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

TL;DR

The paper addresses the finite-temperature limitations of the quantum rotor approach for Bose-Hubbard systems in optical lattices. It develops two complementary analytic schemes—the winding-number expansion and an auxiliary-variable expansion—to obtain a closed-form phase correlator $\gamma(\omega_m)$ compatible with the spherical approximation, enabling a practical, geometry-agnostic treatment of thermal fluctuations. The resulting framework reproduces the melting of Mott lobes starting around $k_B T/U \sim 0.1$ and complete washout by about $0.2$, while demonstrating that the zero-temperature QRA can significantly misestimate phase boundaries even at modest temperatures. This finite-T QRA provides a lightweight analytic tool for strongly correlated lattice bosons and sets the stage for incorporating amplitude (Higgs) fluctuations at higher temperatures and for calculating a broad range of observables.

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 $k_{B}T/U\lesssim 0.2$ 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 $k_{B}T/U\simeq 0.2$ 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.

Figures (6)

• Figure 1: Real (top) and imaginary (bottom) parts of the phase correlator: numerical result (solid line) vs. two-term winding-number approximation (colored symbols at Matsubara frequencies) for $\mu/U=0.4$. Each panel shows two temperatures: a lower one with good agreement (right vertical axis) and a higher one where deviations become noticeable (left vertical axis).
• Figure 2: Real (top) and imaginary (bottom) parts of the phase correlator: numerical result (solid line) vs. two-term auxiliary-variable approximation (colored symbols at Matsubara frequencies) for $\mu/U=0.4$. Each panel shows two temperatures. In the top panel, both temperatures display excellent agreement, indicating that the approximation remains accurate even at high temperature. In the bottom panel, the lower temperature still shows good agreement (right vertical axis), while the higher temperature illustrates the onset of deviation (left vertical axis).
• Figure 3: Decrease of the relative error in the estimation of $\gamma\left(\omega_{m}=0\right)$ in Eq. (\ref{['eq:omega0']}) with the numerically evaluated value of Eq. (\ref{['eq:corr']}) for summation over $n=-300,\dots,300$ while the temperature is being increased.
• Figure 4: Phase diagram at different temperatures.
• Figure 5: Comparison between the critical hopping parameter at $\mu/U=1$ for the zero temperature approximation and the winding number expansion with two terms.