1
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Wei Z, Tan C, Zhang R. Generalized spectral form factor in random matrix theory. Phys Rev E 2024; 109:064208. [PMID: 39020871 DOI: 10.1103/physreve.109.064208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2024] [Accepted: 05/02/2024] [Indexed: 07/19/2024]
Abstract
The spectral form factor (SFF) plays a crucial role in revealing the statistical properties of energy-level distributions in complex systems. It is one of the tools to diagnose quantum chaos and unravel the universal dynamics therein. The definition of SFF in most literature only encapsulates the two-level correlation. In this manuscript, we extend the definition of SSF to include the high-order correlation. Specifically, we introduce the standard deviation of energy levels to define correlation functions, from which the generalized spectral form factor (GSFF) can be obtained by Fourier transforms. GSFF provides a more comprehensive knowledge of the dynamics of chaotic systems. Using random matrices as examples, we demonstrate dynamics features that are encoded in GSFF. Remarkably, the GSFF is complex, and the real and imaginary parts exhibit universal dynamics. For instance, in the two-level correlated case, the real part of GSFF shows a dip-ramp-plateau structure akin to the conventional counterpart, and the imaginary part for different system sizes converges in the long-time limit. For the two-level GSFF, the analytical forms of the real part are obtained and consistent with numerical results. The results of the imaginary part are obtained by numerical calculation. Similar analyses are extended to three-level GSFF.
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Affiliation(s)
- Zhiyang Wei
- MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China
| | | | - Ren Zhang
- MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China
- Hefei National Laboratory, Hefei 230088, China
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2
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Chepiga N, Laflorencie N. Resilient Infinite Randomness Criticality for a Disordered Chain of Interacting Majorana Fermions. PHYSICAL REVIEW LETTERS 2024; 132:056502. [PMID: 38364172 DOI: 10.1103/physrevlett.132.056502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2023] [Revised: 12/22/2023] [Accepted: 01/05/2024] [Indexed: 02/18/2024]
Abstract
The quantum critical properties of interacting fermions in the presence of disorder are still not fully understood. While it is well known that for Dirac fermions, interactions are irrelevant to the noninteracting infinite randomness fixed point (IRFP), the problem remains largely open in the case of Majorana fermions which further display a much richer disorder-free phase diagram. Here, pushing the limits of density matrix renormalization group simulations, we carefully examine the ground state of a Majorana chain with both disorder and interactions. Building on appropriate boundary conditions and key observables such as entanglement, energy gap, and correlations, we strikingly find that the noninteracting Majorana IRFP is very stable against finite interactions, in contrast with previous claims.
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Affiliation(s)
- Natalia Chepiga
- Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
| | - Nicolas Laflorencie
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France
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3
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Royen K, Mondal S, Pollmann F, Heidrich-Meisner F. Enhanced many-body localization in a kinetically constrained model. Phys Rev E 2024; 109:024136. [PMID: 38491625 DOI: 10.1103/physreve.109.024136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2023] [Accepted: 01/23/2024] [Indexed: 03/18/2024]
Abstract
In the study of the thermalization of closed quantum systems, the role of kinetic constraints on the temporal dynamics and the eventual thermalization is attracting significant interest. Kinetic constraints typically lead to long-lived metastable states depending on initial conditions. We consider a model of interacting hardcore bosons with an additional kinetic constraint that was originally devised to capture glassy dynamics at high densities. As a main result, we demonstrate that the system is highly prone to localization in the presence of uncorrelated disorder. Adding disorder quickly triggers long-lived dynamics as evidenced in the time evolution of density autocorrelations. Moreover, the kinetic constraint favors localization also in the eigenstates, where a finite-size transition to a many-body localized phase occurs for much lower disorder strengths than for the same model without a kinetic constraint. Our work sheds light on the intricate interplay of kinetic constraints and localization and may provide additional control over many-body localized phases in the time domain.
