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Łydżba P, Mierzejewski M, Rigol M, Vidmar L. Generalized Thermalization in Quantum-Chaotic Quadratic Hamiltonians. PHYSICAL REVIEW LETTERS 2023; 131:060401. [PMID: 37625057 DOI: 10.1103/physrevlett.131.060401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 07/14/2023] [Indexed: 08/27/2023]
Abstract
Thermalization (generalized thermalization) in nonintegrable (integrable) quantum systems requires two ingredients: equilibration and agreement with the predictions of the Gibbs (generalized Gibbs) ensemble. We prove that observables that exhibit eigenstate thermalization in single-particle sector equilibrate in many-body sectors of quantum-chaotic quadratic models. Remarkably, the same observables do not exhibit eigenstate thermalization in many-body sectors (we establish that there are exponentially many outliers). Hence, the generalized Gibbs ensemble is generally needed to describe their expectation values after equilibration, and it is characterized by Lagrange multipliers that are smooth functions of single-particle energies.
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Affiliation(s)
- Patrycja Łydżba
- Institute of Theoretical Physics, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland
| | - Marcin Mierzejewski
- Institute 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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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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Mierzejewski M, Vidmar L. Quantitative Impact of Integrals of Motion on the Eigenstate Thermalization Hypothesis. PHYSICAL REVIEW LETTERS 2020; 124:040603. [PMID: 32058771 DOI: 10.1103/physrevlett.124.040603] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2019] [Revised: 11/26/2019] [Indexed: 06/10/2023]
Abstract
Even though the eigenstate thermalization hypothesis (ETH) may be introduced as an extension of the random matrix theory, physical Hamiltonians and observables differ from random operators. One of the challenges is to embed local integrals of motion (LIOMs) within the ETH. Here we make steps towards a unified treatment of the ETH in integrable and nonintegrable models with translational invariance. Specifically, we focus on the impact of LIOMs on the fluctuations and structure of the diagonal matrix elements of local observables. We first show that nonvanishing fluctuations entail the presence of LIOMs. Then we introduce a generic protocol to construct observables, subtracted by their projections on LIOMs as well as products of LIOMs. The protocol systematically reduces fluctuations and/or the structure of the diagonal matrix elements. We verify our arguments by numerical results for integrable and nonintegrable models.
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Affiliation(s)
- Marcin Mierzejewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
| | - 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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Więckowski A, Maśka MM, Mierzejewski M. Identification of Majorana Modes in Interacting Systems by Local Integrals of Motion. PHYSICAL REVIEW LETTERS 2018; 120:040504. [PMID: 29437428 DOI: 10.1103/physrevlett.120.040504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2017] [Revised: 09/08/2017] [Indexed: 06/08/2023]
Abstract
Recently, there has been substantial progress in methods of identifying local integrals of motion in interacting integrable models or in systems with many-body localization. We show that one of these approaches can be utilized for constructing local, conserved, Majorana fermions in systems with an arbitrary many-body interaction. As a test case, we first investigate a noninteracting Kitaev model and demonstrate that this approach perfectly reproduces the standard results. Then, we discuss how the many-body interactions influence the spatial structure and the lifetime of the Majorana modes. Finally, we determine the regime for which the information stored in the Majorana correlators is also retained for arbitrarily long times at high temperatures. We show that it is included in the regime with topologically protected soft Majorana modes, but in some cases is significantly smaller.
