Eigenstate properties of the disordered Bose−Hubbard chain

Abstract

Many-body localization (MBL) of a disordered interacting boson system in one dimension is studied numerically at the filling faction one-half. The von Neumann entanglement entropy S_{\mathrm{v}\mathrm{N}} is commonly used to detect the MBL phase transition but remains challenging to be directly measured. Based on the U(1) symmetry from the particle number conservation, S_{\mathrm{v}\mathrm{N}} can be decomposed into the particle number entropy S_N and the configuration entropy S_C. In light of the tendency that the eigenstate’s S_C nears zero in the localized phase, we introduce a quantity describing the deviation of S_N from the ideal thermalization distribution; finite-size scaling analysis illustrates that it shares the same phase transition point with S_{\mathrm{v}\mathrm{N}} but displays the better critical exponents. This observation hints that the phase transition to MBL might largely be determined by S_N and its fluctuations. Notably, the recent experiments [A. Lukin, et al., Science 364, 256 (2019); J. Léonard, et al., Nat. Phys. 19, 481 (2023)] demonstrated that this deviation can potentially be measured through the S_N measurement. Furthermore, our investigations reveal that the thermalized states primarily occupy the low-energy section of the spectrum, as indicated by measures of localization length, gap ratio, and energy density distribution. This low-energy spectrum of the Bose model closely resembles the entire spectrum of the Fermi (or spin XXZ) model, accommodating a transition from the thermalized to the localized states. While, owing to the bosonic statistics, the high-energy spectrum of the model allows the formation of distinct clusters of bosons in the random potential background. We analyze the resulting eigenstate properties and briefly summarize the associated dynamics. To distinguish between the phase regions at the low and high energies, a probing quantity based on the structure of S_{\mathrm{v}\mathrm{N}} is also devised. Our work highlights the importance of symmetry combined with entanglement in the study of MBL. In this regard, for the disordered Heisenberg XXZ chain, the recent pure eigenvalue analyses in [J. Šuntajs, et al., Phys. Rev. E 102, 062144 (2020)] would appear inadequate, while methods used in [A. Morningstar, et al., Phys. Rev. B 105, 174205 (2022)] that spoil the U(1) symmetry could be misleading.

Graphical abstract

Keywords

entanglement entropy decomposition / U(1) symmetry / thermalization-to-localization transition

References

[1] P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 1958, 109(5): 1492

[2] E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan. Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett., 1979, 42(10): 673

[3] D. Vollhardt, P. Wölfle. Diagrammatic, self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions. Phys. Rev. B, 1980, 22(10): 4666

[4] S. John. Electromagnetic absorption in a disordered medium near a photon mobility edge. Phys. Rev. Lett., 1984, 53(22): 2169

[5] K. Arya, Z. B. Su, J. L. Birman. Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena. Phys. Rev. Lett., 1985, 54(14): 1559

[6] Q. J. Chu, Z. Q. Zhang. Localization of phonons in mixed crystals. Phys. Rev. B, 1988, 38(7): 4906

[7] R. Nandkishore, D. A. Huse. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys., 2015, 6(1): 15

[8] D. M. Basko, I. L. Aleiner, B. L. Altshuler. Metal–insulator transition in a weakly interacting many electron system with localized single-particle states. Ann. Phys., 2006, 321(5): 1126

[9] V. Oganesyan, D. A. Huse. Localization of interacting fermions at high temperature. Phys. Rev. B, 2007, 75(15): 155111

[10] M. Žnidarič, T. Prosen, P. Prelovšek. Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B, 2008, 77(6): 064426

[11] J. M. Deutsch. Quantum statistical mechanics in a closed system. Phys. Rev. A, 1991, 43(4): 2046

[12] M. Srednicki. Chaos and quantum thermalization. Phys. Rev. E, 1994, 50(2): 888

[13] M. Serbyn, Z. Papić, D. A. Abanin. Local conservation laws and the structure of the many-body localized states. Phys. Rev. Lett., 2013, 111(12): 127201

[14] A. Chandran, I. H. Kim, G. Vidal, D. A. Abanin. Constructing local integrals of motion in the many-body localized phase. Phys. Rev. B, 2015, 91(8): 085425

[15] S. D. Geraedts, R. N. Bhatt, R. Nandkishore. Emergent local integrals of motion without a complete set of localized eigenstates. Phys. Rev. B, 2017, 95(6): 064204

[16] L. D’Alessio, Y. Kafri, A. Polkovnikov, M. Rigol. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys., 2016, 65(3): 239

[17] A. Lazarides, A. Das, R. Moessner. Fate of many-body localization under periodic driving. Phys. Rev. Lett., 2015, 115(3): 030402

[18] P. Ponte, Z. Papić, F. Huveneers, D. A. Abanin. Many-body localization in periodically driven systems. Phys. Rev. Lett., 2015, 114(14): 140401

[19] D. V. Else, B. Bauer, C. Nayak. Floquet time crystals. Phys. Rev. Lett., 2016, 117(9): 090402