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Affiliation(s)
- Karl Royen
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, D-37077 Göttingen, Germany
| | - Suman Mondal
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, D-37077 Göttingen, Germany
| | - Frank Pollmann
- Physics Department, TUM School of Natural Sciences, Technical University of Munich, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany
| | - Fabian Heidrich-Meisner
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, D-37077 Göttingen, Germany
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4
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Chen W, Lemarié G, Gong J. Critical dynamics of long-range quantum disordered systems. Phys Rev E 2023; 108:054127. [PMID: 38115526 DOI: 10.1103/physreve.108.054127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2023] [Accepted: 10/23/2023] [Indexed: 12/21/2023]
Abstract
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
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Affiliation(s)
- Weitao Chen
- Department of Physics, National University of Singapore, Singapore
- MajuLab, Centre National de la Recherche Scientifique-Université Côte d'Azur-Sorbonne Université-National University of Singapore-Nanyang Technological University (CNRS-UCA-SU-NUS-NTU) International Joint Research Unit, Singapore
- Centre for Quantum Technologies, National University of Singapore, Singapore
| | - Gabriel Lemarié
- MajuLab, Centre National de la Recherche Scientifique-Université Côte d'Azur-Sorbonne Université-National University of Singapore-Nanyang Technological University (CNRS-UCA-SU-NUS-NTU) International Joint Research Unit, Singapore
- Centre for Quantum Technologies, National University of Singapore, Singapore
- Laboratoire de Physique Théorique, Université de Toulouse, Centre National de la Recherche Scientifique, UPS, Toulouse, France
| | - Jiangbin Gong
- Department of Physics, National University of Singapore, Singapore
- MajuLab, Centre National de la Recherche Scientifique-Université Côte d'Azur-Sorbonne Université-National University of Singapore-Nanyang Technological University (CNRS-UCA-SU-NUS-NTU) International Joint Research Unit, Singapore
- Centre for Quantum Technologies, National University of Singapore, Singapore
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5
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Vallejo-Fabila I, Torres-Herrera EJ. Late-time universal distribution functions of observables in one-dimensional many-body quantum systems. Phys Rev E 2023; 108:044102. [PMID: 37978615 DOI: 10.1103/physreve.108.044102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2023] [Accepted: 09/12/2023] [Indexed: 11/19/2023]
Abstract
We study the probability distribution function of the long-time values of observables being time-evolved by Hamiltonians modeling clean and disordered one-dimensional chains of many spin-1/2 particles. In particular, we analyze the return probability and its version for a completely extended initial state, the so-called spectral form factor. We complement our analysis with the spin autocorrelation and connected spin-spin correlation functions, both of interest in experiments with quantum simulators. We show that the distribution function has a universal shape provided the central limit theorem holds. Explicitly, the shape is exponential for the return probability and spectral form factor, meanwhile it is Gaussian for the few-body observables. We also discuss implications over the so-called many-body localization. Remarkably, our approach requires only a single sample of the dynamics and small system sizes, which could be quite advantageous when dealing specially with disordered systems.
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Affiliation(s)
- I Vallejo-Fabila
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Puebla, 72570, México
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6
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Wilming H, Osborne TJ, Decker KSC, Karrasch C. Reviving product states in the disordered Heisenberg chain. Nat Commun 2023; 14:5847. [PMID: 37730793 PMCID: PMC10511451 DOI: 10.1038/s41467-023-41464-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2022] [Accepted: 09/01/2023] [Indexed: 09/22/2023] Open
Abstract
When a generic quantum system is prepared in a simple initial condition, it typically equilibrates toward a state that can be described by a thermal ensemble. A known exception is localized systems that are non-ergodic and do not thermalize; however, local observables are still believed to become stationary. Here we demonstrate that this general picture is incomplete by constructing product states that feature periodic high-fidelity revivals of the full wavefunction and local observables that oscillate indefinitely. The system neither equilibrates nor thermalizes. This is analogous to the phenomenon of weak ergodicity breaking due to many-body scars and challenges aspects of the current phenomenology of many-body localization, such as the logarithmic growth of the entanglement entropy. To support our claim, we combine analytic arguments with large-scale tensor network numerics for the disordered Heisenberg chain. Our results hold for arbitrarily long times in chains of 160 sites up to machine precision.
