Logarithmic growth of local entropy and total correlations in many-body localized dynamics

Fabio Anza1, Francesca Pietracaprina2, and John Goold3

1Complexity Sciences Center, Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616
2Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
3School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
4 pages, 5 figures, comments are welcome


The characterizing feature of a many-body localized phase is the existence of an extensive set of quasi-local conserved quantities with an exponentially localized support. This structure endows the system with the signature logarithmic in time entanglement growth between spatial partitions. This feature differentiates the phase from Anderson localization, in a non-interacting model. Experimentally measuring the entanglement between large partitions of an interacting many-body system requires highly non-local measurements which are currently beyond the reach of experimental technology. In this work we demonstrate that the defining structure of many-body localization can be detected by the dynamics of a simple quantity from quantum information known as the total correlations which is connected to the local entropies. Central to our finding is the necessity to propagate specific initial states, drawn from the Hamiltonian unbiased basis (HUB). The dynamics of the local entropies and total correlations requires only local measurements in space and therefore is potentially experimentally accessible in a range of platforms.

Localization phenomena in interacting quantum systems have recently received a lot of attention from the scientific community, both for their theoretical interest and potential technological applications. Despite that, we still lack a direct experimental evidence of their most important feature: The logarithmic growth in time of entanglement among its components. In this work we show how this problem can be overcome: A smart choice of initial states guarantees that local measurements are sufficient to experimentally reveal this fascinating feature.

► BibTeX data

► References

[1] Dmitry A Abanin and Zlatko Papić. Recent progress in many-body localization. Ann. Phys., 529 (7): 1700169, jul 2017. ISSN 00033804. 10.1002/​andp.201700169.

[2] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys., 91: 021001, May 2019. 10.1103/​RevModPhys.91.021001.

[3] Kartiek Agarwal, Sarang Gopalakrishnan, Michael Knap, Markus Müller, and Eugene Demler. Anomalous diffusion and griffiths effects near the many-body localization transition. Phys. Rev. Lett., 114: 160401, Apr 2015. 10.1103/​PhysRevLett.114.160401.

[4] Fabien Alet and Nicolas Laflorencie. Many-body localization: an introduction and selected topics. Comptes Rendus Physique, 19 (6): 498–525, 2018. 10.1016/​j.crhy.2018.03.003.

[5] Edhud Altman. Many-body localisation and quantum thermalisation. Nature Physics, 14: 979–983, 2018. 10.1038/​s41567-018-0305-7.

[6] Ehud Altman and Ronen Vosk. Universal Dynamics and Renormalization in Many-Body-Localized Systems. Annu. Rev. Condens. Matter Phys., 6 (1): 383–409, mar 2015a. ISSN 1947-5454. 10.1146/​annurev-conmatphys-031214-014701.

[7] Ehud Altman and Ronen Vosk. Universal dynamics and renormalization in many-body-localized systems. Annu. Rev. Condens. Matter Phys., 6 (1): 383–409, 2015b. 10.1146/​annurev-conmatphys-031214-014701.

[8] P. W. Anderson. Absence of Diffusion in Certain Random Lattices. Phys. Rev., 109 (5): 1492–1505, mar 1958. ISSN 0031-899X. 10.1103/​PhysRev.109.1492.

[9] F. Anzà and Vlatko Vedral. Information-theoretic equilibrium and observable thermalization. Sci. Rep., 7: 44066, 2017. ISSN 2045-2322. 10.1038/​srep44066.

[10] Fabio Anza, Christian Gogolin, and Marcus Huber. Eigenstate Thermalization for Degenerate Observables. Phys. Rev. Lett., 120 (15): 150603, apr 2018. ISSN 0031-9007. 10.1103/​PhysRevLett.120.150603.

[11] Jens H. Bardarson, Frank Pollmann, and Joel E. Moore. Unbounded Growth of Entanglement in Models of Many-Body Localization. Phys. Rev. Lett., 109 (1): 017202, jul 2012. ISSN 0031-9007. 10.1103/​PhysRevLett.109.017202.

[12] D.M. Basko, I.L. Aleiner, and B.L. Altshuler. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. (N. Y)., 321 (5): 1126–1205, may 2006. ISSN 00034916. 10.1016/​j.aop.2005.11.014.

[13] Soumya Bera and Arul Lakshminarayan. Local entanglement structure across a many-body localization transition. Phys. Rev. B, 93 (13): 134204, apr 2016. ISSN 2469-9950. 10.1103/​PhysRevB.93.134204.

[14] Steve Campbell, Matthew J. M. Power, and Gabriele De Chiara. Dynamics and asymptotics of correlations in a many-body localized system. Eur. Phys. J. D, 71 (8): 206, aug 2017. ISSN 1434-6060. 10.1140/​epjd/​e2017-80302-8.