[20] N. Y. Yao, A. C. Potter, I. D. Potirniche, A. Vishwanath. Discrete time crystals: Rigidity, criticality, and realizations. Phys. Rev. Lett., 2017, 118(3): 030401

[21] 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, C. Monroe. Observation of a discrete time crystal. Nature, 2017, 543(7644): 217

[22] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, M. D. Lukin. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature, 2017, 543(7644): 221

[23] J. A. Kjäll, J. H. Bardarson, F. Pollmann. Many-body localization in a disordered quantum Ising chain. Phys. Rev. Lett., 2014, 113(10): 107204

[24] M. Brenes, M. Dalmonte, M. Heyl, A. Scardicchio. Many-body localization dynamics from gauge invariance. Phys. Rev. Lett., 2018, 120(3): 030601

[25] E. Levi, M. Heyl, I. Lesanovsky, J. P. Garrahan. Robustness of many-body localization in the presence of dissipation. Phys. Rev. Lett., 2016, 116(23): 237203

[26] M. H. Fischer, M. Maksymenko, E. Altman. Dynamics of a many-body-localized system coupled to a bath. Phys. Rev. Lett., 2016, 116(16): 160401

[27] H. P. Lüschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, U. Schneider. Signatures of many-body localization in a controlled open quantum system. Phys. Rev. X, 2017, 7(1): 011034

[28] J. Ren, Q. Li, W. Li, Z. Cai, X. Wang. Noise-driven universal dynamics towards an infinite temperature state. Phys. Rev. Lett., 2020, 124(13): 130602

[29] M. Žnidarič, A. Scardicchio, V. K. Varma. Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system. Phys. Rev. Lett., 2016, 117(4): 040601

[30] S. P. Kelly, R. Nandkishore, J. Marino. Exploring many-body localization in quantum systems coupled to an environment via Wegner‒Wilson flows. Nucl. Phys. B, 2020, 951: 114886

[31] C. Chamon, A. Hamma, E. R. Mucciolo. Emergent irreversibility and entanglement spectrum statistics. Phys. Rev. Lett., 2014, 112(24): 240501

[32] C. R. Laumann, A. Pal, A. Scardicchio. Many-body mobility edge in a mean-field quantum spin glass. Phys. Rev. Lett., 2014, 113(20): 200405

[33] D. J. Luitz, N. Laflorencie, F. Alet. Many-body localization edge in the random-field Heisenberg chain. Phys. Rev. B, 2015, 91: 081103(R)

[34] I. Mondragon-Shem, A. Pal, T. L. Hughes, C. R. Laumann. Many-body mobility edge due to symmetry-constrained dynamics and strong interactions. Phys. Rev. B, 2015, 92(6): 064203

[35] R. Mondaini, Z. Cai. Many-body self-localization in a translation-invariant Hamiltonian. Phys. Rev. B, 2017, 96(3): 035153

[36] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, I. Bloch. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 2015, 349(6250): 842

[37] J. H. Bardarson, F. Pollmann, J. E. Moore. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett., 2012, 109(1): 017202

[38] M. Serbyn, Z. Papić, D. A. Abanin. Universal slow growth of entanglement in interacting strongly disordered systems. Phys. Rev. Lett., 2013, 110(26): 260601

[39] A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, M. Greiner. Probing entanglement in a many-body-localized system. Science, 2019, 364(6437): 256

[40] J. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, C. Gross. Exploring the many-body localization transition in two dimensions. Science, 2016, 352(6293): 1547

[41] Q. Guo, C. Cheng, Z. H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y. R. Zhang, R. Mondaini, H. Fan, H. Wang. Observation of energy-resolved many-body localization. Nat. Phys., 2021, 17(2): 234

[42] M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, M. Greiner. Quantum critical behaviour at the many-body localization transition. Nature, 2019, 573(7774): 385

[43] J. Léonard, S. Kim, M. Rispoli, A. Lukin, R. Schittko, J. Kwan, E. Demler, D. Sels, M. Greiner. Probing the onset of quantum avalanches in a many-body localized system. Nat. Phys., 2023, 19(4): 481

[44] J. M. Zhang, R. X. Dong. Exact diagonalization: The Bose–Hubbard model as an example. Eur. J. Phys., 2010, 31(3): 591

[45] S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, C. Hubig. Time-evolution methods for matrix-product states. Ann. Phys., 2019, 411: 167998

[46] G. Vidal. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett., 2004, 93(4): 040502

[47] M. P. Kennett. Out-of-equilibrium dynamics of the Bose–Hubbard model. ISRN Cond. Matter Phys., 2013, 393616: 39

[48] P. Sengupta, S. Haas. Quantum glass phases in the disordered Bose–Hubbard model. Phys. Rev. Lett., 2007, 99(5): 050403

[49] V. Gurarie, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, M. Troyer. Phase diagram of the disordered Bose–Hubbard model. Phys. Rev. B, 2009, 80(21): 214519