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Affiliation(s)
- Henrik Wilming
- Leibniz Universität Hannover, Appelstraße 2, 30167, Hannover, Germany.
| | - Tobias J Osborne
- Leibniz Universität Hannover, Appelstraße 2, 30167, Hannover, Germany
| | - Kevin S C Decker
- Technische Universität Braunschweig, Institut für Mathematische Physik, Mendelssohnstraße 3, 38106, Braunschweig, Germany
| | - Christoph Karrasch
- Technische Universität Braunschweig, Institut für Mathematische Physik, Mendelssohnstraße 3, 38106, Braunschweig, Germany
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7
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Long DM, Crowley PJD, Khemani V, Chandran A. Phenomenology of the Prethermal Many-Body Localized Regime. PHYSICAL REVIEW LETTERS 2023; 131:106301. [PMID: 37739351 DOI: 10.1103/physrevlett.131.106301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Revised: 01/30/2023] [Accepted: 08/18/2023] [Indexed: 09/24/2023]
Abstract
The dynamical phase diagram of interacting disordered systems has seen substantial revision over the past few years. Theory must now account for a large prethermal many-body localized regime in which thermalization is extremely slow, but not completely arrested. We derive a quantitative description of these dynamics in short-ranged one-dimensional systems using a model of successive many-body resonances. The model explains the decay timescale of mean autocorrelators, the functional form of the decay-a stretched exponential-and relates the value of the stretch exponent to the broad distribution of resonance timescales. The Jacobi method of matrix diagonalization provides numerical access to this distribution, as well as a conceptual framework for our analysis. The resonance model correctly predicts the stretch exponents for several models in the literature. Successive resonances may also underlie slow thermalization in strongly disordered systems in higher dimensions, or with long-range interactions.
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Affiliation(s)
- David M Long
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Philip J D Crowley
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Vedika Khemani
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - Anushya Chandran
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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8
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Hopjan M, Vidmar L. Scale-Invariant Survival Probability at Eigenstate Transitions. PHYSICAL REVIEW LETTERS 2023; 131:060404. [PMID: 37625072 DOI: 10.1103/physrevlett.131.060404] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2022] [Revised: 04/21/2023] [Accepted: 07/20/2023] [Indexed: 08/27/2023]
Abstract
Understanding quantum phase transitions in highly excited Hamiltonian eigenstates is currently far from being complete. It is particularly important to establish tools for their characterization in time domain. Here, we argue that a scaled survival probability, where time is measured in units of a typical Heisenberg time, exhibits a scale-invariant behavior at eigenstate transitions. We first demonstrate this property in two paradigmatic quadratic models, the one-dimensional Aubry-Andre model and three-dimensional Anderson model. Surprisingly, we then show that similar phenomenology emerges in the interacting avalanche model of ergodicity breaking phase transitions. This establishes an intriguing similarity between localization transition in quadratic systems and ergodicity breaking phase transition in interacting systems.
