1
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Ikeda TN, Sugiura S, Polkovnikov A. Robust Effective Ground State in a Nonintegrable Floquet Quantum Circuit. PHYSICAL REVIEW LETTERS 2024; 133:030401. [PMID: 39094155 DOI: 10.1103/physrevlett.133.030401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2023] [Revised: 05/21/2024] [Accepted: 06/05/2024] [Indexed: 08/04/2024]
Abstract
An external periodic (Floquet) drive is believed to bring any initial state to the featureless infinite temperature state in generic nonintegrable isolated quantum many-body systems in the thermodynamic limit, irrespective of the driving frequency Ω. However, numerical or analytical evidence either proving or disproving this hypothesis is very limited and the issue has remained unsettled. Here, we study the initial state dependence of Floquet heating in a nonintegrable kicked Ising chain of length up to L=30 with an efficient quantum circuit simulator, showing a possible counterexample: the ground state of the effective Floquet Hamiltonian is exceptionally robust against heating, and could stay at finite energy density even after infinitely many Floquet cycles, if the driving period is shorter than a threshold value. This sharp energy localization transition or crossover does not happen for generic excited states. The exceptional robustness of the ground state is interpreted by (i) its isolation in the energy spectrum and (ii) the fact that those states with L-independent ℏΩ energy above the ground state energy of any generic local Hamiltonian, like the approximate Floquet Hamiltonian, are atypical and viewed as a collection of noninteracting quasiparticles. Our finding paves the way for engineering Floquet protocols with finite driving periods realizing long-lived, or possibly even perpetual, Floquet phases by initial state design.
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Affiliation(s)
| | - Sho Sugiura
- Blocq, Inc., Toranomon Hills Business Tower 1-17-1 Toranomon, Minato, Tokyo 105-6416 Japan
- Physics and Informatics Laboratory, NTT Research, Inc., Sunnyvale, California 94085, USA
- Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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2
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Yao Y, Xiang L. Superconducting Quantum Simulation for Many-Body Physics beyond Equilibrium. ENTROPY (BASEL, SWITZERLAND) 2024; 26:592. [PMID: 39056954 PMCID: PMC11275873 DOI: 10.3390/e26070592] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2024] [Revised: 07/07/2024] [Accepted: 07/09/2024] [Indexed: 07/28/2024]
Abstract
Quantum computing is an exciting field that uses quantum principles, such as quantum superposition and entanglement, to tackle complex computational problems. Superconducting quantum circuits, based on Josephson junctions, is one of the most promising physical realizations to achieve the long-term goal of building fault-tolerant quantum computers. The past decade has witnessed the rapid development of this field, where many intermediate-scale multi-qubit experiments emerged to simulate nonequilibrium quantum many-body dynamics that are challenging for classical computers. Here, we review the basic concepts of superconducting quantum simulation and their recent experimental progress in exploring exotic nonequilibrium quantum phenomena emerging in strongly interacting many-body systems, e.g., many-body localization, quantum many-body scars, and discrete time crystals. We further discuss the prospects of quantum simulation experiments to truly solve open problems in nonequilibrium many-body systems.
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Affiliation(s)
- Yunyan Yao
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, Department of Physics, Zhejiang University, Hangzhou 311200, China
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3
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Bertini B, Kos P, Prosen T. Localized Dynamics in the Floquet Quantum East Model. PHYSICAL REVIEW LETTERS 2024; 132:080401. [PMID: 38457727 DOI: 10.1103/physrevlett.132.080401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2023] [Accepted: 01/23/2024] [Indexed: 03/10/2024]
Abstract
We introduce and study the discrete-time version of the quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses. Previous work has established that its continuous-time counterpart displays a disorder-free localization transition signaled by the appearance of an exponentially large (in the volume) family of nonthermal, localized eigenstates. Here we combine analytical and numerical approaches to show that (i) the transition persists for discrete times, in fact, it is present for any finite value of the time step apart from a zero measure set; (ii) it is directly detected by following the nonequilibrium dynamics of the fully polarized state. Our findings imply that the transition is currently observable in state-of-the-art platforms for digital quantum simulation.
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Affiliation(s)
- Bruno Bertini
- School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
| | - Pavel Kos
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
| | - Tomaž Prosen
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
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4
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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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5
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Stephen DT, Hart O, Nandkishore RM. Ergodicity Breaking Provably Robust to Arbitrary Perturbations. PHYSICAL REVIEW LETTERS 2024; 132:040401. [PMID: 38335344 DOI: 10.1103/physrevlett.132.040401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Revised: 02/13/2023] [Accepted: 10/25/2023] [Indexed: 02/12/2024]
Abstract
We present a new route to ergodicity breaking via Hilbert space fragmentation that displays an unprecedented level of robustness. Our construction relies on a single emergent (prethermal) conservation law. In the limit when the conservation law is exact, we prove the emergence of Hilbert space fragmentation with an exponential number of frozen configurations. These configurations are low-entanglement states in the middle of the energy spectrum and therefore constitute examples of quantum many-body scars. We further prove that every frozen configuration is absolutely stable to arbitrary perturbations, to all finite orders in perturbation theory. In contrast to previous constructions, our proof is not limited to symmetric perturbations, or to perturbations with compact support, but also applies to perturbations with long-range tails, and even to arbitrary geometrically nonlocal k-body perturbations, as long as k/L→0 in the thermodynamic limit, where L is linear system size. Additionally, we identify one-form U(1) charges characterizing some nonfrozen sectors, and discuss the dynamics starting from typical initial conditions, which we argue is best interpreted in terms of the magnetohydrodynamics of the emergent one-form symmetry.
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Affiliation(s)
- David T Stephen
- Department of Physics and Center for Theory of Quantum Matter, University of Colorado Boulder, Boulder, Colorado 80309, USA
- Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - Oliver Hart
- Department of Physics and Center for Theory of Quantum Matter, University of Colorado Boulder, Boulder, Colorado 80309, USA
| | - Rahul M Nandkishore
- Department of Physics and Center for Theory of Quantum Matter, University of Colorado Boulder, Boulder, Colorado 80309, USA
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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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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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11
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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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12
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Moudgalya S, Bernevig BA, Regnault N. Quantum many-body scars and Hilbert space fragmentation: a review of exact results. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:086501. [PMID: 35617909 DOI: 10.1088/1361-6633/ac73a0] [Citation(s) in RCA: 28] [Impact Index Per Article: 14.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2022] [Accepted: 05/26/2022] [Indexed: 06/15/2023]
Abstract
The discovery of quantum many-body scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of systems exhibiting QMBS. We cover examples of equally-spaced towers that lead to exact revivals for particular initial states, as well as isolated examples of QMBS. Finally, we review Hilbert space fragmentation, a related phenomenon where systems exhibit a richer variety of ergodic and non-ergodic behaviors, and discuss its connections to QMBS.
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Affiliation(s)
- Sanjay Moudgalya
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, United States of America
- Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, United States of America
| | - B Andrei Bernevig
- Department of Physics, Princeton University, Princeton, NJ 08544, United States of America
- Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
- IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
| | - Nicolas Regnault
- Department of Physics, Princeton University, Princeton, NJ 08544, United States of America
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 75005 Paris, France
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13
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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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