Rise and fall of hidden string order of lattice bosons

TL;DR

This paper explores the properties of a novel phase called the Haldane Insulator in one-dimensional lattice bosons, revealing how local probes can detect non-local order and how symmetry-breaking perturbations and inter-chain couplings influence phase transitions.

Contribution

It introduces the concept of the Haldane Insulator phase in lattice bosons, analyzes its non-local order parameter, and studies the effects of symmetry-breaking and inter-chain coupling on phase transitions.

Findings

String order can be probed with local fields via quantum phase transitions.

Breaking inversion symmetry gaps the critical point between Mott and Haldane phases.

Inter-chain tunneling destroys direct Mott-Haldane transition, creating an intermediate superfluid phase.

Abstract

We investigate the ground state properties of a newly discovered phase of one dimensional lattice bosons with extended interactions (see E. G. Dalla Torre et al., Phys. Rev. Lett. \textbf{97}, 260401 (2006)). The new phase, termed the Haldane Insulator (HI) in analogy with the gapped phase of spin-1 chains, is characterized by a non local order parameter, which can only be written as an infinite string in terms of the bosonic densities. We show that the string order can nevertheless be probed with physical fields that couple locally, via the effect those fields have on the quantum phase transitions separating the exotic phase from the conventional Mott and density wave phases. Using a field theoretical analysis we show that a perturbation which breaks lattice inversion symmetry gaps the critical point separating the Mott and Haldane phases and eliminates the sharp distinction between them. This is remarkable given that neither of these phases involves broken inversion symmetry. We also investigate the evolution of the phase diagram with the tunable coupling between parallel chains in an optical lattice setup. We find that inter-chain tunneling destroys the direct phase transition between the Mott and Haldane insulators by establishing an intermediate superfluid phase. On the other hand coupling the chains only by weak repulsive interactions does not modify the structure of the phase diagram. The theoretical predictions are confirmed with numerical calculations using the Density Matrix Renormalization Group (DMRG).