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Affiliation(s)
| | - Maciej M Maśka
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
| | - Marcin Mierzejewski
- Department of Theoretical Physics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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Lange F, Lenarčič Z, Rosch A. Pumping approximately integrable systems. Nat Commun 2017; 8:15767. [PMID: 28598444 PMCID: PMC5472763 DOI: 10.1038/ncomms15767] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2016] [Accepted: 04/24/2017] [Indexed: 11/09/2022] Open
Abstract
Weak perturbations can drive an interacting many-particle system far from its initial equilibrium state if one is able to pump into degrees of freedom approximately protected by conservation laws. This concept has for example been used to realize Bose–Einstein condensates of photons, magnons and excitons. Integrable quantum systems, like the one-dimensional Heisenberg model, are characterized by an infinite set of conservation laws. Here, we develop a theory of weakly driven integrable systems and show that pumping can induce large spin or heat currents even in the presence of integrability breaking perturbations, since it activates local and quasi-local approximate conserved quantities. The resulting steady state is qualitatively captured by a truncated generalized Gibbs ensemble with Lagrange parameters that depend on the structure but not on the overall amplitude of perturbations nor the initial state. We suggest to use spin-chain materials driven by terahertz radiation to realize integrability-based spin and heat pumps. Integrable models have an infinite number of conserved quantities but most realizations suffer from integrability breaking perturbations. Here the authors show that weakly driving such a system by periodic perturbations leads to large nonlinear responses governed by the approximate conservation laws.
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Affiliation(s)
- Florian Lange
- Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77a, D-50937 Cologne, Germany
| | - Zala Lenarčič
- Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77a, D-50937 Cologne, Germany
| | - Achim Rosch
- Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77a, D-50937 Cologne, Germany
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Karrasch C, Kennes DM, Heidrich-Meisner F. Thermal Conductivity of the One-Dimensional Fermi-Hubbard Model. PHYSICAL REVIEW LETTERS 2016; 117:116401. [PMID: 27661705 DOI: 10.1103/physrevlett.117.116401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2015] [Indexed: 06/06/2023]
Abstract
We study the thermal conductivity of the one-dimensional Fermi-Hubbard model at a finite temperature using a density matrix renormalization group approach. The integrability of this model gives rise to ballistic thermal transport. We calculate the temperature dependence of the thermal Drude weight at half filling for various interaction strengths. The finite-frequency contributions originating from the fact that the energy current is not a conserved quantity are investigated as well. We report evidence that breaking the integrability through a nearest-neighbor interaction leads to vanishing Drude weights and diffusive energy transport. Moreover, we demonstrate that energy spreads ballistically in local quenches with initially inhomogeneous energy density profiles in the integrable case. We discuss the relevance of our results for thermalization in ultracold quantum-gas experiments and for transport measurements with quasi-one-dimensional materials.
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Affiliation(s)
- C Karrasch
- Department of Physics, University of California, Berkeley, California 95720, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - D M Kennes
- Institut für Theorie der Statistischen Physik, RWTH Aachen University and JARA-Fundamentals of Future Information Technology, 52056 Aachen, Germany
| | - F Heidrich-Meisner
- Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, 80333 München, Germany
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Ilievski E, De Nardis J, Wouters B, Caux JS, Essler FHL, Prosen T. Complete Generalized Gibbs Ensembles in an Interacting Theory. PHYSICAL REVIEW LETTERS 2015; 115:157201. [PMID: 26550747 DOI: 10.1103/physrevlett.115.157201] [Citation(s) in RCA: 65] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2015] [Indexed: 06/05/2023]
Abstract
In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-1/2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method.
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Affiliation(s)
- E Ilievski
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - J De Nardis
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - B Wouters
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - J-S Caux
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - F H L Essler
- The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
| | - T Prosen
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
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Ilievski E, Medenjak M, Prosen T. Quasilocal Conserved Operators in the Isotropic Heisenberg Spin-1/2 Chain. PHYSICAL REVIEW LETTERS 2015; 115:120601. [PMID: 26430976 DOI: 10.1103/physrevlett.115.120601] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2015] [Indexed: 06/05/2023]
Abstract
Composing higher auxiliary-spin transfer matrices and their derivatives, we construct a family of quasilocal conserved operators of isotropic Heisenberg spin-1/2 chain and rigorously establish their linear independence from the well-known set of local conserved charges.
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Affiliation(s)
- Enej Ilievski
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
| | - Marko Medenjak
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Tomaž Prosen
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
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