Bimodal entanglement entropy distribution in the many-body localization transition

Yu, Xiongjie; Luitz, David J.; Clark, Bryan K.

doi:10.1103/physrevb.94.184202

Bimodal entanglement entropy distribution in the many-body localization transition

Journal Article · Tue Nov 15 00:00:00 EST 2016 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.94.184202· OSTI ID:1535788

Yu, Xiongjie ^[1]; Luitz, David J. ^[2]; Clark, Bryan K. ^[2]

Univ. of Illinois at Urbana-Champaign, IL (United States); DOE/OSTI
Univ. of Illinois at Urbana-Champaign, IL (United States)

We introduce the cut-averaged entanglement entropy in disordered periodic spin chains and prove it to be a concave function of subsystem size for individual eigenstates. This allows us to identify the entanglement scaling as a function of subsystem size for individual states in inhomogeneous systems. Using this quantity, we probe the critical region between the many-body localized (MBL) and ergodic phases in finite systems. In the middle of the spectrum, we show evidence for bimodality of the entanglement distribution in the MBL critical region, finding both volume law and area law eigenstates over disorder realizations as well as within single disorder realizations. The disorder-averaged entanglement entropy in this region then scales as a volume law with a coefficient below its thermal value. We discover in the critical region, as we approach the thermodynamic limit, that the cut-averaged entanglement entropy density falls on a one-parameter family of curves. Finally, we also show that without averaging over cuts the slope of the entanglement entropy vs subsystem size can be negative at intermediate and strong disorder, caused by rare localized regions in the system.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Illinois at Urbana-Champaign, IL (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC)

Grant/Contract Number:: SC0008692

OSTI ID:: 1535788

Alternate ID(s):: OSTI ID: 1332141

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 18 Vol. 94; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (62)

