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Orgad D, Oganesyan V, Gopalakrishnan S. Dynamical Transitions from Slow to Fast Relaxation in Random Open Quantum Systems. PHYSICAL REVIEW LETTERS 2024; 132:040403. [PMID: 38335340 DOI: 10.1103/physrevlett.132.040403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2022] [Accepted: 01/08/2024] [Indexed: 02/12/2024]
Abstract
We explore the effects of spatial locality on the dynamics of random quantum systems subject to a Markovian noise. To this end, we study a model in which the system Hamiltonian and its couplings to the noise are random matrices whose entries decay as power laws of distance, with distinct exponents α_{H}, α_{L}. The steady state is always featureless, but the rate at which it is approached exhibits three phases depending on α_{H} and α_{L}: a phase where the approach is asymptotically exponential as a result of a gap in the spectrum of the Lindblad superoperator that generates the dynamics, and two gapless phases with subexponential relaxation, distinguished by the manner in which the gap decreases with system size. Within perturbation theory, the phase boundaries in the (α_{H},α_{L}) plane differ for weak and strong decoherence, suggesting phase transitions as a function of noise strength. We identify nonperturbative effects that prevent such phase transitions in the thermodynamic limit.
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Affiliation(s)
- Dror Orgad
- Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
| | - Vadim Oganesyan
- Department of Physics and Astronomy, College of Staten Island, CUNY, Staten Island, New York 10314, USA
- Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
| | - Sarang Gopalakrishnan
- Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08540, USA
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2
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Nakerst G, Denisov S, Haque M. Random sparse generators of Markovian evolution and their spectral properties. Phys Rev E 2023; 108:014102. [PMID: 37583175 DOI: 10.1103/physreve.108.014102] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 06/07/2023] [Indexed: 08/17/2023]
Abstract
The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter φ, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with D vertices (number of system states) and 2φD edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as ∼φ, while its variance is ∼sqrt[φ]. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of D. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, φ=const, and find that starting already at φ⩾2, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.
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Affiliation(s)
- Goran Nakerst
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Sergey Denisov
- NordSTAR - Nordic Center for Sustainable and Trustworthy AI Research, Pilestredet 52, N-0166, Oslo, Norway
- Department of Computer Science, Oslo Metropolitan University, N-0130 Oslo, Norway
| | - Masudul Haque
- 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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Cipolloni G, Kudler-Flam J. Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble. PHYSICAL REVIEW LETTERS 2023; 130:010401. [PMID: 36669222 DOI: 10.1103/physrevlett.130.010401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2022] [Revised: 10/22/2022] [Accepted: 12/13/2022] [Indexed: 06/17/2023]
Abstract
Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems, and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality, and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.
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Affiliation(s)
- Giorgio Cipolloni
- Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
| | - Jonah Kudler-Flam
- Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
- School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540 USA
- Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
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Akemann G, Mielke A, Päßler P. Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger β. Phys Rev E 2022; 106:014146. [PMID: 35974587 DOI: 10.1103/physreve.106.014146] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Accepted: 07/07/2022] [Indexed: 06/15/2023]
Abstract
A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.
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Affiliation(s)
- Gernot Akemann
- Faculty of Physics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
| | - Adam Mielke
- Technical University of Denmark, Asmussens Allé, Building 303B, 2800 Kgs. Lyngby, Denmark
| | - Patricia Päßler
- Faculty of Physics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
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Cornelius J, Xu Z, Saxena A, Chenu A, Del Campo A. Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos. PHYSICAL REVIEW LETTERS 2022; 128:190402. [PMID: 35622025 DOI: 10.1103/physrevlett.128.190402] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2021] [Accepted: 04/18/2022] [Indexed: 06/15/2023]
Abstract
The dynamical signatures of quantum chaos in an isolated system are captured by the spectral form factor, which exhibits as a function of time a dip, a ramp, and a plateau, with the ramp being governed by the correlations in the level spacing distribution. While decoherence generally suppresses these dynamical signatures, the nonlinear non-Hermitian evolution with balanced gain and loss (BGL) in an energy-dephasing scenario can enhance manifestations of quantum chaos. In the Sachdev-Ye-Kitaev model and random matrix Hamiltonians, BGL increases the span of the ramp, lowering the dip as well as the value of the plateau, providing an experimentally realizable physical mechanism for spectral filtering. The chaos enhancement due to BGL is optimal over a family of filter functions that can be engineered with fluctuating Hamiltonians.
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Affiliation(s)
- Julien Cornelius
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Zhenyu Xu
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Avadh Saxena
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Aurélia Chenu
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Adolfo Del Campo
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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Li J, Prosen T, Chan A. Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos. PHYSICAL REVIEW LETTERS 2021; 127:170602. [PMID: 34739275 DOI: 10.1103/physrevlett.127.170602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Accepted: 09/24/2021] [Indexed: 06/13/2023]
Abstract
We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and nonunitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy scale (and timescale). Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems, respectively. For dissipative quantum chaotic systems, we show that the DSFF exhibits an exact rotational symmetry in its complex time argument τ. Analogous to the spectral form factor (SFF) behavior for Gaussian unitary ensemble, the DSFF for GinUE shows a "dip-ramp-plateau" behavior in |τ|: the DSFF initially decreases, increases at intermediate timescales, and saturates after a generalized Heisenberg time, which scales as the inverse mean level spacing. Remarkably, for large matrix size, the "ramp" of the DSFF for GinUE increases quadratically in |τ|, in contrast to the linear ramp in the SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that the DSFF takes a constant value, except for a region in complex time whose size and behavior depend on the eigenvalue density. Numerically, we verify the above claims and additionally show that the DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behavior, except for a region in the complex time plane of measure zero in the limit of large matrix size. As a physical example, we consider the quantum kicked top model with dissipation and show that it falls under the Ginibre universality class and Poisson as the "kick" is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices or Markov chains and show that these models again fall under the Ginibre universality class.