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Affiliation(s)
- Miroslav Hopjan
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
| | - 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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9
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Kloss B, Halimeh JC, Lazarides A, Bar Lev Y. Absence of localization in interacting spin chains with a discrete symmetry. Nat Commun 2023; 14:3778. [PMID: 37355694 DOI: 10.1038/s41467-023-39468-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2022] [Accepted: 06/08/2023] [Indexed: 06/26/2023] Open
Abstract
Novel paradigms of strong ergodicity breaking have recently attracted significant attention in condensed matter physics. Understanding the exact conditions required for their emergence or breakdown not only sheds more light on thermalization and its absence in closed quantum many-body systems, but it also has potential benefits for applications in quantum information technology. A case of particular interest is many-body localization whose conditions are not yet fully settled. Here, we prove that spin chains symmetric under a combination of mirror and spin-flip symmetries and with a non-degenerate spectrum show finite spin transport at zero total magnetization and infinite temperature. We demonstrate this numerically using two prominent examples: the Stark many-body localization system (Stark-MBL) and the symmetrized many-body localization system (symmetrized-MBL). We provide evidence of delocalization at all energy densities and show that delocalization persists when the symmetry is broken. We use our results to construct two localized systems which, when coupled, delocalize each other. Our work demonstrates the dramatic effect symmetries can have on disordered systems, proves that the existence of exact resonances is not a sufficient condition for delocalization, and opens the door to generalization to higher spatial dimensions and different conservation laws.
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Affiliation(s)
- Benedikt Kloss
- Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Ave, New York, NY, 10010, USA.
| | - Jad C Halimeh
- Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333, München, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799, München, Germany
| | - Achilleas Lazarides
- Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK
| | - Yevgeny Bar Lev
- Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel
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10
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Xu WF, Rao WJ. Non-ergodic extended regime in random matrix ensembles: insights from eigenvalue spectra. Sci Rep 2023; 13:634. [PMID: 36635352 PMCID: PMC9837147 DOI: 10.1038/s41598-023-27751-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2022] [Accepted: 01/06/2023] [Indexed: 01/13/2023] Open
Abstract
The non-ergodic extended (NEE) regime in physical and random matrix (RM) models has attracted a lot of attention in recent years. Formally, NEE regime is characterized by its fractal wavefunctions and long-range spectral correlations such as number variance or spectral form factor. More recently, it's proposed that this regime can be conveniently revealed through the eigenvalue spectra by means of singular-value-decomposition (SVD), whose results display a super-Poissonian behavior that reflects the minibands structure of NEE regime. In this work, we employ SVD to a number of RM models, and show it not only qualitatively reveals the NEE regime, but also quantitatively locates the ergodic-NEE transition point. With SVD, we further suggest the NEE regime in a new RM model-the sparse RM model.
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Affiliation(s)
- Wang-Fang Xu
- School of Science, Hangzhou Dianzi University, Hangzhou, 310027, China
- China Academy for Rural Development and School of Public Affairs, Zhejiang University, Hangzhou, 310027, China
| | - W J Rao
- School of Science, Hangzhou Dianzi University, Hangzhou, 310027, China.
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11
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Krajewski B, Vidmar L, Bonča J, Mierzejewski M. Restoring Ergodicity in a Strongly Disordered Interacting Chain. PHYSICAL REVIEW LETTERS 2022; 129:260601. [PMID: 36608198 DOI: 10.1103/physrevlett.129.260601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Accepted: 12/06/2022] [Indexed: 06/17/2023]
Abstract
We consider a chain of interacting fermions with random disorder that was intensively studied in the context of many-body localization. We show that only a small fraction of the two-body interaction represents a true local perturbation to the Anderson insulator. While this true perturbation is nonzero at any finite disorder strength W, it decreases with increasing W. This establishes a view that the strongly disordered system should be viewed as a weakly perturbed integrable model, i.e., a weakly perturbed Anderson insulator. As a consequence, the latter can hardly be distinguished from a strictly integrable system in finite-size calculations at large W. We then introduce a rescaled model in which the true perturbation is of the same order of magnitude as the other terms of the Hamiltonian, and show that the system remains ergodic at arbitrary large disorder.