Low-energy transition in spectral statistics of two-dimensional interacting fermions Song, Pil Hun; Shepelyansky, Dima L. Physical Review B, Vol. 61, Issue 23 https://doi.org/10.1103/physrevb.61.15546	journal	June 2000
Efficient variational diagonalization of fully many-body localized Hamiltonians Pollmann, Frank; Khemani, Vedika; Cirac, J. Ignacio Physical Review B, Vol. 94, Issue 4 https://doi.org/10.1103/physrevb.94.041116	journal	July 2016
Integrability and Quantum Chaos in Spin Glass Shards Georgeot, B.; Shepelyansky, D. L. Physical Review Letters, Vol. 81, Issue 23 https://doi.org/10.1103/physrevlett.81.5129	journal	December 1998
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling Amestoy, Patrick R.; Duff, Iain S.; L'Excellent, Jean-Yves SIAM Journal on Matrix Analysis and Applications, Vol. 23, Issue 1 https://doi.org/10.1137/s0895479899358194	journal	January 2001
Encoding the structure of many-body localization with matrix product operators Pekker, David; Clark, Bryan K. arXiv https://doi.org/10.48550/arxiv.1410.2224	text	January 2014
Signatures of the many-body localization transition in the dynamics of entanglement and bipartite fluctuations Singh, Rajeev; Bardarson, Jens H.; Pollmann, Frank arXiv https://doi.org/10.48550/arxiv.1508.05045	text	January 2015
Obtaining highly-excited eigenstates of many-body localized Hamiltonians by the density matrix renormalization group Khemani, Vedika; Pollmann, Frank; Sondhi, S. L. arXiv https://doi.org/10.48550/arxiv.1509.00483	text	January 2015
Interacting electrons in disordered wires: Anderson localization and low-temperature transport Gornyi, I. V.; Mirlin, A. D.; Polyakov, D. G. arXiv https://doi.org/10.48550/arxiv.cond-mat/0506411	text	January 2005
Inequalities for Quantum Entropy: A Review with Conditions for Equality Ruskai, Mary Beth arXiv https://doi.org/10.48550/arxiv.quant-ph/0205064	text	January 2002
Completely positive maps and entropy inequalities Lindblad, Göran Communications in Mathematical Physics, Vol. 40, Issue 2 https://doi.org/10.1007/BF01609396	journal	June 1975
On Many-Body Localization for Quantum Spin Chains Imbrie, John Z. Journal of Statistical Physics, Vol. 163, Issue 5 https://doi.org/10.1007/s10955-016-1508-x	journal	April 2016
Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states Basko, D. M.; Aleiner, I. L.; Altshuler, B. L. Annals of Physics, Vol. 321, Issue 5 https://doi.org/10.1016/j.aop.2005.11.014	journal	May 2006
Hybrid scheduling for the parallel solution of linear systems Amestoy, Patrick R.; Guermouche, Abdou; L’Excellent, Jean-Yves Parallel Computing, Vol. 32, Issue 2 https://doi.org/10.1016/j.parco.2005.07.004	journal	February 2006
Thermalization and its mechanism for generic isolated quantum systems Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim Nature, Vol. 452, Issue 7189 https://doi.org/10.1038/nature06838	journal	April 2008
Inequalities for quantum entropy: A review with conditions for equality Ruskai, Mary Beth Journal of Mathematical Physics, Vol. 43, Issue 9 https://doi.org/10.1063/1.1497701	journal	September 2002
Proof of the strong subadditivity of quantum‐mechanical entropy Lieb, Elliott H.; Ruskai, Mary Beth Journal of Mathematical Physics, Vol. 14, Issue 12 https://doi.org/10.1063/1.1666274	journal	December 1973
From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics D'Alessio, Luca; Kafri, Yariv; Polkovnikov, Anatoli Advances in Physics, Vol. 65, Issue 3 https://doi.org/10.1080/00018732.2016.1198134	journal	May 2016
Signatures of the many-body localization transition in the dynamics of entanglement and bipartite fluctuations Singh, Rajeev; Bardarson, Jens H.; Pollmann, Frank New Journal of Physics, Vol. 18, Issue 2 https://doi.org/10.1088/1367-2630/18/2/023046	journal	February 2016
Area laws in a many-body localized state and its implications for topological order Bauer, Bela; Nayak, Chetan Journal of Statistical Mechanics: Theory and Experiment, Vol. 2013, Issue 09 https://doi.org/10.1088/1742-5468/2013/09/P09005	journal	September 2013
Level repulsion exponent β for many-body localization transitions and for Anderson localization transitions via Dyson Brownian motion Monthus, Cécile Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 3 https://doi.org/10.1088/1742-5468/2016/03/033113	journal	March 2016
Many-body-localization transition: strong multifractality spectrum for matrix elements of local operators Monthus, Cécile Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 7 https://doi.org/10.1088/1742-5468/2016/07/073301	journal	July 2016