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Affiliation(s)
- Jiachen Li
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Tomaž Prosen
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Amos Chan
- Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
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Tarnowski W, Yusipov I, Laptyeva T, Denisov S, Chruściński D, Życzkowski K. Random generators of Markovian evolution: A quantum-classical transition by superdecoherence. Phys Rev E 2021; 104:034118. [PMID: 34654129 DOI: 10.1103/physreve.104.034118] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2021] [Accepted: 08/27/2021] [Indexed: 11/07/2022]
Abstract
Continuous-time Markovian evolution appears to be manifestly different in classical and quantum worlds. We consider ensembles of random generators of N-dimensional Markovian evolution, quantum and classical ones, and evaluate their universal spectral properties. We then show how the two types of generators can be related by superdecoherence. In analogy with the mechanism of decoherence, which transforms a quantum state into a classical one, superdecoherence can be used to transform a Lindblad operator (generator of quantum evolution) into a Kolmogorov operator (generator of classical evolution). We inspect spectra of random Lindblad operators undergoing superdecoherence and demonstrate that, in the limit of complete superdecoherence, the resulting operators exhibit spectral density typical to random Kolmogorov operators. By gradually increasing strength of superdecoherence, we observe a sharp quantum-to-classical transition. Furthermore, we define an inverse procedure of supercoherification that is a generalization of the scheme used to construct a quantum state out of a classical one. Finally, we study microscopic correlation between neighboring eigenvalues through the complex spacing ratios and observe the horseshoe distribution, emblematic of the Ginibre universality class, for both types of random generators. Remarkably, it survives both superdecoherence and supercoherification.
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Affiliation(s)
- W Tarnowski
- Institute of Theoretical Physics, Uniwersytet Jagielloński, 30-348 Kraków, Poland
| | - I Yusipov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia
| | - T Laptyeva
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia
| | - S Denisov
- Department of Computer Science, Oslo Metropolitan University, N-0130 Oslo, Norway
| | - D Chruściński
- Institute of Physics, Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University, 87-100 Toruń, Poland
| | - K Życzkowski
- Institute of Theoretical Physics, Uniwersytet Jagielloński, 30-348 Kraków, Poland.,Centrum Fizyki Teoretycznej PAN, 02-668 Warszawa, Poland
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Popkov V, Presilla C. Full Spectrum of the Liouvillian of Open Dissipative Quantum Systems in the Zeno Limit. PHYSICAL REVIEW LETTERS 2021; 126:190402. [PMID: 34047584 DOI: 10.1103/physrevlett.126.190402] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2021] [Accepted: 04/12/2021] [Indexed: 06/12/2023]
Abstract
We consider an open quantum system with dissipation, described by a Lindblad Master equation (LME). For dissipation locally acting and sufficiently strong, a separation of the relaxation timescales occurs, which, in terms of the eigenvalues of the Liouvillian, implies a grouping of the latter in distinct vertical stripes in the complex plane at positions determined by the eigenvalues of the dissipator. We derive effective LME equations describing the modes within each stripe separately, and solve them perturbatively, obtaining for the full set of eigenvalues and eigenstates of the Liouvillian explicit expressions correct at order 1/Γ included, where Γ is the strength of the dissipation. As an example, we apply our general results to quantum XYZ spin chains coupled, at one boundary, to a dissipative bath of polarization.
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Affiliation(s)
- Vladislav Popkov
- Department of Physics, University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany
| | - Carlo Presilla
- Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, Roma 00185, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, Roma 00185, Italy
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Wang K, Piazza F, Luitz DJ. Hierarchy of Relaxation Timescales in Local Random Liouvillians. PHYSICAL REVIEW LETTERS 2020; 124:100604. [PMID: 32216400 DOI: 10.1103/physrevlett.124.100604] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Revised: 01/20/2020] [Accepted: 02/25/2020] [Indexed: 06/10/2023]
Abstract
To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size ℓ with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality (n=ℓ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140403]. However, for local Liouvillians (n<ℓ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit. Our findings for n locality generalize immediately to the case of spatial locality, introducing further splitting of timescales due to the additional structure.
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Affiliation(s)
- Kevin Wang
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - Francesco Piazza
- Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01167 Dresden, Germany
| | - David J Luitz
- Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01167 Dresden, Germany
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10
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Lieu S, McGinley M, Cooper NR. Tenfold Way for Quadratic Lindbladians. PHYSICAL REVIEW LETTERS 2020; 124:040401. [PMID: 32058773 DOI: 10.1103/physrevlett.124.040401] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2019] [Revised: 11/14/2019] [Indexed: 06/10/2023]
Abstract
We uncover a topological classification applicable to open fermionic systems governed by a general class of Lindblad master equations. These "quadratic Lindbladians" can be captured by a non-Hermitian single-particle matrix which describes internal dynamics as well as system-environment coupling. We show that this matrix must belong to one of ten non-Hermitian Bernard-LeClair symmetry classes which reduce to the Altland-Zirnbauer classes in the closed limit. The Lindblad spectrum admits a topological classification, which we show results in gapless edge excitations with finite lifetimes. Unlike previous studies of purely Hamiltonian or purely dissipative evolution, these topological edge modes are unconnected to the form of the steady state. We provide one-dimensional examples where the addition of dissipators can either preserve or destroy the closed classification of a model, highlighting the sensitivity of topological properties to details of the system-environment coupling.
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Affiliation(s)
- Simon Lieu
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Max McGinley
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Nigel R Cooper
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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