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Burke PC, Nakerst G, Haque M. Structure of the Hamiltonian of mean force. Phys Rev E 2024; 110:014111. [PMID: 39160947 DOI: 10.1103/physreve.110.014111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2023] [Accepted: 06/06/2024] [Indexed: 08/21/2024]
Abstract
The Hamiltonian of mean force is an effective Hamiltonian that allows a quantum system, nonweakly coupled to an environment, to be written in an effective Gibbs state. We present results on the structure of the Hamiltonian of mean force in extended quantum systems with local interactions. We show that its spatial structure exhibits a "skin effect"-its difference from the system Hamiltonian dies off exponentially with distance from the system-environment boundary. For spin systems, we identify the terms that can appear in the Hamiltonian of mean force at different orders in the inverse temperature.
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Affiliation(s)
- Phillip C Burke
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
- Centre for Quantum Engineering, Science, and Technology, University College Dublin, Dublin 4, Ireland
- Department of Theoretical Physics, Maynooth University, Maynooth, Kildare, W23 F2H6, Ireland
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2
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Zhang X, Jiang W, Deng J, Wang K, Chen J, Zhang P, Ren W, Dong H, Xu S, Gao Y, Jin F, Zhu X, Guo Q, Li H, Song C, Gorshkov AV, Iadecola T, Liu F, Gong ZX, Wang Z, Deng DL, Wang H. Digital quantum simulation of Floquet symmetry-protected topological phases. Nature 2022; 607:468-473. [PMID: 35859194 PMCID: PMC9300455 DOI: 10.1038/s41586-022-04854-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Accepted: 05/11/2022] [Indexed: 11/09/2022]
Abstract
Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals1-8, in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions9,10, solid-state spin systems11-15, ultracold atoms16,17 and superconducting qubits18-20. Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors21.
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Affiliation(s)
- Xu Zhang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Wenjie Jiang
- Center for Quantum Information, IIIS, Tsinghua University, Beijing, China
| | - Jinfeng Deng
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Ke Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Jiachen Chen
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Pengfei Zhang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Wenhui Ren
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Hang Dong
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Shibo Xu
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Yu Gao
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Feitong Jin
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Xuhao Zhu
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
| | - Qiujiang Guo
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
- Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Hekang Li
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
- Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Chao Song
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
- Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
| | - Alexey V Gorshkov
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland and NIST, College Park, MD, USA
| | - Thomas Iadecola
- Department of Physics and Astronomy, Iowa State University, Ames, IA, USA
- Ames Laboratory, Ames, IA, USA
| | - Fangli Liu
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland and NIST, College Park, MD, USA
- QuEra Computing Inc., Boston, MA, USA
| | - Zhe-Xuan Gong
- Department of Physics, Colorado School of Mines, Golden, CO, USA
- National Institute of Standards and Technology, Boulder, CO, USA
| | - Zhen Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China.
- Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China.
| | - Dong-Ling Deng
- Center for Quantum Information, IIIS, Tsinghua University, Beijing, China.
- Shanghai Qi Zhi Institute, Shanghai, China.
| | - H Wang
- Department of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Interdisciplinary Center for Quantum Information, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China
- Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies, Hangzhou, China
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Morera I, Polls A, Juliá-Díaz B. Entanglement structure of the two-component Bose-Hubbard model as a quantum simulator of a Heisenberg chain. Sci Rep 2019; 9:9424. [PMID: 31263117 PMCID: PMC6603041 DOI: 10.1038/s41598-019-45737-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Accepted: 06/12/2019] [Indexed: 11/09/2022] Open
Abstract
We consider a quantum simulator of the Heisenberg chain with ferromagnetic interactions based on the two-component 1D Bose-Hubbard model at filling equal to two in the strong coupling regime. The entanglement properties of the ground state of the two-component Bose-Hubbard model are compared to those of the effective spin model as the interspecies interaction approaches the intraspecies one. A numerical study of the entanglement properties of the two-component Bose-Hubbard model is supplemented with analytical expressions derived from the effective spin Hamiltonian. When the pure ferromagnetic Heisenberg chain is considered, the entanglement properties of the effective Hamiltonian are not properly predicted by the quantum simulator.