[15] Jae-yoon Choi, Sebastian Hild, Johannes Zeiher, Peter Schauß, Antonio Rubio-Abadal, Tarik Yefsah, Vedika Khemani, David A Huse, Immanuel Bloch, and Christian Gross. Exploring the many-body localization transition in two dimensions. Science, 352 (6293): 1547–1552, 2016. 10.1126/​science.aaf8834.

[16] Giuseppe De Tomasi, Soumya Bera, Jens H. Bardarson, and Frank Pollmann. Quantum Mutual Information as a Probe for Many-Body Localization. Phys. Rev. Lett., 118 (1): 016804, jan 2017. ISSN 0031-9007. 10.1103/​PhysRevLett.118.016804.

[17] J. Goold, C. Gogolin, S. R. Clark, J. Eisert, A. Scardicchio, and A. Silva. Total correlations of the diagonal ensemble herald the many-body localization transition. Phys. Rev. B, 92 (18): 180202, nov 2015. ISSN 1098-0121. 10.1103/​PhysRevB.92.180202.

[18] P. W. Hess, P. Becker, H. B. Kaplan, A. Kyprianidis, A. C. Lee, B. Neyenhuis, G. Pagano, P. Richerme, C. Senko, J. Smith, W. L. Tan, J. Zhang, and C. Monroe. Non-thermalization in trapped atomic ion spin chains. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375 (2108): 20170107, dec 2017. ISSN 1364-503X. 10.1098/​rsta.2017.0107.

[19] David A Huse, Rahul Nandkishore, and Vadim Oganesyan. Phenomenology of fully many-body-localized systems. Physical Review B, 90 (17): 174202, 2014. 10.1103/​PhysRevB.90.174202.

[20] Fernando Iemini, Angelo Russomanno, Davide Rossini, Antonello Scardicchio, and Rosario Fazio. Signatures of many-body localization in the dynamics of two-site entanglement. Phys. Rev. B, 94 (21): 214206, dec 2016. ISSN 2469-9950. 10.1103/​PhysRevB.94.214206.

[21] John Z. Imbrie, Valentina Ros, and Antonello Scardicchio. Local integrals of motion in many-body localized systems. Ann. Phys., 529 (7): 1600278, jul 2017. ISSN 00033804. 10.1002/​andp.201600278.

[22] Rajibul Islam, Ruichao Ma, Philipp M Preiss, M Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528 (7580): 77, 2015. 10.1038/​nature15750.

[23] F Liu, R Lundgren, P Titum, G Pagano, J Zhang, C Monroe, and A Gorshkov. Confined Quasiparticle Dynamics in Long-Range Interacting Quantum Spin Chains. Phys. Rev. Lett., 122: 150601, 2019. ISSN 1079-7114. 10.1103/​PhysRevLett.122.150601.

[24] David J Luitz and Yevgeny Bar Lev. The ergodic side of the many-body localization transition. Annalen der Physik, 529 (7): 1600350, 2017. 10.1002/​andp.201600350.

[25] David J Luitz, Nicolas Laflorencie, and Fabien Alet. Extended slow dynamical regime close to the many-body localization transition. Phys. Rev. B, 93 (6): 060201, feb 2016. ISSN 2469-9950. 10.1103/​PhysRevB.93.060201.

[26] Alexander Lukin, Matthew Rispoli, Robert Schittko, M. Eric Tai, Adam M. Kaufman, Soonwon Choi, Vedika Khemani, Julian Léonard, and Markus Greiner. Probing entanglement in a many-body–localized system. Science, 364 (6437): 256–260, 2019a. ISSN 0036-8075. 10.1126/​science.aau0818.

[27] Alexander Lukin, Matthew Rispoli, Robert Schittko, M. Eric Tai, Adam M. Kaufman, Soonwon Choi, Vedika Khemani, Julian Léonard, and Markus Greiner. Probing entanglement in a many-body–localized system. Science, 364 (6437): 256–260, 2019b. ISSN 0036-8075. 10.1126/​science.aau0818.

[28] Juan Jose Mendoza-Arenas, M Žnidarič, Vipin Kerala Varma, John Goold, Stephen R Clark, and Antonello Scardicchio. Asymmetry in energy versus spin transport in certain interacting disordered systems. Physical Review B, 99 (9): 094435, 2019. 10.1103/​PhysRevB.99.094435.

[29] Kavan Modi, Tomasz Paterek, Wonmin Son, Vlatko Vedral, and Mark Williamson. Unified view of quantum and classical correlations. Physical review letters, 104 (8): 080501, 2010. 10.1103/​PhysRevLett.104.080501.

[30] Rahul Nandkishore and David A Huse. Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Annu. Rev. Condens. Matter Phys., 6 (1): 15–38, mar 2015. ISSN 1947-5454. 10.1146/​annurev-conmatphys-031214-014726.

[31] F Pietracaprina, C Gogolin, and J Goold. Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems. Phys. Rev. B, 95 (12): 125118, mar 2017. ISSN 2469-9950. 10.1103/​PhysRevB.95.125118.