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Affiliation(s)
- B Krajewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
| | - L 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
| | - J Bonča
- 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
| | - M Mierzejewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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12
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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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13
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Šuntajs J, Vidmar L. Ergodicity Breaking Transition in Zero Dimensions. PHYSICAL REVIEW LETTERS 2022; 129:060602. [PMID: 36018665 DOI: 10.1103/physrevlett.129.060602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Accepted: 07/11/2022] [Indexed: 06/15/2023]
Abstract
It is of great current interest to establish toy models of ergodicity breaking transitions in quantum many-body systems. Here, we study a model that is expected to exhibit an ergodic to nonergodic transition in the thermodynamic limit upon tuning the coupling between an ergodic quantum dot and distant particles with spin-1/2. The model is effectively zero dimensional; however, a variant of the model was proposed by De Roeck and Huveneers to describe the avalanche mechanism of ergodicity breaking transition in one-dimensional disordered spin chains. We show that exact numerical results based on the spectral form factor calculation accurately agree with theoretical predictions, and hence unambiguously confirm existence of the ergodicity breaking transition in this model. We benchmark specific properties that represent hallmarks of the ergodicity breaking transition in finite systems.
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Affiliation(s)
- Jan Šuntajs
- 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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14
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Wang J, Lamann MH, Richter J, Steinigeweg R, Dymarsky A, Gemmer J. Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time. PHYSICAL REVIEW LETTERS 2022; 128:180601. [PMID: 35594081 DOI: 10.1103/physrevlett.128.180601] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2021] [Accepted: 04/14/2022] [Indexed: 06/15/2023]
Abstract
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
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Affiliation(s)
- Jiaozi Wang
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Mats H Lamann
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Jonas Richter
- Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Robin Steinigeweg
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
| | - Anatoly Dymarsky
- Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny 141701, Russia
- Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 121205, Russia
- Department of Physics, University of Kentucky, Lexington, Kentucky 40506, USA
| | - Jochen Gemmer
- Department of Physics, University of Osnabrück, D-49076 Osnabrück, Germany
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15
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Sels D, Polkovnikov A. Dynamical obstruction to localization in a disordered spin chain. Phys Rev E 2021; 104:054105. [PMID: 34942719 DOI: 10.1103/physreve.104.054105] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2021] [Accepted: 10/05/2021] [Indexed: 11/07/2022]
Abstract
We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior, we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low-frequency asymptotes of the spectral function. We identify a region of maximal chaos-with exponentially enhanced susceptibility-which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport, and we argue that the presence of such slow dynamics highly constrains any possible localization transition in the thermodynamic limit. Rather, the results are more consistent with absence of the localized phase.
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Affiliation(s)
- Dries Sels
- Department of Physics, New York University, New York, New York 10003, USA.,Center for Computational Quantum Physics, Flatiron Institute, New York 10010, New York, USA
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16
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Guo Q, Cheng C, Li H, Xu S, Zhang P, Wang Z, Song C, Liu W, Ren W, Dong H, Mondaini R, Wang H. Stark Many-Body Localization on a Superconducting Quantum Processor. PHYSICAL REVIEW LETTERS 2021; 127:240502. [PMID: 34951777 DOI: 10.1103/physrevlett.127.240502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2021] [Accepted: 11/08/2021] [Indexed: 06/14/2023]
Abstract
Quantum emulators, owing to their large degree of tunability and control, allow the observation of fine aspects of closed quantum many-body systems, as either the regime where thermalization takes place or when it is halted by the presence of disorder. The latter, dubbed many-body localization (MBL) phenomenon, describes the nonergodic behavior that is dynamically identified by the preservation of local information and slow entanglement growth. Here, we provide a precise observation of this same phenomenology in the case where the quenched on-site energy landscape is not disordered, but rather linearly varied, emulating the Stark MBL. To this end, we construct a quantum device composed of 29 functional superconducting qubits, faithfully reproducing the relaxation dynamics of a nonintegrable spin model. At large Stark potentials, local observables display periodic Bloch oscillations, a manifesting characteristic of the fragmentation of the Hilbert space in sectors that conserve dipole moments. The flexible programmability of our quantum emulator highlights its potential in helping the understanding of nontrivial quantum many-body problems, in direct complement to simulations in classical computers.
