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arXiv:1604.05664 · Physical Review A

Spectral analysis of two-dimensional Bose-Hubbard models

David Fischer, Darius Hoffmann, Sandro Wimberger

Published Apr 22, 2016• 16 citations
arXivHTMLPDF

Abstract

One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple two-dimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest-neighbor couplings at each site.

DOI
10.1103/physreva.93.043620

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One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple two-dimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest-neighbor couplings at each site.
Motivated by the role that spectral properties play for the dynamical evolution of a quantum many-body system, we investigate the level spacing statistics of the extended Bose–Hubbard model. In particular, we focus on the distribution of the ratio of adjacent level spacings, useful at large interactions, to distinguish between chaotic and non-chaotic regimes. After revisiting the bare Bose–Hubbard model, we study the effect of two different perturbations: next-nearest-neighbor hopping and nearest-neighbor interaction. The system size dependence is investigated together with the effect of the proximity to integrable points or lines. Lastly, we discuss the consequences of a cutoff in the number of onsite bosons on the level statistics.
In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.
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.
We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor K(t) analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, t_{Th}. We obtain a striking dependence of t_{Th} on the spatial dimension d and size of the system. For d>1, t_{Th} is finite in the thermodynamic limit and set by the intersite coupling strength. By contrast, in one dimension t_{Th} diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form. Lastly, our Floquet model exhibits a many-body localization transition, and we discuss the behavior of the spectral form factor in the localized phase.
Abstract This paper is a first step toward rigorously establishing the validity of mean-field approximations for the dynamics of quantum systems in a large dimensional limit. Our result is a derivation of the large d mean-field limit of the Bose–Hubbard model on a square lattice. The Bose–Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator phase transition. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.