Cold bosons in optical lattices: a tutorial for exact diagonalization

Abstract

Exact diagonalization (ED) techniques are a powerful method for studying many-body problems. Here, we apply this method to systems of few bosons in an optical lattice, and use it to demonstrate the emergence of interesting quantum phenomena such as fragmentation and coherence. Starting with a standard Bose-Hubbard Hamiltonian, we first revise the characterisation of the superfluid to Mott insulator (MI) transitions. We then consider an inhomogeneous lattice, where one potential minimum is made much deeper than the others. The MI phase due to repulsive on-site interactions then competes with the trapping of all atoms in the deep potential. Finally, we turn our attention to attractively interacting systems, and discuss the appearance of strongly correlated phases and the onset of localisation for a slightly biased lattice. The article is intended to serve as a tutorial for ED of Bose-Hubbard models.

Superfluid weight and polarization amplitude in the one-dimensional bosonic Hubbard model

“…An important aspect of the method we use is the indexing of the states, which is based on the Lehmer combinatorial code [45,46], a way to order permutations. This procedure is also called Ponomarev ordering [47][48][49] and has been implemented in the BHM by Raventos et al [50].…”

Various topological Mott insulators and topological bulk charge pumping in strongly-interacting boson system in one-dimensional superlattice

“…We identify the TMIs by using the exact diagonalization [22,23,30], which is an efficient method to study the bulk properties of the system. Figure 2 (3), we employed the methods proposed in [31] and used the discretized adiabatic parameter space with N×N mesh.…”

“…In particular, strongly-correlated bosonic topological states with high Chern number in the bulk have not been clarified yet in a global parameter regime, nor it is understood well how the competition between the superlattice potential and the on-site repulsion determines the ground state of the system. Also, there is one important question, i.e., how the topological phases in the obtained phase diagrams are related with the topological charge pumping as bulk topological properties.In this paper, we shall study the above problems in the strongly-interacting boson system by using the exact diagonalization [22,23], and show explicitly relation between the equilibrium topological phases and the topological charge pumping in the adiabatic process by following [24]. From the relation, the global phase diagram plays a role of a guide for detecting various topological charge pumping in the experiments.This paper is organized as follows.…”

“…In this paper, we shall study the above problems in the strongly-interacting boson system by using the exact diagonalization [22,23], and show explicitly relation between the equilibrium topological phases and the topological charge pumping in the adiabatic process by following [24]. From the relation, the global phase diagram plays a role of a guide for detecting various topological charge pumping in the experiments.…”

Topological order versus many-body localization in periodically modulated spin chains

“…In particular, we study robustness of the ground-state topological properties against the LP, and how the WS localization influences the topological ground state. In this paper, we mainly use a numerical exact diagonalization (ED) [23][24][25][26], since we need to investigate properties of the whole energy eigenstates. Numerically, we will clarify the phase diagram in the presence of the LP and on-site disorder, by the energy-level statistics [27,28] to detect the WS localization, and investigate in detail the behavior of the topological edge modes under the LP.…”