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Affiliation(s)
- Qiujiang Guo
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Chen Cheng
- School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
- Beijing Computational Science Research Center, Beijing 100094, China
| | - Hekang Li
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Shibo Xu
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Pengfei Zhang
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Zhen Wang
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Chao Song
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Wuxin Liu
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Wenhui Ren
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Hang Dong
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
| | - Rubem Mondaini
- Beijing Computational Science Research Center, Beijing 100094, China
| | - H Wang
- Department of Physics and Hangzhou Innovation Center, Interdisciplinary Center for Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
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Vidmar L, Krajewski B, Bonča J, Mierzejewski M. Phenomenology of Spectral Functions in Disordered Spin Chains at Infinite Temperature. PHYSICAL REVIEW LETTERS 2021; 127:230603. [PMID: 34936803 DOI: 10.1103/physrevlett.127.230603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2021] [Accepted: 11/01/2021] [Indexed: 06/14/2023]
Abstract
Studies of disordered spin chains have recently experienced a renewed interest, inspired by the question to which extent the exact numerical calculations comply with the existence of a many-body localization phase transition. For the paradigmatic random field Heisenberg spin chains, many intriguing features were observed when the disorder is considerable compared to the spin interaction strength. Here, we introduce a phenomenological theory that may explain some of those features. The theory is based on the proximity to the noninteracting limit, in which the system is an Anderson insulator. Taking the spin imbalance as an exemplary observable, we demonstrate that the proximity to the local integrals of motion of the Anderson insulator determines the dynamics of the observable at infinite temperature. In finite interacting systems our theory quantitatively describes its integrated spectral function for a wide range of disorders.
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Affiliation(s)
- 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
| | - Bartosz Krajewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
| | - Janez Bonča
- 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
| | - Marcin Mierzejewski
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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18
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Sierant P, Lazo EG, Dalmonte M, Scardicchio A, Zakrzewski J. Constraint-Induced Delocalization. PHYSICAL REVIEW LETTERS 2021; 127:126603. [PMID: 34597060 DOI: 10.1103/physrevlett.127.126603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2021] [Accepted: 08/12/2021] [Indexed: 06/13/2023]
Abstract
We study the impact of quenched disorder on the dynamics of locally constrained quantum spin chains, that describe 1D arrays of Rydberg atoms in both the frozen (Ising-type) and dressed (XY-type) regime. Performing large-scale numerical experiments, we observe no trace of many-body localization even at large disorder. Analyzing the role of quenched disorder terms in constrained systems we show that they act in two, distinct and competing ways: as an on-site disorder term for the basic excitations of the system, and as an interaction between excitations. The two contributions are of the same order, and as they compete (one towards localization, the other against it), one does never enter a truly strong disorder, weak interaction limit, where many-body localization occurs. Such a mechanism is further clarified in the case of XY-type constrained models: there, a term which would represent a bona fide local quenched disorder term acting on the excitations of the clean model must be written as a series of nonlocal terms in the unconstrained variables. Our observations provide a simple picture to interpret the role of quenched disorder that could be immediately extended to other constrained models or quenched gauge theories.