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Affiliation(s)
- I Morera
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028, Barcelona, Spain. .,Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona, 08028, Spain.
| | - Artur Polls
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028, Barcelona, Spain.,Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona, 08028, Spain
| | - Bruno Juliá-Díaz
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028, Barcelona, Spain.,Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona, 08028, Spain.,ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
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Serbyn M, Michailidis AA, Abanin DA, Papić Z. Power-Law Entanglement Spectrum in Many-Body Localized Phases. PHYSICAL REVIEW LETTERS 2016; 117:160601. [PMID: 27792374 DOI: 10.1103/physrevlett.117.160601] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2016] [Indexed: 06/06/2023]
Abstract
The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.
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Affiliation(s)
- Maksym Serbyn
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Alexios A Michailidis
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Dmitry A Abanin
- Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland
| | - Z Papić
- School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
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Lundgren R, Blair J, Greiter M, Läuchli A, Fiete GA, Thomale R. Momentum-space entanglement spectrum of bosons and fermions with interactions. PHYSICAL REVIEW LETTERS 2014; 113:256404. [PMID: 25554899 DOI: 10.1103/physrevlett.113.256404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2014] [Indexed: 06/04/2023]
Abstract
We study the momentum space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain with analytical methods and exact diagonalization. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entanglement gap observed in Thomale et al. [Phys. Rev. Lett. 105, 116805 (2010)] depends on the scaling dimension of the conformal field theory as varied by the XXZ anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how these shifted transition points in the entanglement spectra may support the numerical study of phase transitions in the momentum space density matrix renormalization group.
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Affiliation(s)
- Rex Lundgren
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Jonathan Blair
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Martin Greiter
- Insitute for Theoritical Physics, Univesity of Würzburg, D-97074 Würzburg, Germany
| | - Andreas Läuchli
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
| | - Gregory A Fiete
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Ronny Thomale
- Insitute for Theoritical Physics, Univesity of Würzburg, D-97074 Würzburg, Germany
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Chandran A, Khemani V, Sondhi SL. How universal is the entanglement spectrum? PHYSICAL REVIEW LETTERS 2014; 113:060501. [PMID: 25148308 DOI: 10.1103/physrevlett.113.060501] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2014] [Indexed: 05/10/2023]
Abstract
It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this Letter, we show that this belief is false in general. Most significantly, we show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low-energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the low-energy ES and the Rényi entropies can mislead entirely, while for quantum Hall systems, the ES has much less universal content than assumed to date.
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Affiliation(s)
- Anushya Chandran
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA and Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario N2L 2Y5, Canada
| | - Vedika Khemani
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - S L Sondhi
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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Hu S, Turner AM, Penc K, Pollmann F. Berry-phase-induced dimerization in one-dimensional quadrupolar systems. PHYSICAL REVIEW LETTERS 2014; 113:027202. [PMID: 25062224 DOI: 10.1103/physrevlett.113.027202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2014] [Indexed: 06/03/2023]
Abstract
We investigate the effect of the Berry phase on quadrupoles that occur, for example, in the low-energy description of spin models. Specifically, we study here the one-dimensional bilinear-biquadratic spin-one model. An open question for many years about this model is whether it has a nondimerized fluctuating nematic phase. The dimerization has recently been proposed to be related to Berry phases of the quantum fluctuations. We use an effective low-energy description to calculate the scaling of the dimerization according to this theory and then verify the predictions using large scale density-matrix renormalization group simulations, giving good evidence that the state is dimerized all the way up to its transition into the ferromagnetic phase. We furthermore discuss the multiplet structure found in the entanglement spectrum of the ground state wave functions.
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Affiliation(s)
- Shijie Hu
- Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany
| | - Ari M Turner
- Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Karlo Penc
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Frank Pollmann
- Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany
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