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arXiv:2512.16364 · Physical review. B./Physical review. B

Finite-temperature quantum rotor approach for ultracold bosons in optical lattices

M. Rodríguez Martín, T. A. Zaleski

Published Nov 18, 2025• 0 citations
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Abstract

Interacting bosons in optical lattices directly expose quantum phases in a clean, highly controllable environment. This requires engineering systems with very low entropies, but the resulting temperature--interaction ratios $T/U$ of present experiments remain well above the domain where zero-temperature theories are expected to be reliable. The quantum-rotor approach (QRA), while analytically powerful and extremely flexible, inherits ground-state phase correlations and therefore breaks down once thermal winding of the phase field becomes significant. Here we construct a finite-temperature extension of QRA by (i) performing resummation of winding-number contributions for temperatures ${k}_{B}T/U\ensuremath{\lesssim}0.2$ and (ii) developing an auxiliary-variable expansion that remains accurate toward the classical limit. The resulting closed expression for the phase correlator is inserted into the standard spherical-approximation QRA without sacrificing the method's flexibility with respect to lattice geometry and dimensionality. The approach reproduces the shrinkage of Mott lobes from $T=0$ up to ${k}_{B}T/U\ensuremath{\simeq}0.2$ in quantitative agreement with theoretical predictions and with in-situ imaging experiments. This finite-T QRA thus supplies an analytic, computationally light tool for strongly correlated lattice bosons and sets the stage for amplitude-fluctuation upgrades required at higher temperatures.

DOI
10.1103/nmz6-87l6

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Interacting bosons in optical lattices directly expose quantum phases in a clean, highly controllable environment. This requires engineering systems with very low entropies, but the resulting temperature--interaction ratios T/UT/UT/U of present experiments remain well above the domain where zero-temperature theories are expected to be reliable. The quantum-rotor approach (QRA), while analytically powerful and extremely flexible, inherits ground-state phase correlations and therefore breaks down once thermal winding of the phase field becomes significant. Here we construct a finite-temperature extension of QRA by (i) performing resummation of winding-number contributions for temperatures kBT/U\ensuremath≲0.2{k}_{B}T/U\ensuremath{\lesssim}0.2kB​T/U\ensuremath≲0.2 and (ii) developing an auxiliary-variable expansion that remains accurate toward the classical limit. The resulting closed expression for the phase correlator is inserted into the standard spherical-approximation QRA without sacrificing the method's flexibility with respect to lattice geometry and dimensionality. The approach reproduces the shrinkage of Mott lobes from T=0T=0T=0 up to kBT/U\ensuremath≃0.2{k}_{B}T/U\ensuremath{\simeq}0.2kB​T/U\ensuremath≃0.2 in quantitative agreement with theoretical predictions and with in-situ imaging experiments. This finite-T QRA thus supplies an analytic, computationally light tool for strongly correlated lattice bosons and sets the stage for amplitude-fluctuation upgrades required at higher temperatures.
We discuss the finite-temperature phase diagram in the three-dimensional Bose–Hubbard (BH) model in the strong correlation regime, relevant for Bose–Einstein condensates in optical lattices, by employing a quantum rotor approach. In systems with strong on-site repulsive interactions, the rotor U (1) phase variable dual to the local boson density emerges as an important collective field. After establishing the connection between the rotor construction and the on-site interaction in the BH model a robust effective action formalism is developed which allows us to study the superfluid phase transition in various temperature-interaction regimes.
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Abstract The physics of the Bose–Hubbard (BH) model is the subject of intensive studies in recent years, since it has been realized that BH Hamiltonian can be applied to systems of cold atoms confined in periodic optical lattice potential, where the effects on interactions are strongly enhanced. Our aim is to extend the widely used mean‐field treatment of the superfluid–Mott insulator (SF–MI) transition in the BH model. Our method also improves the strong‐coupling expansion that works well only for sufficiently large insulating gap. The key point of the approach is to consider the representation of strongly interacting bosons as particles with attached U(1) gauge group flux tubes, which constitutes the quantum rotor description. The effective action formalism allows us to cast the problem in terms of the phaseonly action and obtain an analytical formulas for critical lines. Finally, we calculate the zero‐temperature SF–MI phase diagrams for three‐dimensional BH model and compare our results with the outcome of numerical Monte Carlo simulations. We found a very good agreement for the quantitative results regarding the details of the lobe in the phase diagrams showing the quantum transition from superfluid to the Mott insulating phase (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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