Spectral analysis of two-dimensional Bose-Hubbard models

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 twodimensional 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.

Chaos in the Bose–Hubbard model and random two-body Hamiltonians

“…The non-integrability of H resulting from the combination of interaction and tunneling leads to the emergence of a chaotic phase that leaves an imprint in the spectral and eigenvector properties [14,18,52,53,63].…”

“…Understanding the conditions for the emergence of ergodicity [37] or its absence [38], and the ensuing complex dynamics in many-particle quantum systems is currently a topic of intense research, with ultracold atoms in optical potentials as a prominent experimental platform [39][40][41][42][43][44][45][46][47][48][49][50][51], and interacting bosons on regular lattices as a paradigmatic theoretical model, exhibiting signatures of chaos in its spectrum [14,18,52,53], its eigenvectors [14,21,52,[54][55][56] and its time evolution [57][58][59][60][61][62].…”

Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems

“…These include (i) an analysis of level fluctuations using GOE and related ensembles in trapped interacting boson systems, diffuse van der Walls clusters and molecular resonances in erbium isotopes; and (ii) statistical relaxation in interaction quench dynamics of ultra-cold bosons and thermalization in isolated quantum systems. Before turning to these, let us add that there have also been important studies on cold atoms using RMT by the Heidelberg group [ 84 , 85 , 86 , 87 ].…”

Chaos and ergodicity across the energy spectrum of interacting bosons

“…Since the majority of dynamical systems features mixed rather than strictly chaotic dynamics [12][13][14][15][16][17], one therefore expects detectable deviations from RMT ergodicity [18,19], in particular at the level of the eigenvectors' structural properties-which reflect the underlying phase space structure [12][13][14][15][16]20]. This holds on the level of single as well as of many-body quantum systems, with engineered ensembles of ultracold atoms [21][22][23][24][25][26] as a modern playground: Notably interacting bosons on a regular lattice provide a paradigmatic experimental setting to explore the questions above [27][28][29][30][31]; they feature chaos on the level of spectral [32][33][34][35] and eigenvector properties [32,33,[36][37][38][39] as well as quench dynamics [40][41][42][43][44].…”