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arXiv:2506.03629 · Physical review. B./Physical review. B

One-dimensional Bose-Hubbard model with long-range hopping

E. Orignac

Published Jul 14, 2025• 2 citations
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Abstract

Interacting one-dimensional bosons with long range hopping decaying as a power law $r^{-α}$ with distance $r$ are considered with the renormalization group and the self-consistent harmonic approximation. For $α\ge 3$, the ground state is always a Tomonaga-Luttinger liquid, whereas for $α<3$, 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 $α<2$, and for $2<α<3$, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.

DOI
10.1103/sn2d-l7xr

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We study the superfluid--Bose-glass transition in a one-dimensional lattice boson model with power-law decaying hopping amplitude ${t}_{i\ensuremath{-}j}\ensuremath{\sim}1/{|i\ensuremath{-}j|}^{\ensuremath{\alpha}}$, using bosonization and the nonperturbative functional renormalization group (FRG). When $\ensuremath{\alpha}$ is smaller than a critical value ${\ensuremath{\alpha}}_{c}<3$, the U(1) symmetry is spontaneously broken, which leads to a density mode with nonlinear dispersion and dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}1)/2$; the superfluid phase is then stable for sufficiently weak disorder, contrary to the case of short-range hopping where disorder is a relevant perturbation when the Luttinger parameter is smaller than 3/23/23/2. In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion (z=1z=1z=1) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.
We study one-dimensional fermionic and bosonic gases with repulsive power-law interactions 1/|x|(β), with β>1, in the framework of Tomonaga-Luttinger liquid (TLL) theory. We obtain an accurate analytical expression linking the TLL parameter to the microscopic Hamiltonian, for arbitrary β and strength of the interactions. In the presence of a small periodic potential, power-law interactions make the TLL unstable towards the formation of a cascade of lattice solids with fractional filling, a "Luttinger staircase." Several of these quantum phases and phase transitions are realized with ground state polar molecules and weakly bound magnetic Feshbach molecules.
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