Analytically Solvable Renormalization Group for the Many-Body Localization Transition

Goremykina, Anna; Vasseur, Romain; Serbyn, Maksym

doi:10.1103/PhysRevLett.122.040601

Analytically Solvable Renormalization Group for the Many-Body Localization Transition

Journal Article · Mon Jan 28 00:00:00 EST 2019 · Physical Review Letters

DOI:https://doi.org/10.1103/PhysRevLett.122.040601· OSTI ID:1509473

Goremykina, Anna ^[1]; Vasseur, Romain ^[2]; Serbyn, Maksym ^[3]

Univ. of Geneva (Switzerland); Inst. of Science and Technology Austria, Klosterneuburg (Austria); UMass Amherst
Univ. of Massachusetts, Amherst, MA (United States)
Inst. of Science and Technology Austria, Klosterneuburg (Austria)

We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Massachusetts, Amherst, MA (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division

Grant/Contract Number:: SC0019168

OSTI ID:: 1509473

Alternate ID(s):: OSTI ID: 1492475

Journal Information:: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 4 Vol. 122; ISSN 0031-9007; ISSN PRLTAO

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (70)

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
Many-body localization: An introduction and selected topics Alet, Fabien; Laflorencie, Nicolas Comptes Rendus Physique, Vol. 19, Issue 6 https://doi.org/10.1016/j.crhy.2018.03.003	journal	September 2018
Localization and topology protected quantum coherence at the edge of hot matter Bahri, Yasaman; Vosk, Ronen; Altman, Ehud Nature Communications, Vol. 6, Issue 1 https://doi.org/10.1038/ncomms8341	journal	July 2015
Ordering, metastability and phase transitions in two-dimensional systems Kosterlitz, J M; Thouless, D J Journal of Physics C: Solid State Physics, Vol. 6, Issue 7, p. 1181-1203 https://doi.org/10.1088/0022-3719/6/7/010	journal	April 1973
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
Nonequilibrium quantum dynamics and transport: from integrability to many-body localization Vasseur, Romain; Moore, Joel E. Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6 https://doi.org/10.1088/1742-5468/2016/06/064010	journal	June 2016
Interactions and the Anderson transition Fleishman, L.; Anderson, P. W. Physical Review B, Vol. 21, Issue 6 https://doi.org/10.1103/PhysRevB.21.2366	journal	March 1980
Low-temperature properties of the random Heisenberg antiferromagnetic chain Dasgupta, Chandan; Ma, Shang-keng Physical Review B, Vol. 22, Issue 3 https://doi.org/10.1103/PhysRevB.22.1305	journal	August 1980
Random antiferromagnetic quantum spin chains Fisher, Daniel S. Physical Review B, Vol. 50, Issue 6 https://doi.org/10.1103/PhysRevB.50.3799	journal	August 1994
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
Low-frequency conductivity in many-body localized systems Gopalakrishnan, Sarang; Müller, Markus; Khemani, Vedika Physical Review B, Vol. 92, Issue 10 https://doi.org/10.1103/PhysRevB.92.104202	journal	September 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
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
Many-body localization phase transition: A simplified strong-randomness approximate renormalization group Zhang, Liangsheng; Zhao, Bo; Devakul, Trithep Physical Review B, Vol. 93, Issue 22 https://doi.org/10.1103/PhysRevB.93.224201	journal	June 2016
Thouless energy and multifractality across the many-body localization transition Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. Physical Review B, Vol. 96, Issue 10 https://doi.org/10.1103/PhysRevB.96.104201	journal	September 2017
Phase-ordering kinetics of one-dimensional nonconserved scalar systems Rutenberg, A. D.; Bray, A. J. Physical Review E, Vol. 50, Issue 3 https://doi.org/10.1103/PhysRevE.50.1900	journal	September 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
Interferometric Probes of Many-Body Localization Serbyn, M.; Knap, M.; Gopalakrishnan, S. Physical Review Letters, Vol. 113, Issue 14 https://doi.org/10.1103/PhysRevLett.113.147204	journal	October 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
Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain Imbrie, John Z. Physical Review Letters, Vol. 117, Issue 2 https://doi.org/10.1103/PhysRevLett.117.027201	journal	July 2016
Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System Žnidarič, Marko; Scardicchio, Antonello; Varma, Vipin Kerala Physical Review Letters, Vol. 117, Issue 4 https://doi.org/10.1103/PhysRevLett.117.040601	journal	July 2016
Scaling Theory of Entanglement at the Many-Body Localization Transition Dumitrescu, Philipp T.; Vasseur, Romain; Potter, Andrew C. Physical Review Letters, Vol. 119, Issue 11 https://doi.org/10.1103/PhysRevLett.119.110604	journal	September 2017
Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems Lüschen, Henrik P.; Bordia, Pranjal; Scherg, Sebastian Physical Review Letters, Vol. 119, Issue 26 https://doi.org/10.1103/PhysRevLett.119.260401	journal	December 2017
Many-Body Delocalization as a Quantum Avalanche Thiery, Thimothée; Huveneers, François; Müller, Markus Physical Review Letters, Vol. 121, Issue 14 https://doi.org/10.1103/PhysRevLett.121.140601	journal	October 2018
Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet Griffiths, Robert B. Physical Review Letters, Vol. 23, Issue 1 https://doi.org/10.1103/PhysRevLett.23.17	journal	July 1969
Random Antiferromagnetic Chain Ma, Shang-keng; Dasgupta, Chandan; Hu, Chin-kun Physical Review Letters, Vol. 43, Issue 19 https://doi.org/10.1103/PhysRevLett.43.1434	journal	November 1979
Quasiparticle Lifetime in a Finite System: A Nonperturbative Approach Altshuler, Boris L.; Gefen, Yuval; Kamenev, Alex Physical Review Letters, Vol. 78, Issue 14 https://doi.org/10.1103/PhysRevLett.78.2803	journal	April 1997
Phase Transition in a System of One-Dimensional Bosons with Strong Disorder Altman, Ehud; Kafri, Yariv; Polkovnikov, Anatoli Physical Review Letters, Vol. 93, Issue 15 https://doi.org/10.1103/PhysRevLett.93.150402	journal	October 2004
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
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
Critical Properties of the Many-Body Localization Transition Khemani, Vedika; Lim, S. P.; Sheng, D. N. Physical Review X, Vol. 7, Issue 2 https://doi.org/10.1103/PhysRevX.7.021013	journal	April 2017
Random Antiferromagnetic Chain. Ma, Shang-keng; Dasgupta, Chandan; Hu, Chin-kun Physical Review Letters, Vol. 43, Issue 25 https://doi.org/10.1103/physrevlett.43.1899.2	journal	December 1979
Observation of many-body localization of interacting fermions in a quasirandom optical lattice Schreiber, M.; Hodgman, S. S.; Bordia, P. Science, Vol. 349, Issue 6250 https://doi.org/10.1126/science.aaa7432	journal	July 2015
Exploring the many-body localization transition in two dimensions Choi, J. -y.; Hild, S.; Zeiher, J. Science, Vol. 352, Issue 6293 https://doi.org/10.1126/science.aaf8834	journal	June 2016
Universal Dynamics and Renormalization in Many-Body-Localized Systems Altman, Ehud; Vosk, Ronen Annual Review of Condensed Matter Physics, Vol. 6, Issue 1 https://doi.org/10.1146/annurev-conmatphys-031214-014701	journal	March 2015
Many-Body Localization and Thermalization in Quantum Statistical Mechanics Nandkishore, Rahul; Huse, David A. Annual Review of Condensed Matter Physics, Vol. 6, Issue 1 https://doi.org/10.1146/annurev-conmatphys-031214-014726	journal	March 2015
Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems. Lüschen, Henrik P.; Bordia, Pranjal; Scherg, Sebastian Apollo - University of Cambridge Repository https://doi.org/10.17863/cam.16811	text	January 2017
QUANTUM GASES. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Schreiber, Michael; Hodgman, Sean S.; Bordia, Pranjal Apollo - University of Cambridge Repository https://doi.org/10.17863/cam.45758	text	January 2015
The many-body localization phase transition Pal, Arijeet; Huse, David A. arXiv https://doi.org/10.48550/arxiv.1010.1992	text	January 2010
Area laws in a many-body localized state and its implications for topological order Bauer, Bela; Nayak, Chetan arXiv https://doi.org/10.48550/arxiv.1306.5753	text	January 2013
Localization and topology protected quantum coherence at the edge of 'hot' matter Bahri, Yasaman; Vosk, Ronen; Altman, Ehud arXiv https://doi.org/10.48550/arxiv.1307.4092	preprint	January 2013
