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Sinha S, Ray S, Sinha S. Classical route to ergodicity and scarring in collective quantum systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:163001. [PMID: 38190726 DOI: 10.1088/1361-648x/ad1bf5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2023] [Accepted: 01/08/2024] [Indexed: 01/10/2024]
Abstract
Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems.
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Affiliation(s)
- Sudip Sinha
- Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia 741246, India
| | - Sayak Ray
- Physikalisches Institut, Universität Bonn, Nußallee 12, 53115 Bonn, Germany
| | - Subhasis Sinha
- Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia 741246, India
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2
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Nakerst G, Haque M. Chaos in the three-site Bose-Hubbard model: Classical versus quantum. Phys Rev E 2023; 107:024210. [PMID: 36932617 DOI: 10.1103/physreve.107.024210] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2022] [Accepted: 01/30/2023] [Indexed: 06/18/2023]
Abstract
We consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.
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Affiliation(s)
- Goran Nakerst
- Department of Theoretical Physics, Maynooth University, County Kildare, Ireland
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Masudul Haque
- Department of Theoretical Physics, Maynooth University, County Kildare, Ireland
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
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3
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Mondal D, Sinha S, Sinha S. Quantum transitions, ergodicity, and quantum scars in the coupled top model. Phys Rev E 2022; 105:014130. [PMID: 35193322 DOI: 10.1103/physreve.105.014130] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 01/10/2022] [Indexed: 06/14/2023]
Abstract
We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.
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Affiliation(s)
- Debabrata Mondal
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia-741246, India
| | - Sudip Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia-741246, India
| | - S Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia-741246, India
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4
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Wang Q, Pérez-Bernal F. Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis. Phys Rev E 2021; 104:034119. [PMID: 34654165 DOI: 10.1103/physreve.104.034119] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2021] [Accepted: 08/31/2021] [Indexed: 11/07/2022]
Abstract
Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.
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Affiliation(s)
- Qian Wang
- Department of Physics, Zhejiang Normal University, Jinhua 321004, China and CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia
| | - Francisco Pérez-Bernal
- Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física, Matemáticas y Computación, Universidad de Huelva, Huelva 21071, Spain and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain
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Sen A, Sen D, Sengupta K. Analytic approaches to periodically driven closed quantum systems: methods and applications. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:443003. [PMID: 34359051 DOI: 10.1088/1361-648x/ac1b61] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Accepted: 08/06/2021] [Indexed: 06/13/2023]
Abstract
We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed are presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus (FM) expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a FM resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.
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Affiliation(s)
- Arnab Sen
- School of Physical Sciences, Indian Association for the Cultivation of Science, 2A and 2B Raja S C Mullick Road, Jadavpur 700032, India
| | - Diptiman Sen
- Center for High Energy Physics and Department of Physics, Indian Institute of Science, Bengaluru 560012, India
| | - K Sengupta
- School of Physical Sciences, Indian Association for the Cultivation of Science, 2A and 2B Raja S C Mullick Road, Jadavpur 700032, India
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Mori T, Zhao H, Mintert F, Knolle J, Moessner R. Rigorous Bounds on the Heating Rate in Thue-Morse Quasiperiodically and Randomly Driven Quantum Many-Body Systems. PHYSICAL REVIEW LETTERS 2021; 127:050602. [PMID: 34397234 DOI: 10.1103/physrevlett.127.050602] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2021] [Accepted: 07/07/2021] [Indexed: 06/13/2023]
Abstract
The nonequilibrium quantum dynamics of closed many-body systems is a rich yet challenging field. While recent progress for periodically driven (Floquet) systems has yielded a number of rigorous results, our understanding on quantum many-body systems driven by rapidly varying but aperiodic and quasiperiodic driving is still limited. Here, we derive rigorous, nonperturbative, bounds on the heating rate in quantum many-body systems under Thue-Morse quasiperiodic driving and under random multipolar driving, the latter being a tunably randomized variant of the former. In the process, we derive a static effective Hamiltonian that describes the transient prethermal state, including the dynamics of local observables. Our bound for Thue-Morse quasiperiodic driving suggests that the heating time scales like (ω/g)^{-C ln(ω/g)} with a positive constant C and a typical energy scale g of the Hamiltonian, in agreement with our numerical simulations.
