One dimensional Bose-Hubbard model with long range hopping

TL;DR

This paper investigates the phase behavior of one-dimensional bosons with long-range hopping, revealing conditions under which different quantum phases and symmetry breakings occur depending on the decay exponent and temperature.

Contribution

It provides a detailed analysis of the ground state and finite temperature phases of the long-range Bose-Hubbard model using renormalization group and harmonic approximation methods.

Findings

Ground state is a Tomonaga-Luttinger liquid for $eta ext{ } extgreater{} ext{ }3$

Long range order with symmetry breaking appears for $eta ext{ } extless{} ext{ }3$ at weak repulsion

At positive temperature, symmetry breaking is limited to $eta ext{ } extless{} ext{ }2$

Abstract

Interacting one-dimensional bosons with long range hopping decaying as a power law $r^{-\alpha}$ with distance $r$ are considered with the renormalization group and the self-consistent harmonic approximation. For $\alpha\ge 3$, the ground state is always a Tomonaga-Luttinger liquid, whereas for $\alpha <3$, a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to $\alpha<2$, and for $2<\alpha<3$, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.