[32] V Ros, M Müller, and A Scardicchio. Integrals of motion in the many-body localized phase. Nucl. Phys. B, 891: 420–465, feb 2015. ISSN 05503213. 10.1016/​j.nuclphysb.2014.12.014.

[33] Antonello Scardicchio and Thimothée Thiery. Perturbation theory approaches to Anderson and Many-Body Localization: some lecture notes. oct 2017. https:/​/​arxiv.org/​abs/​1710.01234.

[34] Michael Schreiber, Sean S Hodgman, Pranjal Bordia, Henrik P Lüschen, Mark H Fischer, Ronen Vosk, Ehud Altman, Ulrich Schneider, and Immanuel Bloch. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 349 (6250): 842–845, 2015. 10.1126/​science.aaa7432.

[35] Maximilian Schulz, Scott Richard Taylor, Christpher Andrew Hooley, and Antonello Scardicchio. Energy transport in a disordered spin chain with broken u (1) symmetry: Diffusion, subdiffusion, and many-body localization. Physical Review B, 98 (18): 180201, 2018. 10.1103/​PhysRevB.98.180201.

[36] M. Serbyn, M. Knap, S. Gopalakrishnan, Z. Papić, N. Y. Yao, C. R. Laumann, D. A. Abanin, M. D. Lukin, and E. A. Demler. Interferometric Probes of Many-Body Localization. Phys. Rev. Lett., 113 (14): 147204, oct 2014a. ISSN 0031-9007. 10.1103/​PhysRevLett.113.147204.

[37] Maksym Serbyn, Z. Papić, and Dmitry A Abanin. Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems. Phys. Rev. Lett., 110 (26): 260601, jun 2013. ISSN 0031-9007. 10.1103/​PhysRevLett.110.260601.

[38] Maksym Serbyn, Z. Papić, and D. A. Abanin. Quantum quenches in the many-body localized phase. Phys. Rev. B, 90 (17): 174302, nov 2014b. ISSN 1098-0121. 10.1103/​PhysRevB.90.174302.

[39] Roger B Sidje. Expokit: a software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS), 24 (1): 130–156, 1998. 10.1145/​285861.285868.

[40] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe. Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys., 12 (10): 907–911, oct 2016a. ISSN 1745-2473. 10.1038/​nphys3783.

[41] Jacob Smith, Aaron Lee, Philip Richerme, Brian Neyenhuis, Paul W Hess, Philipp Hauke, Markus Heyl, David A Huse, and Christopher Monroe. Many-body localization in a quantum simulator with programmable random disorder. Nature Physics, 12 (10): 907, 2016b. 10.1038/​nphys3783.

[42] V K Varma, A Lerose, F Pietracaprina, J Goold, and A Scardicchio. Energy diffusion in the ergodic phase of a many body localizable spin chain. J. Stat. Mech. Theory Exp., 2017 (5): 053101, may 2017. ISSN 1742-5468. 10.1088/​1742-5468/​aa668b.

[43] Ronen Vosk, David A. Huse, and Ehud Altman. Theory of the Many-Body Localization Transition in One-Dimensional Systems. Phys. Rev. X, 5 (3): 031032, sep 2015. 10.1103/​PhysRevX.5.031032.

[44] Ken Xuan Wei, Chandrasekhar Ramanathan, and Paola Cappellaro. Exploring Localization in Nuclear Spin Chains. Phys. Rev. Lett., 120 (7): 070501, feb 2018a. ISSN 0031-9007. 10.1103/​PhysRevLett.120.070501.

[45] Ken Xuan Wei, Chandrasekhar Ramanathan, and Paola Cappellaro. Exploring localization in nuclear spin chains. Physical review letters, 120 (7): 070501, 2018b. 10.1103/​PhysRevLett.120.070501.

[46] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe. Observation of a discrete time crystal. Nature, 543 (7644): 217–220, mar 2017a. ISSN 0028-0836. 10.1038/​nature21413.

[47] J Zhang, G Pagano, P W Hess, A Kyprianidis, P Becker, H Kaplan, A V Gorshkov, Z.-X Gong, and C Monroe. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature, 551 (7682): 601–604, nov 2017b. ISSN 0028-0836. 10.1038/​nature24654.

[48] Marko Žnidarič, Antonello Scardicchio, and Vipin Kerala Varma. Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System. Phys. Rev. Lett., 117 (4): 040601, jul 2016. ISSN 0031-9007. 10.1103/​PhysRevLett.117.040601.

[49] Marko Žnidarič, Juan Jose Mendoza-Arenas, Stephen R Clark, and John Goold. Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system. Ann. Phys. (Berl.), 529 (7): 1600298, 2017. 10.1002/​andp.201600298.

Cited by

[1] Sheng-Wen Li and C. P. Sun, "Hierarchy recurrences in local relaxation", Physical Review A 103 4, 042201 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-18 05:02:18). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2022-05-18 05:02:18).