Some convexity and subadditivity properties of entropy Lieb, Elliott H. Bulletin of the American Mathematical Society, Vol. 81, Issue 1 https://doi.org/10.1090/S0002-9904-1975-13621-4	journal	January 1975
Absence of Diffusion in Certain Random Lattices Anderson, P. W. Physical Review, Vol. 109, Issue 5 https://doi.org/10.1103/PhysRev.109.1492	journal	March 1958
Quantum statistical mechanics in a closed system Deutsch, J. M. Physical Review A, Vol. 43, Issue 4 https://doi.org/10.1103/PhysRevA.43.2046	journal	February 1991
Low-energy transition in spectral statistics of two-dimensional interacting fermions Song, Pil Hun; Shepelyansky, Dima L. Physical Review B, Vol. 61, Issue 23 https://doi.org/10.1103/PhysRevB.61.15546	journal	June 2000
Localization of interacting fermions at high temperature Oganesyan, Vadim; Huse, David A. Physical Review B, Vol. 75, Issue 15 https://doi.org/10.1103/PhysRevB.75.155111	journal	April 2007
Many-body localization in the Heisenberg X X Z magnet in a random field Žnidarič, Marko; Prosen, Tomaž; Prelovšek, Peter Physical Review B, Vol. 77, Issue 6 https://doi.org/10.1103/PhysRevB.77.064426	journal	February 2008
Many-body localization phase transition Pal, Arijeet; Huse, David A. Physical Review B, Vol. 82, Issue 17 https://doi.org/10.1103/PhysRevB.82.174411	journal	November 2010
Phenomenology of fully many-body-localized systems Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim Physical Review B, Vol. 90, Issue 17 https://doi.org/10.1103/PhysRevB.90.174202	journal	November 2014
Mean-field theory of nearly many-body localized metals Gopalakrishnan, Sarang; Nandkishore, Rahul Physical Review B, Vol. 90, Issue 22 https://doi.org/10.1103/PhysRevB.90.224203	journal	December 2014
Many-body localization edge in the random-field Heisenberg chain Luitz, David J.; Laflorencie, Nicolas; Alet, Fabien Physical Review B, Vol. 91, Issue 8 https://doi.org/10.1103/PhysRevB.91.081103	journal	February 2015
Many-body localization transition in Rokhsar-Kivelson-type wave functions Chen, Xiao; Yu, Xiongjie; Cho, Gil Young Physical Review B, Vol. 92, Issue 21 https://doi.org/10.1103/PhysRevB.92.214204	journal	December 2015
Spectral statistics across the many-body localization transition Serbyn, Maksym; Moore, Joel E. Physical Review B, Vol. 93, Issue 4 https://doi.org/10.1103/PhysRevB.93.041424	journal	January 2016
Extended slow dynamical regime close to the many-body localization transition Luitz, David J.; Laflorencie, Nicolas; Alet, Fabien Physical Review B, Vol. 93, Issue 6 https://doi.org/10.1103/PhysRevB.93.060201	journal	February 2016
Long tail distributions near the many-body localization transition Luitz, David J. Physical Review B, Vol. 93, Issue 13 https://doi.org/10.1103/PhysRevB.93.134201	journal	April 2016
Local entanglement structure across a many-body localization transition Bera, Soumya; Lakshminarayan, Arul Physical Review B, Vol. 93, Issue 13 https://doi.org/10.1103/PhysRevB.93.134204	journal	April 2016
Griffiths effects and slow dynamics in nearly many-body localized systems Gopalakrishnan, Sarang; Agarwal, Kartiek; Demler, Eugene A. Physical Review B, Vol. 93, Issue 13 https://doi.org/10.1103/PhysRevB.93.134206	journal	April 2016
Many-body localization and thermalization: Insights from the entanglement spectrum Geraedts, Scott D.; Nandkishore, Rahul; Regnault, Nicolas Physical Review B, Vol. 93, Issue 17 https://doi.org/10.1103/PhysRevB.93.174202	journal	May 2016
Entanglement scaling of excited states in large one-dimensional many-body localized systems Kennes, D. M.; Karrasch, C. Physical Review B, Vol. 93, Issue 24 https://doi.org/10.1103/PhysRevB.93.245129	journal	June 2016
Efficient variational diagonalization of fully many-body localized Hamiltonians Pollmann, Frank; Khemani, Vedika; Cirac, J. Ignacio Physical Review B, Vol. 94, Issue 4 https://doi.org/10.1103/PhysRevB.94.041116	journal	July 2016
Many-body localization and transition by density matrix renormalization group and exact diagonalization studies Lim, S. P.; Sheng, D. N. Physical Review B, Vol. 94, Issue 4 https://doi.org/10.1103/PhysRevB.94.045111	journal	July 2016
Chaos and quantum thermalization Srednicki, Mark Physical Review E, Vol. 50, Issue 2 https://doi.org/10.1103/PhysRevE.50.888	journal	August 1994
