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Vardi A. Chaos and bipartite entanglement between Bose-Josephson junctions. Phys Rev E 2022; 106:064210. [PMID: 36671102 DOI: 10.1103/physreve.106.064210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2022] [Accepted: 12/02/2022] [Indexed: 12/24/2022]
Abstract
The entanglement between two weakly coupled bosonic Josephson junctions is studied in relation to the classical mixed phasespace structure of the system, containing symmetry-related regular islands separated by chaos. The symmetry-resolved entanglement spectrum and bipartite entanglement entropy of the system's energy eigenstates are calculated and compared to their expected structure for random states that exhibit complete or partial ergodicity. The entanglement spectra of chaos-supported eigenstates match the microcanonical structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic invariant that restricts ergodization on the energy shell. The symmetry-resolved entanglement entropy of these quasistochastic states consists of a mean-field maximum entanglement term and a fluctuation correction due to the finite size of the constituent subsystems. The total bipartite entanglement entropy of the eigenstates correlates with their chaoticity. Island-supported eigenstates are macroscopic Schrödinger cat states for particles and excitations with substantially lower entanglement.
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Affiliation(s)
- Amichay Vardi
- Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
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2
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Ulčakar I, Vidmar L. Tight-binding billiards. Phys Rev E 2022; 106:034118. [PMID: 36266801 DOI: 10.1103/physreve.106.034118] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. Here we introduce tight-binding billiards in two dimensions, which are described by noninteracting spinless fermions on a disorder-free square lattice subject to curved open (hard-wall) boundaries. We show that many properties of tight-binding billiards match those of quantum-chaotic quadratic Hamiltonians: The average entanglement entropy of many-body eigenstates approaches the random matrix theory predictions and one-body observables in single-particle eigenstates obey the single-particle eigenstate thermalization hypothesis. On the other hand, a degenerate subset of single-particle eigenstates at zero energy (i.e., the zero modes) can be described as chiral particles whose wave functions are confined to one of the sublattices.
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Affiliation(s)
- Iris Ulčakar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia and Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - Lev Vidmar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia and Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
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3
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Haque M, McClarty PA, Khaymovich IM. Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles. Phys Rev E 2022; 105:014109. [PMID: 35193274 DOI: 10.1103/physreve.105.014109] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2020] [Accepted: 12/14/2021] [Indexed: 06/14/2023]
Abstract
Eigenstates of local many-body interacting systems that are far from spectral edges are thought to be ergodic and close to being random states. This is consistent with the eigenstate thermalization hypothesis and volume-law scaling of entanglement. We point out that systematic departures from complete randomness are generically present in midspectrum eigenstates, and focus on the departure of the entanglement entropy from the random-state prediction. We show that the departure is (partly) due to spatial correlations and due to orthogonality to the eigenstates at the spectral edge, which imposes structure on the midspectrum eigenstates.
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Affiliation(s)
- Masudul Haque
- Department of Theoretical Physics, Maynooth University, County Kildare, Ireland
- Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Paul A McClarty
- Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
| | - Ivan M Khaymovich
- Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
- Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia
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Miao Q, Barthel T. Eigenstate Entanglement: Crossover from the Ground State to Volume Laws. PHYSICAL REVIEW LETTERS 2021; 127:040603. [PMID: 34355950 DOI: 10.1103/physrevlett.127.040603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2019] [Revised: 05/27/2021] [Accepted: 06/14/2021] [Indexed: 06/13/2023]
Abstract
For the typical quantum many-body systems that obey the eigenstate thermalization hypothesis (ETH), we argue that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function. The ETH implies that the crossover functions can be deduced from subsystem entropies of thermal ensembles and have universal properties. These functions capture the full crossover from the ground-state entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For critical one-dimensional systems, a universal scaling function follows from conformal field theory and can be adapted for nonlinear dispersions. We use it to also deduce the crossover scaling function for Fermi liquids in d>1 dimensions. The analytical results are complemented by numerics for large noninteracting systems of fermions in d≤3 dimensions and have also been confirmed for bosonic systems and nonintegrable spin chains.
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Affiliation(s)
- Qiang Miao
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
| | - Thomas Barthel
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
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Łydżba P, Rigol M, Vidmar L. Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians. PHYSICAL REVIEW LETTERS 2020; 125:180604. [PMID: 33196274 DOI: 10.1103/physrevlett.125.180604] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2020] [Accepted: 10/02/2020] [Indexed: 06/11/2023]
Abstract
The eigenstate entanglement entropy is a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) has a volume-law coefficient that generally depends on the subsystem fraction. In contrast, it is maximal (subsystem fraction independent) in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a function of the subsystem fraction. We test it against numerical results for the quadratic Sachdev-Ye-Kitaev model and show that it describes the results for the power-law random banded matrix model (in the delocalized regime). We show that localization in quasimomentum space produces (small) deviations from our analytic predictions.
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Affiliation(s)
- Patrycja Łydżba
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
- Department of Theoretical Physics, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland
| | - Marcos Rigol
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Lev Vidmar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
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LeBlond T, Mallayya K, Vidmar L, Rigol M. Entanglement and matrix elements of observables in interacting integrable systems. Phys Rev E 2019; 100:062134. [PMID: 31962410 DOI: 10.1103/physreve.100.062134] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2019] [Indexed: 06/10/2023]
Abstract
We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.
