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arXiv:2303.14825 · Physical review. B./Physical review. B

Symmetry- and energy-resolved entanglement dynamics in a disordered Bose-Hubbard model

Jie Chen, Chun Chen, Xiaoqun Wang

Published Aug 13, 2025• 2 citations
arXivHTMLPDF

Abstract

The phenomenology of many-body localization (MBL) develops mainly from tackling the one-dimensional spin or fermion systems. The situation when interacting bosons get clustered in a random potential remains less explored. Using numerical quantum quenches with special emphasis on the integration of both symmetry and energy resolutions, we comprehensively study the dynamics of symmetry-resolved entanglement in a disordered Bose-Hubbard (dBH) model, concentrating on the two types of inhomogeneous initial states to target the lower- and higher-energy sections of its dynamical phase diagram. (i) Motivated by the recent experiment [A. Lukin et al., Science 364, 256 (2019)] which focused on the lower-energy dynamical behaviors of the dBH chain, we first show that at low energies, for a thermalizing state, although the second law of thermodynamics prohibits the decrease of the total entropy over time, for part of the channel-resolved entropies, a long-term entropic reduction may arise at weak disorder. (ii) A companion channel-resolved analysis at strong disorder further hints that the prior observed double-logarithmic growth of the number entropy might not directly indicate the breakdown of MBL in spin or fermion chains, providing a refreshing perspective on this major controversy in the community. (iii) From time-evolving the line-shape low-energy product state, we subsequently reveal an abrupt formation of a novel ``entropy imbalance pattern'' across the different symmetry channels. Intriguingly, this imbalance melts in the strong-disorder limit. We conjecture that the melting of the entropic pattern, together with the freezing of a concurrent particle-density wave, embodies a dual trait inherent to MBL. (iv) Conversely, the higher-energy section of the dynamical phase diagram is where the dBH model differs most significantly from the spin or fermion systems. This parametric space was not included in the previous literature. Specifically, we find a cluster MBL regime, unique to the Bose statistics, emerging from the higher-energy section. This cluster MBL regime, realizable even at weak disorder, does not appear to suffer from the finite-size drift and is distinguished by its absence of the hallmark of MBL---the unbounded growth of the entanglement entropy. Our theoretical predictions are, by and large, testable via the present experimental facilities.

