Leggett's bound and superfluidity in strongly interacting bosons

TL;DR

This work probes the accuracy of Leggett's density-based bound for the superfluid fraction in strongly interacting bosons across a 2D-1D dimensional crossover at low temperature. By combining ab initio quantum Monte Carlo, Gross-Pitaevskii, and field-theory analyses (including SCHA), the authors show that the Leggett bound remains a reliable estimator for the transverse superfluid fraction $f_s^y$ even beyond mean-field, and they elucidate the scaling with interchain coupling via a Tomonaga-Luttinger framework. They also identify two counterexamples where the bound loses predictive power, highlighting that the bound tracks superfluid suppression only when the mechanism is tied to density modulation. The results have practical implications for experimental probes of superfluidity in cold-atom systems and delineate the bound's limits in strongly correlated and quasi-1D regimes.

Abstract

A density-based superfluid bound called Leggett's bound has been proved to be a good estimator of the superfluid fraction for cold atomic gases in the mean-field regime. Here, we investigate the accuracy of such bound in the strongly interacting regime, where the mean-field approach fails. Combining quantum Monte Carlo, Gross-Pitaevskii equation and field-theory calculations, we demonstrate that the bound serves as a reliable estimator of the superfluid fraction for strongly interacting bosons at 2D-1D dimensional crossover at low temperatures. By further presenting two counterexamples where the bound predicts trivial results, we shed light on the conditions under which the Leggett's bound serves as a good predictor.

Figures (4)

• Figure 1: Comparison between the exact superfluid fraction $f_s^y$ via QMC winding number (blue circles) and the Leggett's bound $f_{\uparrow,s}^y$ (solid lines). We use QMC at low temperature $k_BT\simeq 0.005E_r$ and GPE density profiles at zero temperature $k_BT=0$ (blue and red solid lines, respectively) for varying lattice depths $V_y$. Panels (a) and (b) correspond to weak ($\tilde{g}_{\textrm{\tiny 2D}}\simeq0.018$) and strong ($\tilde{g}_{\textrm{\tiny 2D}}\simeq1.364$) interaction regimes, respectively. The QMC bound shows excellent agreement with the exact $f_s^y$ in both cases. As expected, the GPE bound is more accurate in the weak interaction case.
• Figure 2: Scaling analysis of $f_s^y$ for small $t_y$ from QMC data. Panels (a) and (b) correspond to weak ($\tilde{g}_{\textrm{\tiny 2D}}\simeq0.018$) and strong ($\tilde{g}_{\textrm{\tiny 2D}}\simeq1.364$) interaction regimes, respectively, for a system size of $L_x=25a, L_y=10a$ and $k_B T\simeq 0.005 E_r$. The black dashed lines are linear fits for small $t_y$ of the exact SF fraction computed with QMC winding number (light blue circles) and the blue solid line is the QMC bound. We find the scaling exponents to be in excellent agreement with the SCHA predictions $\nu(K)=\frac{4K}{4K-1}$. For weak interactions ($K=10$) we have $\nu^{\text{QMC}}_\text{weak}=1.06 \pm 0.09$ and for strong interactions ($K=1$) we find $\nu^{\text{QMC}}_\text{strong}=1.33 \pm 0.07$. Fitting the QMC bounds, we have $\nu^{\text{QMC bound}}_{\text{weak}} = 0.96 \pm 0.01$ and $\nu^{\text{QMC bound}}_{\text{strong}} = 1.00 \pm 0.03$.
• Figure 3: Two cases where the bound is no more a quantitative SF estimator (solid line). In panel (a1) we study the longitudinal SF fraction $f_s^x$ at intermediate temperature $k_BT=0.09E_r$ and strong interactions $\tilde{g}_{\textrm{\tiny 2D}}\simeq1.364$ with system size of $L_x=L_y=12a$. (a2) shows the corresponded density profile $n(x)$ for the case $V_y=10E_r$. It leads to the constant bound (blue solid curve in (a1) which is not compatible with the $f_s^x$ (light blue balls). In panel (b1) we consider large enough $V_y$ such that the system is strictly 1D. We further consider small $V_x=2E_r$ in the Tonks-Girardeau limit , with the 1D dimensionless coupling constant $\tilde{g}_{\textrm{\tiny 1D}}=7$ and temperature $k_BT\simeq0.004E_r$ and system sizes $L_x=50a$. The system shows a transition from 1D superfluid state to 1D Mott insulator state by plotting the superfluid fraction $f_s^x$ as a function of the chemical potential $\mu$. We show the results from both the QMC winding (light blue balls) and Leggett's bound (blue solid curve). In panel (b2) we present one example of the density profile $n(x)$ at $\mu=1.45E_r$.
• Figure S1: Comparison of QMC calculations for different temperatures $k_BT=0.0034E_r$ (green), $k_BT=0.005E_r$ (blue) and $k_BT=0.01E_r$ (red) in strong interaction regime $\tilde{g}_{\textrm{\tiny 2D}}\simeq1.364$, with the system size $L_x=25a, L_y=10a$. The solid lines are Leggett's bounds, with the linear fit in log-log scale of the superfluid fraction $f_s^y$ (black dashed line) computed via QMC winding number at $k_BT=0.005E_r$ in the range $V_y\in[15,20]E_r$ (blue circles).