Leggett's bound and superfluidity in strongly interacting bosons

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.

💡 Research Summary

This paper investigates whether Leggett’s density‑based upper bound, originally shown to be a good estimator of the superfluid fraction in weakly interacting Bose gases, remains reliable in the strongly interacting regime where mean‑field theory fails. The authors focus on a two‑dimensional Bose gas subjected to a one‑dimensional optical lattice that can be tuned from a shallow 2D configuration to an array of quasi‑1D tubes, thus exploring a 2D‑1D dimensional crossover at very low temperature.

Three complementary theoretical tools are employed. (i) Path‑integral quantum Monte Carlo (QMC) with the worm algorithm is used to compute the winding‑number superfluid fraction (f_i^s) and the real‑space density profile (n(x,y)) for a range of interaction strengths (\tilde g_{2D}) (both weak, (\sim0.018), and strong, (\sim1.36)) and lattice depths (V_y). (ii) Gross‑Pitaevskii equation (GPE) calculations provide the mean‑field density and the corresponding Leggett bound for comparison in the weak‑interaction limit. (iii) In the deep lattice limit, where the system reduces to weakly coupled 1D chains, a Tomonaga‑Luttinger liquid description combined with the self‑consistent harmonic approximation (SCHA) yields an analytical scaling law for the superfluid fraction, (f_y^{\rm SCHA}\sim (t_y/E_r)^{\nu(K)}) with (\nu(K)=4K/(4K-1)) and (K) the Luttinger parameter.

The first major result, illustrated in Fig. 1, shows that the QMC‑derived Leggett bound (f_y^{\uparrow}) tracks the exact QMC superfluid fraction (f_y^s) across the whole range of lattice depths for both weak and strong interactions. In the weak‑interaction case the GPE bound also agrees well, while in the strong‑interaction case the GPE substantially overestimates the superfluid fraction, confirming that the density‑only bound remains accurate even when mean‑field theory breaks down.

Next, the authors examine the scaling of (f_y) as the inter‑tube hopping (t_y) becomes small. QMC data for the weakly interacting gas give a scaling exponent (\nu_{\rm QMC}^{\rm weak}=1.06\pm0.09), consistent with the SCHA prediction for (K\approx10) ((\nu\approx1.03)). For the strongly interacting case the exponent is (\nu_{\rm QMC}^{\rm strong}=1.33\pm0.07), matching the theoretical value (\nu=4/3) for hard‑core bosons ((K=1)). However, when the same data are analyzed using the Leggett bound, the extracted exponents are (\nu_{\rm bound}^{\rm weak}=0.96\pm0.01) and (\nu_{\rm bound}^{\rm strong}=1.00\pm0.03). Thus, while the bound remains a quantitatively good estimator for realistic hopping amplitudes, it fails to reproduce the correct asymptotic scaling in the limit (t_y\to0), especially for strong interactions where quantum fluctuations dominate.

To delineate the limits of applicability, two counter‑examples are presented. (1) At an intermediate temperature ((k_B T=0.09E_r)) with moderate interaction ((\tilde g_{2D}=1.36)), the longitudinal superfluid fraction (f_x^s) decreases as the lattice depth grows, yet the density profile along (x) shows virtually no modulation. Consequently the Leggett bound stays constant and cannot capture the temperature‑induced reduction of superfluidity. (2) In a strictly 1D regime with strong interactions ((\tilde g_{1D}=7)) and a very weak periodic potential ((V_x=2E_r)), a pinning Mott transition eliminates the superfluid fraction over a range of chemical potentials, while the density modulation remains small. The Leggett bound therefore predicts a large, almost unchanged value ((\sim0.83)), completely missing the transition. These examples demonstrate that the bound works reliably only when variations in the superfluid fraction are directly linked to observable density modulations.

In conclusion, Leggett’s bound is an accurate and practical estimator of the transverse superfluid fraction for strongly interacting bosons undergoing a 2D‑1D crossover at low temperature, both in weak and strong coupling regimes. The bound’s scaling agrees with QMC and SCHA predictions for weak interactions, but deviates for strong interactions as the system approaches the 1D limit. Moreover, the bound fails in situations where superfluid suppression originates from thermal fluctuations or subtle many‑body effects that do not manifest as pronounced density variations. The work thus provides clear criteria for when the density‑based Leggett bound can be trusted and highlights the necessity of complementary measurements (e.g., winding‑number or phase‑twist response) in regimes where the bound becomes unreliable. This insight is directly relevant to current quantum‑gas‑microscope experiments, where high‑resolution density imaging is readily available.