[50] L. Pollet, N. V. Prokof’ev, B. V. Svistunov, M. Troyer. Absence of a direct superfluid to Mott insulator transition in disordered Bose systems. Phys. Rev. Lett., 2009, 103(14): 140402

[51] M. Gerster, M. Rizzi, F. Tschirsich, P. Silvi, R. Fazio, S. Montangero. Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model. New J. Phys., 2016, 18(1): 015015

[52] S. Hu, Y. Wen, Y. Yu, B. Normand, X. Wang. Quantized squeezing and even‒odd asymmetry of trapped bosons. Phys. Rev. A, 2009, 80(6): 063624

[53] J. Šuntajs, J. Bonča, T. Prosen, L. Vidmar. Quantum chaos challenges many-body localization. Phys. Rev. E, 2020, 102(6): 062144

[54] A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, D. A. Huse. Avalanches and many-body resonances in many-body localized systems. Phys. Rev. B, 2022, 105(17): 174205

[55] B. Widom. Surface tension and molecular correlations near the critical point. J. Chem. Phys., 1965, 43(11): 3892

[56] T. Vojta. Phases and phase transitions in disordered quantum systems. AIP Conf. Proc., 2013, 1550: 188

[57] T. Orell, A. A. Michailidis, M. Serbyn, M. Silveri. Probing the many-body localization phase transition with superconducting circuits. Phys. Rev. B, 2019, 100(13): 134504

[58] S. X. Zhang, H. Yao. Universal properties of many body localization transitions in quasiperiodic systems. Phys. Rev. Lett., 2018, 121(20): 206601

[59] V. Khemani, D. N. Sheng, D. A. Huse. Two universality classes for the many-body localization transition. Phys. Rev. Lett., 2017, 119(7): 075702

[60] J. Šuntajs, J. Bonča, T. Prosen, L. Vidmar. Ergodicity breaking transition in finite disordered spin chains. Phys. Rev. B, 2020, 102(6): 064207

[61] P. Sierant, D. Delande, J. Zakrzewski. Many-body localization for randomly interacting bosons. Acta Phys. Pol. A, 2017, 132(6): 1707

[62] P. Sierant, D. Delande, J. Zakrzewski. Many-body localization due to random interactions. Phys. Rev. A, 2017, 95: 021601(R)

[63] T. Orell, A. A. Michailidis, M. Serbyn, M. Silveri. Probing the many-body localization phase transition with superconducting circuits. Phys. Rev. B, 2019, 100(13): 134504

[64] Y. Y. Atas, E. Bogomolny, O. Giraud, G. Roux. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Phys. Rev. Lett., 2013, 110(8): 084101

[65] J.ChenC. ChenX.Wang, Energy- and symmetry-resolved entanglement dynamics in disordered Bose−Hubbard chain, arXiv: 2303.14825 (2023)

[66] D. J. Luitz, Y. B. Lev. Absence of slow particle transport in the many-body localized phase. Phys. Rev. B, 2020, 102: 100202(R)

[67] M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, J. Sirker. Evidence for unbounded growth of the number entropy in many-body localized phases. Phys. Rev. Lett., 2020, 124(24): 243601

[68] D. J. Luitz, N. Laflorencie, F. Alet. Extended slow dynamical regime close to the many-body localization transition. Phys. Rev. B, 2016, 93: 060201(R)

[69] T. Kohlert, S. Scherg, X. Li, H. P. Lüschen, S. Das Sarma, I. Bloch, M. Aidelsburger. Observation of many-body localization in a one-dimensional system with a single-particle mobility edge. Phys. Rev. Lett., 2019, 122(17): 170403

[70] K. Agarwal, E. Altman, E. Demler, S. Gopalakrishnan, D. A. Huse, M. Knap. Rareregion effects and dynamics near the many-body localization transition. Ann. Phys., 2017, 529(7): 1600326

[71] V. Khemani, D. N. Sheng, D. A. Huse. Two universality classes for the many-body localization transition. Phys. Rev. Lett., 2017, 119(7): 075702

[72] V. Khemani, S. P. Lim, D. N. Sheng, D. A. Huse. Critical properties of the many-body localization transition. Phys. Rev. X, 2017, 7(2): 021013

[73] S. Schierenberg, F. Bruckmann, T. Wettig. Wigner surmise for mixed symmetry classes in random matrix theory. Phys. Rev. E, 2012, 85(6): 061130

[74] S. D. Geraedts, R. Nandkishore, N. Regnault. Many-body localization and thermalization: Insights from the entanglement spectrum. Phys. Rev. B, 2016, 93(17): 174202

Acknowledgements

We thank Zi Cai, Changfeng Chen, Shijie Hu, Haiqing Lin, Mingpu Qin, Jie Ren, and Wei Su for fruitful discussions. This work is supported by the Ministry of Science and Technology of China (Grant No. 2016YFA0300500) and the National Natural Science Foundation of China (No. 11974244). C. C. acknowledges the support from the SJTU start-up fund and the sponsorship from Yangyang Development fund.

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