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Affiliation(s)
- Piotr Sierant
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- Institute of Theoretical Physics, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Eduardo Gonzalez Lazo
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea, 265, 34136 Trieste, Italy
| | - Marcello Dalmonte
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea, 265, 34136 Trieste, Italy
| | - Antonello Scardicchio
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
| | - Jakub Zakrzewski
- Institute of Theoretical Physics, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
- Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
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19
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Corps ÁL, Relaño A. Long-range level correlations in quantum systems with finite Hilbert space dimension. Phys Rev E 2021; 103:012208. [PMID: 33601518 DOI: 10.1103/physreve.103.012208] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2020] [Accepted: 12/17/2020] [Indexed: 11/07/2022]
Abstract
We study the spectral statistics of quantum systems with finite Hilbert spaces. We derive a theorem showing that eigenlevels in such systems cannot be globally uncorrelated, even in the case of fully integrable dynamics, as a consequence of the unfolding procedure. We provide an analytic expression for the power spectrum of the δ_{n} statistic for a model of intermediate statistics with level repulsion but independent spacings, and we show both numerically and analytically that the result is spoiled by the unfolding procedure. Then, we provide a simple model to account for this phenomenon, and test it by means of numerics on the disordered XXZ chain, the paradigmatic model of many-body localization, and the rational Gaudin-Richardson model, a prototypical model for quantum integrability.
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Affiliation(s)
- Ángel L Corps
- Departamento de Estructura de la Materia, Física Térmica y Electrónica and GISC, Universidad Complutense de Madrid, Avenida Complutense s/n, E-28040 Madrid, Spain
| | - Armando Relaño
- Departamento de Estructura de la Materia, Física Térmica y Electrónica and GISC, Universidad Complutense de Madrid, Avenida Complutense s/n, E-28040 Madrid, Spain
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20
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Šuntajs J, Bonča J, Prosen T, Vidmar L. Quantum chaos challenges many-body localization. Phys Rev E 2020; 102:062144. [PMID: 33466008 DOI: 10.1103/physreve.102.062144] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 12/02/2020] [Indexed: 06/12/2023]
Abstract
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
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Affiliation(s)
- Jan Šuntajs
- Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia
| | - Janez Bonča
- 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
| | - Tomaž Prosen
- 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
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
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21
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Sierant P, Lewenstein M, Zakrzewski J. Polynomially Filtered Exact Diagonalization Approach to Many-Body Localization. PHYSICAL REVIEW LETTERS 2020; 125:156601. [PMID: 33095617 DOI: 10.1103/physrevlett.125.156601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2020] [Accepted: 09/13/2020] [Indexed: 06/11/2023]
Abstract
Polynomially filtered exact diagonalization method (POLFED) for large sparse matrices is introduced. The algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix. The memory requirements scale better with system size than in the state-of-the-art shift-invert approach. The potential of POLFED is demonstrated examining many-body localization transition in 1D interacting quantum spin-1/2 chains. We investigate the disorder strength and system size scaling of Thouless time. System size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite-size effects. We discuss possible scenarios regarding the many-body localization transition obtaining estimates for the critical disorder strength.
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Affiliation(s)
- Piotr Sierant
- Institute of Theoretical Physics, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
- ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
| | - Maciej Lewenstein
- ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
- ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
| | - Jakub Zakrzewski
- Institute of Theoretical Physics, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
- Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
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22
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Buijsman W, Cheianov V, Gritsev V. Sensitivity of the spectral form factor to short-range level statistics. Phys Rev E 2020; 102:042216. [PMID: 33212613 DOI: 10.1103/physreve.102.042216] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2020] [Accepted: 10/06/2020] [Indexed: 06/11/2023]
Abstract
The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behavior is commonly interpreted as a characterization of two-point correlations at large separation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally, we indicate that each expansion coefficient of the two-point correlation function imposes a constraint on the properly weighted time-integrated spectral form factor. We discuss how these constraints can affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a probe, which eliminates the effect of the constraint imposed by the self-correlation. The use of this probe is demonstrated for a model of randomly incomplete spectra and a Floquet model supporting many-body localization.
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Affiliation(s)
- Wouter Buijsman
- Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, P.O. Box 94485, 1090 GL Amsterdam, The Netherlands
| | - Vadim Cheianov
- Instituut-Lorentz and Delta Institute for Theoretical Physics, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
| | - Vladimir Gritsev
- Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, P.O. Box 94485, 1090 GL Amsterdam, The Netherlands
- Russian Quantum Center, Skolkovo, Moscow 143025, Russia
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