DOI
10.1103/88sq-cm27

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The phenomenology of many-body localization (MBL) develops mainly from tackling the one-dimensional spin or fermion systems. The situation when interacting bosons get clustered in a random potential remains less explored. Using numerical quantum quenches with special emphasis on the integration of both symmetry and energy resolutions, we comprehensively study the dynamics of symmetry-resolved entanglement in a disordered Bose-Hubbard (dBH) model, concentrating on the two types of inhomogeneous initial states to target the lower- and higher-energy sections of its dynamical phase diagram. (i) Motivated by the recent experiment [A. Lukin et al., Science 364, 256 (2019)] which focused on the lower-energy dynamical behaviors of the dBH chain, we first show that at low energies, for a thermalizing state, although the second law of thermodynamics prohibits the decrease of the total entropy over time, for part of the channel-resolved entropies, a long-term entropic reduction may arise at weak disorder. (ii) A companion channel-resolved analysis at strong disorder further hints that the prior observed double-logarithmic growth of the number entropy might not directly indicate the breakdown of MBL in spin or fermion chains, providing a refreshing perspective on this major controversy in the community. (iii) From time-evolving the line-shape low-energy product state, we subsequently reveal an abrupt formation of a novel ``entropy imbalance pattern'' across the different symmetry channels. Intriguingly, this imbalance melts in the strong-disorder limit. We conjecture that the melting of the entropic pattern, together with the freezing of a concurrent particle-density wave, embodies a dual trait inherent to MBL. (iv) Conversely, the higher-energy section of the dynamical phase diagram is where the dBH model differs most significantly from the spin or fermion systems. This parametric space was not included in the previous literature. Specifically, we find a cluster MBL regime, unique to the Bose statistics, emerging from the higher-energy section. This cluster MBL regime, realizable even at weak disorder, does not appear to suffer from the finite-size drift and is distinguished by its absence of the hallmark of MBL---the unbounded growth of the entanglement entropy. Our theoretical predictions are, by and large, testable via the present experimental facilities.
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 SvN 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, SvN can be decomposed into the particle number entropy SN and the configuration entropy SC. In light of the tendency that the eigenstate’s SC nears zero in the localized phase, we introduce a quantity describing the deviation of SN from the ideal thermalization distribution; finite-size scaling analysis illustrates that it shares the same phase transition point with SvN but displays the better critical exponents. This observation hints that the phase transition to MBL might largely be determined by SN 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 SN 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 SvN 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. Suntajs, 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.
Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.
We investigate the non-equilibrium dynamics of the disordered one-dimensional Fermi-Hubbard model with a focus on many-body localization. The system is initialized in a charge-density-wave-state, and its time evolution is analyzed through sublattice imbalance (spin and charge), and bipartite entanglement entropy. A clear crossover from ergodic to non-ergodic behavior is observed with increasing disorder strength. In the weak disorder regime, rapid decay of imbalance and the fast growth of entanglement indicate efficient thermalization. In contrast, a strong disorder leads to persistent imbalance and slow dynamics, signaling the breakdown of ergodicity. The charge and spin sectors exhibit distinct relaxation behavior, providing evidence for partial decoupling between these degrees of freedom. Furthermore, in the interacting regime, the entanglement entropy shows slow logarithmic growth, reflecting the dephasing-driven dynamics characteristic of the many-body localized phase. These results highlight the interplay between disorder and interactions in determining the dynamical properties of the system and establish robust signatures of many-body localization in the Fermi-Hubbard model.
We study the potential influence of the particle multi-occupations on the stability of many-body localization in the disordered Bose-Hubbard model. Within the higher-energy section of the dynamical phase diagram, we find that there is no apparent finite-size boundary drift between the thermal phase and the many-body localized regime. We substantiate this observation by introducing the Van Vleck perturbation theory into the field of many-body localization. The appropriateness of this method rests largely on the peculiar Hilbert-space structure enabled by the particles' Bose statistics. The situation is reversed in the lower-energy section of the dynamical phase diagram, where the significant finite-size boundary drift pushes the putative many-body localized regime up to the greater disorder strengths. We utilize the algebraic projection method to make a connection linking the disordered Bose-Hubbard model in the lower-energy section to an intricate disordered spin chain model. This issue of the finite-size drift could hence be analogous to what happens in the disordered Heisenberg chain. Both trends might be traced back to the particles' intrinsic or emergent single-occupancy constraint such as the spin-1/21/21/2, hard-core boson, or spinless fermion degrees of freedom.
The law of statistical physics dictates that generic closed quantum many-body systems initialized in nonequilibrium will thermalize under their own dynamics. However, the emergence of many-body localization (MBL) owing to the interplay between interaction and disorder, which is in stark contrast to Anderson localization, which only addresses noninteracting particles in the presence of disorder, greatly challenges this concept, because it prevents the systems from evolving to the ergodic thermalized state. One critical evidence of MBL is the long-time logarithmic growth of entanglement entropy, and a direct observation of it is still elusive due to the experimental challenges in multiqubit single-shot measurement and quantum state tomography. Here we present an experiment fully emulating the MBL dynamics with a 10-qubit superconducting quantum processor, which represents a spin-1/2 XY model featuring programmable disorder and long-range spin-spin interactions. We provide essential signatures of MBL, such as the imbalance due to the initial nonequilibrium, the violation of eigenstate thermalization hypothesis, and, more importantly, the direct evidence of the long-time logarithmic growth of entanglement entropy. Our results lay solid foundations for precisely simulating the intriguing physics of quantum many-body systems on the platform of large-scale multiqubit superconducting quantum processors.
Disordered quantum systems undergoing a many-body localization (MBL) transition fail to reach thermal equilibrium under their own dynamics. Distinguishing between asymptotically localized or delocalized dynamics based on numerical results is, however, nontrivial due to finite-size effects. Numerical linked cluster expansions (NLCE) provide a means to tackle quantum systems directly in the thermodynamic limit but are challenging for models without translational invariance. Here we demonstrate that NLCE provides a powerful tool to explore MBL by simulating quench dynamics in disordered spin-1/21/21/2 two-leg ladders and Fermi-Hubbard chains. Combining NLCE with an efficient real-time evolution of pure states, we obtain converged results for the decay of the imbalance on long time scales and show that, especially for intermediate disorder below the putative MBL transition, NLCE outperforms direct simulations of finite systems with open or periodic boundaries. Furthermore, while spin is delocalized even in strongly disordered Hubbard chains with frozen charge, we unveil that an additional tilted potential leads to a drastic slowdown of the spin imbalance and nonergodic behavior on accessible times. Our work sheds light on MBL in systems beyond the well-studied disordered Heisenberg chain and emphasizes the usefulness of NLCE for this purpose.