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Affiliation(s)
- Tyler LeBlond
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Krishnanand Mallayya
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Lev Vidmar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - Marcos Rigol
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
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Hetterich D, Matveeva P. Construction and Universal Application of Entanglement-Erasing Partner States. PHYSICAL REVIEW LETTERS 2019; 123:160503. [PMID: 31702360 DOI: 10.1103/physrevlett.123.160503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2019] [Indexed: 06/10/2023]
Abstract
We investigate the subadditivity of the bipartite entanglement entropy (EE) of many-particle states, represented by Slater determinants, with respect to single particle excitations. In this setting, subadditivity can be phrased as erasure of EE, i.e., as a relative decrease in EE when adding excitations to the quantum state. We identify sets of single particle states that yield zero EE if jointly excited. Such states we dub entanglement erasing partner states (EEPS). These EEPS reveal a mechanism that describes how to disentangle two subspaces of a Hilbert space by exciting additional states. We demonstrate this general finding in Anderson and many-body localized models. The studied concept of entanglement erasure further enables us to derive the EE of Slater determinants in the free tight binding model. Here, our analytical findings show surprisingly good agreement with numerical results of the interacting XXX chain. The described EEPS further impose a universal, i.e., model independent, erasure of EE for randomly excited Slater determinants. This feature allows us to compute many-particle EE by means of the associated single particle states and the filling ratio. This novel finding can be employed to drastically reduce the computational effort in free models.
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Affiliation(s)
- Daniel Hetterich
- Institute for Theoretical Physics, University of Würzburg, 97074 Würzburg, Germany
- Department of Physics, Technical University of München, 85748 Garching, Germany
| | - Polina Matveeva
- Department of Physics and Research Center Optimas, Technical University of Kaiserslautern, 67663 Kaiserslautern, Germany
- Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
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Abstract
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system-a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V_{A}/V, while the Renyi entropy densities depend nonlinearly on V_{A}/V. Surprisingly, Renyi entropies S_{n} for n>1 are convex functions of the subsystem size, with a volume law coefficient that depends on V_{A}/V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum-mechanical systems, and is closely related to the eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.
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Affiliation(s)
- Tsung-Cheng Lu
- Department of Physics, University of California at San Diego, La Jolla, California 92093, USA
| | - Tarun Grover
- Department of Physics, University of California at San Diego, La Jolla, California 92093, USA
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Vidmar L, Hackl L, Bianchi E, Rigol M. Volume Law and Quantum Criticality in the Entanglement Entropy of Excited Eigenstates of the Quantum Ising Model. PHYSICAL REVIEW LETTERS 2018; 121:220602. [PMID: 30547632 DOI: 10.1103/physrevlett.121.220602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2018] [Revised: 10/12/2018] [Indexed: 06/09/2023]
Abstract
Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit); i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.
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Affiliation(s)
- Lev Vidmar
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - Lucas Hackl
- Max Planck Institute of Quantum Optics, Hans-Kopfermann-Straße 1, D-85748 Garching bei München, Germany
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
- Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Eugenio Bianchi
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
- Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Marcos Rigol
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
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Castro-Alvaredo OA, De Fazio C, Doyon B, Szécsényi IM. Entanglement Content of Quasiparticle Excitations. PHYSICAL REVIEW LETTERS 2018; 121:170602. [PMID: 30411951 DOI: 10.1103/physrevlett.121.170602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2018] [Revised: 09/20/2018] [Indexed: 06/08/2023]
Abstract
We investigate the quantum entanglement content of quasiparticle excitations in extended many-body systems. We show that such excitations give an additive contribution to the bipartite von Neumann and Rényi entanglement entropies that takes a simple, universal form. It is largely independent of the momenta and masses of the excitations and of the geometry, dimension, and connectedness of the entanglement region. The result has a natural quantum information theoretic interpretation as the entanglement of a state where each quasiparticle is associated with two qubits representing their presence within and without the entanglement region, taking into account quantum (in)distinguishability. This applies to any excited state composed of finite numbers of quasiparticles with finite de Broglie wavelengths or finite intrinsic correlation length. This includes particle excitations in massive quantum field theory and gapped lattice systems, and certain highly excited states in conformal field theory and gapless models. We derive this result analytically in one-dimensional massive bosonic and fermionic free field theories and for simple setups in higher dimensions. We provide numerical evidence for the harmonic chain and the two-dimensional harmonic lattice in all regimes where the conditions above apply. Finally, we provide supporting calculations for integrable spin chain models and other interacting cases without particle production. Our results point to new possibilities for creating entangled states using many-body quantum systems.
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Affiliation(s)
- Olalla A Castro-Alvaredo
- Department of Mathematics, City, University of London, 10 Northampton Square, EC1V 0HB, London, United Kingdom
| | - Cecilia De Fazio
- Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
| | - Benjamin Doyon
- Department of Mathematics, King's College London, Strand WC2R 2LS, London, United Kingdom
| | - István M Szécsényi
- Department of Mathematics, City, University of London, 10 Northampton Square, EC1V 0HB, London, United Kingdom
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Vidmar L, Hackl L, Bianchi E, Rigol M. Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians. PHYSICAL REVIEW LETTERS 2017; 119:020601. [PMID: 28753340 DOI: 10.1103/physrevlett.119.020601] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2017] [Indexed: 06/07/2023]
Abstract
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291], Page proved that the average entanglement entropy of subsystems of random pure states is S_{ave}≃lnD_{A}-(1/2)D_{A}^{2}/D for 1≪D_{A}≤sqrt[D], where D_{A} and D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models lnD_{A}-(lnD_{A})^{2}/lnD≤⟨S⟩≤lnD_{A}-[1/(2ln2)](lnD_{A})^{2}/lnD. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnD_{A} in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.
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Affiliation(s)
- Lev Vidmar
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Lucas Hackl
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
- Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Eugenio Bianchi
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
- Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Marcos Rigol
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
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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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