Local Conservation Laws and the Structure of the Many-Body Localized States Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. Physical Review Letters, Vol. 111, Issue 12 https://doi.org/10.1103/PhysRevLett.111.127201	journal	September 2013
Many-Body Localization in a Disordered Quantum Ising Chain Kjäll, Jonas A.; Bardarson, Jens H.; Pollmann, Frank Physical Review Letters, Vol. 113, Issue 10 https://doi.org/10.1103/PhysRevLett.113.107204	journal	September 2014
Absence of Diffusion in an Interacting System of Spinless Fermions on a One-Dimensional Disordered Lattice Bar Lev, Yevgeny; Cohen, Guy; Reichman, David R. Physical Review Letters, Vol. 114, Issue 10 https://doi.org/10.1103/PhysRevLett.114.100601	journal	March 2015
Anomalous Diffusion and Griffiths Effects Near the Many-Body Localization Transition Agarwal, Kartiek; Gopalakrishnan, Sarang; Knap, Michael Physical Review Letters, Vol. 114, Issue 16 https://doi.org/10.1103/PhysRevLett.114.160401	journal	April 2015
Many-Body Localization Characterized from a One-Particle Perspective Bera, Soumya; Schomerus, Henning; Heidrich-Meisner, Fabian Physical Review Letters, Vol. 115, Issue 4 https://doi.org/10.1103/PhysRevLett.115.046603	journal	July 2015
Early Breakdown of Area-Law Entanglement at the Many-Body Delocalization Transition Devakul, Trithep; Singh, Rajiv R. P. Physical Review Letters, Vol. 115, Issue 18 https://doi.org/10.1103/PhysRevLett.115.187201	journal	October 2015
Obtaining Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach Khemani, Vedika; Pollmann, Frank; Sondhi, S. L. Physical Review Letters, Vol. 116, Issue 24 https://doi.org/10.1103/PhysRevLett.116.247204	journal	June 2016
Anomalous Thermalization in Ergodic Systems Luitz, David J.; Bar Lev, Yevgeny Physical Review Letters, Vol. 117, Issue 17 https://doi.org/10.1103/PhysRevLett.117.170404	journal	October 2016
Quantum chaos in spin-fermion models Montambaux, Gilles; Poilblanc, Didier; Bellissard, Jean Physical Review Letters, Vol. 70, Issue 4 https://doi.org/10.1103/PhysRevLett.70.497	journal	January 1993
Emergence of Quantum Chaos in Finite Interacting Fermi Systems Jacquod, Ph.; Shepelyansky, D. L. Physical Review Letters, Vol. 79, Issue 10 https://doi.org/10.1103/PhysRevLett.79.1837	journal	September 1997
Integrability and Quantum Chaos in Spin Glass Shards Georgeot, B.; Shepelyansky, D. L. Physical Review Letters, Vol. 81, Issue 23 https://doi.org/10.1103/PhysRevLett.81.5129	journal	December 1998
Interacting Electrons in Disordered Wires: Anderson Localization and Low- T Transport Gornyi, I. V.; Mirlin, A. D.; Polyakov, D. G. Physical Review Letters, Vol. 95, Issue 20 https://doi.org/10.1103/PhysRevLett.95.206603	journal	November 2005
Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality Pekker, David; Refael, Gil; Altman, Ehud Physical Review X, Vol. 4, Issue 1 https://doi.org/10.1103/PhysRevX.4.011052	journal	March 2014
Theory of the Many-Body Localization Transition in One-Dimensional Systems Vosk, Ronen; Huse, David A.; Altman, Ehud Physical Review X, Vol. 5, Issue 3 https://doi.org/10.1103/PhysRevX.5.031032	journal	September 2015
Universal Properties of Many-Body Delocalization Transitions Potter, Andrew C.; Vasseur, Romain; Parameswaran, S. A. Physical Review X, Vol. 5, Issue 3 https://doi.org/10.1103/PhysRevX.5.031033	journal	September 2015
Criterion for Many-Body Localization-Delocalization Phase Transition Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. Physical Review X, Vol. 5, Issue 4 https://doi.org/10.1103/PhysRevX.5.041047	journal	December 2015
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling Amestoy, Patrick R.; Duff, Iain S.; L'Excellent, Jean-Yves SIAM Journal on Matrix Analysis and Applications, Vol. 23, Issue 1 https://doi.org/10.1137/S0895479899358194	journal	January 2001
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems Hernandez, Vicente; Roman, Jose E.; Vidal, Vicente ACM Transactions on Mathematical Software, Vol. 31, Issue 3 https://doi.org/10.1145/1089014.1089019	journal	September 2005
Ergodicity breaking in a model showing many-body localization De Luca, A.; Scardicchio, A. EPL (Europhysics Letters), Vol. 101, Issue 3 https://doi.org/10.1209/0295-5075/101/37003	journal	February 2013
Many-Body-Localization Transition in the Strong Disorder Limit: Entanglement Entropy from the Statistics of Rare Extensive Resonances Monthus, Cécile Entropy, Vol. 18, Issue 4 https://doi.org/10.3390/e18040122	journal	April 2016

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