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Affiliation(s)
- Takashi Mori
- RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
| | - Hongzheng Zhao
- Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
| | - Florian Mintert
- Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
| | - Johannes Knolle
- Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
- Department of Physics TQM, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
| | - Roderich Moessner
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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Sinha S, Ray S, Sinha S. Fingerprint of chaos and quantum scars in kicked Dicke model: an out-of-time-order correlator study. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:174005. [PMID: 33530075 DOI: 10.1088/1361-648x/abe26b] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2020] [Accepted: 02/02/2021] [Indexed: 06/12/2023]
Abstract
We investigate the onset of chaos in a periodically kicked Dicke model (KDM), using the out-of-time-order correlator (OTOC) as a diagnostic tool, in both the oscillator and the spin subspaces. In the large spin limit, the classical Hamiltonian map is constructed, which allows us to investigate the corresponding phase space dynamics and to compute the Lyapunov exponent. We show that the growth rate of the OTOC for the canonically conjugate coordinates of the oscillator is able to capture the Lyapunov exponent in the chaotic regime. The onset of chaos is further investigated using the saturation value of the OTOC, that can serve as an alternate indicator of chaos in a generic interacting quantum system. This is also supported by a system independent effective random matrix model. We further identify the quantum scars in KDM and detect their dynamical signature by using the OTOC dynamics. The relevance of the present study in the context of ongoing cold atom experiments is also discussed.
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Affiliation(s)
- Sudip Sinha
- Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia 741246, India
| | - Sayak Ray
- Physikalisches Institut, Universität Bonn, Nussallee 12, 53115 Bonn, Germany
| | - Subhasis Sinha
- Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia 741246, India
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Long DM, Crowley PJD, Chandran A. Nonadiabatic Topological Energy Pumps with Quasiperiodic Driving. PHYSICAL REVIEW LETTERS 2021; 126:106805. [PMID: 33784118 DOI: 10.1103/physrevlett.126.106805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2020] [Accepted: 02/12/2021] [Indexed: 06/12/2023]
Abstract
We derive a topological classification of the steady states of d-dimensional lattice models driven by D incommensurate tones. Mapping to a unifying (d+D)-dimensional localized model in frequency space reveals anomalous localized topological phases (ALTPs) with no static analog. While the formal classification is determined by d+D, the observable signatures of each ALTP depend on the spatial dimension d. For each d, with d+D=3, we identify a quantized circulating current and corresponding topological edge states. The edge states for a driven wire (d=1) function as a quantized, nonadiabatic energy pump between the drives. We design concrete models of quasiperiodically driven qubits and wires that achieve ALTPs of several topological classes. Our results provide a route to experimentally access higher dimensional ALTPs in driven low-dimensional systems.
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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, Boston University, Boston, Massachusetts 02215, USA
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Anushya Chandran
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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Mondal D, Sinha S, Sinha S. Chaos and quantum scars in a coupled top model. Phys Rev E 2020; 102:020101. [PMID: 32942427 DOI: 10.1103/physreve.102.020101] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2020] [Accepted: 08/03/2020] [Indexed: 06/11/2023]
Abstract
We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits a variety of interesting phenomena such as a quantum phase transition (QPT), a dynamical transition, and excited-state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveal ergodic behavior at the center of the energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscent of unstable collective dynamics even in the presence of an interaction. The statistical properties of such scarred states deviate from the ergodic limit corresponding to the random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, the oscillatory behavior in the dynamics of the unequal time commutator and survival probability is observed as the dynamical signature of a quantum scar, which can be relevant for its detection.
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Affiliation(s)
- Debabrata Mondal
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741246, India
| | - Sudip Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741246, India
| | - S Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741246, India
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