Dirty bosons on the Cayley tree: Bose-Einstein condensation versus ergodicity breaking

Abstract

Building on large-scale quantum Monte Carlo simulations, we investigate the zero-temperature phase diagram of hard-core bosons in a random potential on site-centered Cayley trees with branching number K = 2. In order to follow how the Bose-Einstein condensate (BEC) is affected by the disorder, we focus on both the zero-momentum density, probing the quantum coherence, and the one-body density matrix (1BDM) whose largest eigenvalue monitors the off-diagonal long-range order. We further study its associated eigenstate which brings useful information about the real-space properties of this leading eigenmode. Upon increasing randomness, we find that the system undergoes a quantum phase transition at finite disorder strength between a long-range ordered BEC state, fully ergodic at large scale, and a new disordered Bose glass regime showing conventional localization for the coherence fraction while the 1BDM displays a non-trivial algebraic vanishing BEC density together with a non-ergodic occupation in real-space. These peculiar properties can be analytically captured by a simple toy-model on the Cayley tree which provides a physical picture of the Bose glass regime.

The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid

“…Solutions on the finite coordination number Bethe lattice provide a better approximation to thermodynamic quantities than the mean-field approximation (corresponding to the infinite dimensional limit) [26][27][28]. Models studied on the Bethe lattice include classical and quantum spin models [29][30][31][32][33][34][35], spin glass systems [36][37][38][39], the Bose Hubbard model [40], and models of Anderson localization [41][42][43][44][45]. The fermionic Hubbard model on the finite version of the z = 3 Bethe lattice (known as a Cayley tree) has also been studied previously using a variant of the density matrix renormalization group (DMRG) algorithm [46], but only the case of half filling was studied (which is a charge insulator) and only local ground state quantities given (energy, staggered magnetization and its fluctuations, and neighboring spin correlations).…”