Many body localization and thermalization in quantum statistical mechanics Nandkishore, Rahul; Huse, David A. arXiv https://doi.org/10.48550/arxiv.1404.0686	text	January 2014
A mean-field theory of nearly many-body localized metals Gopalakrishnan, Sarang; Nandkishore, Rahul arXiv https://doi.org/10.48550/arxiv.1405.1036	text	January 2014
Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice Lev, Yevgeny Bar; Cohen, Guy; Reichman, David R. arXiv https://doi.org/10.48550/arxiv.1407.7535	text	January 2014
Phenomenology of fully many-body-localized systems Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim arXiv https://doi.org/10.48550/arxiv.1408.4297	text	January 2014
Many-body localization edge in the random-field Heisenberg chain Luitz, David J.; Laflorencie, Nicolas; Alet, Fabien arXiv https://doi.org/10.48550/arxiv.1411.0660	text	January 2014
Theory of the many-body localization transition in one dimensional systems Vosk, Ronen; Huse, David A.; Altman, Ehud arXiv https://doi.org/10.48550/arxiv.1412.3117	text	January 2014
Universal properties of many-body delocalization transitions Potter, Andrew C.; Vasseur, Romain; Parameswaran, S. A. arXiv https://doi.org/10.48550/arxiv.1501.03501	text	January 2015
Observation of many-body localization of interacting fermions in a quasi-random optical lattice Schreiber, Michael; Hodgman, Sean S.; Bordia, Pranjal arXiv https://doi.org/10.48550/arxiv.1501.05661	text	January 2015
Low-frequency conductivity in many-body localized systems Gopalakrishnan, Sarang; Mueller, Markus; Khemani, Vedika arXiv https://doi.org/10.48550/arxiv.1502.07712	text	January 2015
A criterion for many-body localization-delocalization phase transition Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. arXiv https://doi.org/10.48550/arxiv.1507.01635	text	January 2015
Extended slow dynamical regime close to the many-body localization transition Luitz, David J.; Laflorencie, Nicolas; Alet, Fabien arXiv https://doi.org/10.48550/arxiv.1511.05141	text	January 2015
Many body localization and thermalization: insights from the entanglement spectrum Geraedts, Scott D.; Nandkishore, Rahul; Regnault, Nicolas arXiv https://doi.org/10.48550/arxiv.1603.00880	text	January 2016
Many-body localization phase transition: A simplified strong-randomness approximate renormalization group Zhang, Liangsheng; Zhao, Bo; Devakul, Trithep arXiv https://doi.org/10.48550/arxiv.1603.02296	text	January 2016
Nonequilibrium quantum dynamics and transport: from integrability to many-body localization Vasseur, Romain; Moore, Joel E. arXiv https://doi.org/10.48550/arxiv.1603.06618	text	January 2016
Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system Znidaric, Marko; Scardicchio, Antonello; Varma, Vipin Kerala arXiv https://doi.org/10.48550/arxiv.1604.08567	text	January 2016
Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain Imbrie, John Z. arXiv https://doi.org/10.48550/arxiv.1605.03003	text	January 2016
Critical Properties of the Many-Body Localization Transition Khemani, Vedika; Lim, S. P.; Sheng, D. N. arXiv https://doi.org/10.48550/arxiv.1607.05756	text	January 2016
Thouless energy and multifractality across the many-body localization transition Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. arXiv https://doi.org/10.48550/arxiv.1610.02389	text	January 2016
Scaling Theory of Entanglement at the Many-Body Localization Transition Dumitrescu, Philipp T.; Vasseur, Romain; Potter, Andrew C. arXiv https://doi.org/10.48550/arxiv.1701.04827	text	January 2017
Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states Basko, D. M.; Aleiner, I. L.; Altshuler, B. L. arXiv https://doi.org/10.48550/arxiv.cond-mat/0506617	text	January 2005
Phase Ordering Kinetics of One-Dimensional Non-Conserved Scalar Systems Rutenberg, A. D.; Bray, A. J. arXiv https://doi.org/10.48550/arxiv.cond-mat/9404046	text	January 1994
Quasiparticle Lifetime in a Finite System: A Non--Perturbative Approach Altshuler, B. L.; Gefen, Y.; Kamenev, A. arXiv https://doi.org/10.48550/arxiv.cond-mat/9609132	text	January 1996

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