, University of California Davis, California 95616, USA
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Inglis S, Melko RG. Wang-Landau method for calculating Rényi entropies in finite-temperature quantum Monte Carlo simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:013306. [PMID: 23410459 DOI: 10.1103/physreve.87.013306] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) simulations for the purpose of calculating the Rényi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analog to the density of states for stochastic series expansion QMC, allowing a direct calculation of Rényi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, a 2D transverse field Ising model, and a three-dimensional Heisenberg model, confirming a critical scaling of the mutual information in cases with a finite-temperature transition. We discuss the benefits and limitations of broad sampling techniques compared to standard importance sampling methods.
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Affiliation(s)
- Stephen Inglis
- Department of Physics and Astronomy, University of Waterloo, 200 University Avenue, Ontario, Canada, N2L 3G1.
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Alba V, Haque M, Läuchli AM. Boundary-locality and perturbative structure of entanglement spectra in gapped systems. PHYSICAL REVIEW LETTERS 2012; 108:227201. [PMID: 23003644 DOI: 10.1103/physrevlett.108.227201] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2011] [Indexed: 06/01/2023]
Abstract
The entanglement between two parts of a many-body system can be characterized in detail by the entanglement spectrum. Focusing on gapped phases of several one-dimensional systems, we show how this spectrum is dominated by contributions from the boundary between the parts. This contradicts the view of an "entanglement Hamiltonian" as a bulk entity. The boundary-local nature of the entanglement spectrum is clarified through its hierarchical level structure, through the combination of two single-boundary spectra to form a two-boundary spectrum, and finally through consideration of dominant eigenfunctions of the entanglement Hamiltonian. We show consequences of boundary-locality for perturbative calculations of the entanglement spectrum.
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Affiliation(s)
- Vincenzo Alba
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany
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