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arXiv:2304.11260 · Nuclear Physics B

Contour-time approach to the disordered Bose-Hubbard model in the strong coupling regime

Ali Mokhtari-Jazi, Matthew Fitzpatrick, Malcolm P. Kennett

Published Nov 8, 2023• 5 citations
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Abstract

There has been considerable interest in the disordered Bose Hubbard model (BHM) in recent years, particularly in the context of thermalization and many-body localization. We develop a two-particle irreducible (2PI) strong-coupling approach to the disordered BHM that allows us to treat both equilibrium and out-of-equilibrium situations. We obtain equations of motion for spatio-temporal correlations and explore their equilibrium solutions. We study the equilibrium phase diagram as a function of disorder strength and discuss applications of the formalism to out-of-equilibrium situations. We also note that the disorder strengths where the emergence of non-ergodic dynamics was observed in a recent experiment [Choi et al. Science 352, 1547 (2016)] appear to correspond to the Mott insulator – Bose Glass phase boundary.

DOI
10.1016/j.nuclphysb.2023.116386

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There has been considerable interest in the disordered Bose Hubbard model (BHM) in recent years, particularly in the context of thermalization and many-body localization. We develop a two-particle irreducible (2PI) strong-coupling approach to the disordered BHM that allows us to treat both equilibrium and out-of-equilibrium situations. We obtain equations of motion for spatio-temporal correlations and explore their equilibrium solutions. We study the equilibrium phase diagram as a function of disorder strength and discuss applications of the formalism to out-of-equilibrium situations. We also note that the disorder strengths where the emergence of non-ergodic dynamics was observed in a recent experiment [Choi et al. Science 352, 1547 (2016)] appear to correspond to the Mott insulator – Bose Glass phase boundary.
We develop a formalism that allows the study of correlations in space and time in both the superfluid and Mott insulating phases of the Bose–Hubbard Model. Specifically, we obtain a two particle irreducible effective action within the contour-time formalism that allows for both equilibrium and out of equilibrium phenomena. We derive equations of motion for both the superfluid order parameter and two-point correlation functions. To assess the accuracy of this formalism, we study the equilibrium solution of the equations of motion and compare our results to existing strong coupling methods as well as exact methods where possible. We discuss applications of this formalism to out of equilibrium situations.
Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years.These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time.This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically.Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally.The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model.The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases.In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM.Specifically, we obtain a two-particle irreducible (2PI) effective action within the contourtime formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena.We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions.First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible.We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches.We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix.We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions.We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality.We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible.Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function.We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.
We study the out-of-equilibrium dynamics of the Bose-Hubbard model for two-component bosons using a strong-coupling approach within the closed-time-path formalism and develop an effective theory for the action of this problem. We obtain equations of motion for the superfluid order parameters of both boson species for both the superfluid and Mott-insulating phases and study these in the low-frequency, long-wavelength limit during a quantum quench for various initial conditions. We find that an additional degree of freedom for bosons leads to a richer phase diagram and out-of-equilibrium dynamics than the single-component situation.
Lieb-Robinson and related bounds set an upper limit on the speed at which information propagates in nonrelativistic quantum systems. Experimentally, light-cone-like spreading has been observed for correlations in the Bose-Hubbard model (BHM) after a quantum quench. Using a two-particle irreducible (2PI) strong-coupling approach to out-of-equilibrium dynamics in the BHM we calculate both the group and phase velocities for the spreading of single-particle correlations in one, two, and three dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the speed of spreading of single-particle correlations in both the one- and two-dimensional BHM realized with ultracold atoms. They also are consistent with the claim that the phase velocity rather than the group velocity was observed in recent experiments in two dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and explore how the anisotropy in the velocity varies across the phase diagram of the BHM. Our results establish the 2PI strong-coupling approach as a powerful tool to study out-of-equilibrium dynamics in the BHM in dimensions greater than one.
The question of how the low-energy properties of disordered quantum systems may be connected to exotic localization phenomena at high energy is a key open question in the context of quantum glasses and many-body localization. In the preceding Letter [L. Villa, S. J. Thomson, and L. Sanchez-Palencia, preceding Letter, Phys. Rev. A 104, L021301 (2021)] we have shown that key features of the excitation spectrum of a disordered system can be efficiently probed from out-of-equilibrium dynamics following a quantum quench, providing distinctive signatures of the various phases. Here we extend this work by providing a more-in-depth study of the behavior of the quench spectral functions associated with different observables and investigating an extended parameter regime. We provide a detailed introduction to quench spectroscopy for disordered systems and show how spectral properties can be probed using both local operators and two-point correlation functions. We benchmark the technique using the one-dimensional Bose-Hubbard model in the presence of a random external potential, focusing on the low-lying excitations, and demonstrate that quench spectroscopy can distinguish the Mott insulator, superfluid, and Bose glass phases. We then explicitly reconstruct the zero-temperature phase diagram of the disordered Bose-Hubbard at fixed filling using two independent methods, experimentally accessible via both time-of-flight imaging and quantum gas microscopy, respectively, and demonstrate that quench spectroscopy can give valuable insights into the distribution of rare regions within disordered systems.
We numerically investigate the two-dimensional Bose-Hubbard model with local onsite disorder, where the competition between disorder and short-range interactions leads to the emergence of a Bose glass (BG) phase between the Mott insulator (MI) and superfluid (SF) phases. In order to analyze the inhomogeneous system we employ real-space bosonic dynamical mean-field theory (RBDMFT) and perform an ensemble average over disorder realizations. To distinguish the MI from the BG phase, we compare the Edwards-Anderson order parameter and the compressibility with the energy-gap condition. To identify the insulator to SF transition, we apply a percolation analysis to the condensate order parameter. In qualitative accordance with the theorem of inclusions we always find an intermediate BG phase between the SF and MI. However, the quantitative comparison indicates significant deviations between the MI to BG phase boundary expected in the thermodynamic limit and the one obtained for a finite system size. Additionally, RBMDFT is capable of reliably calculating spectral information throughout the phase diagram. Analyzing the spectral function reveals evidence for analytically predicted damped localized modes in the dispersion relation in the strong-coupling regime.