Stability of current-carrying states in hard-core bosons with long-range hopping on a square lattice

TL;DR

The study probes the stability of current-carrying states in a hard-core Bose-Hubbard model with long-range hopping decaying as $|\mathbf{r}|^{-\alpha}$ on a square lattice. Using a mean-field treatment mapped to a spin-1/2 XY model, it computes the ground-state phase diagram and excitation spectra to identify Landau and dynamical instabilities as functions of $\alpha$, uncovering that critical quasi-momenta vanish at $\alpha=3$. A key result is the unconventional scaling $K_c a \propto \Delta^{1+\Delta}$ with $\Delta=\alpha-3$ for the dynamical-instability threshold near $\alpha=3$, reflecting the long-range nature of hopping. The findings highlight how long-range hopping suppresses superfluid stability and provide guidance for experimental platforms such as Rydberg-atom arrays and trapped ions, while suggesting avenues for studying finite-temperature effects and the superfluid fraction in these systems.

Abstract

We investigate the stability of current-carrying states with quasi-momentum $K$ in the Bose-condensed phase of the hard-core Bose-Hubbard model on a square lattice, where particles transfer between two sites separated by distance $r$ with hopping amplitude decaying algebraically with $r$ as $\propto r^{-α}$. Using a mean-field theory, we analyze the excitation spectrum and determine the critical quasi-momenta associated with Landau and dynamical instabilities. We find that the long-range hopping suppresses the critical quasi-momenta and makes them vanish at $α=3$. Near $α=3$, we show that the critical quasi-momentum $K_{\mathrm{c}}$ for the dynamical instability exhibits the scaling behavior $K_\mathrm{c} \propto Δ^{1+Δ}$ with $Δ=α-3$, where the scaling exponent explicitly depends on $Δ$, as a consequence of the long-range nature of the hopping.

Figures (5)

• Figure 1: Ground-state phase diagram for $\mathbf{K}=\mathbf{0}$: Critical field $\left(h/J\right)_{\text{c}}$ versus the long-range interaction exponent $\alpha$. The inner region represents the XY ferromagnetic phase, while outer region corresponds to the polarized phase induced by the strong field $h/J$. In terms of the HCB, these phases correspond to the Bose-condensed phase and the Mott-insulator phase, respectively. The thin dashed lines represent $\left(h/J\right)=\pm 4$
• Figure 2: Excitation spectra $\omega_\alpha(\mathbf{q},{\bf K})/J$ at $\alpha = 4$ and $n = 0.5$, where ${\bf K}={\bf e}_x K$. Panels (a) and (b) show the real and imaginary parts of the spectrum for $Ka = 1.0$ and $Ka = 1.3$, respectively.
• Figure 3: Stability phase diagram of the Bose-condensed state in the $(\alpha,Ka)$-plane. Panels (a) and (b) correspond to $n = 0.5$ and $0.7$. Data points with red and blue interpolating curves represent the critical quasi-momenta at which the dynamical instability (DI) and Landau instability (LI) set in, respectively.
• Figure 4: Single-particle energy band $\epsilon_{\alpha}$ defined in Eq. (\ref{['energyband']}) for different decay exponents $\alpha$. The horizontal axis represents quasi-momentum $Ka$ of the BEC. The dashed black, blue, and red lines correspond to the nearest-neighbor hopping case ($\alpha\rightarrow\infty$), $\alpha = 5$, and $\alpha = 3$, respectively.
• Figure 5: Comparison between the analytical prediction and numerical results for the DI critical quasi-momentum $K_{\mathrm{c}}a$ near $\alpha = 3$. Symbols denote numerical results obtained from the excitation spectrum of Eq. (\ref{['Excitationspectra']}). The red dashed line represents the analytical scaling $K_{\rm c}a \propto \Delta^{1 + \Delta}$ with $\Delta = \alpha - 3$, while the blue dotted line shows the linear scaling $K_{\rm c}a \propto \Delta$ for reference. The proportionality constants in both analytical curves are fixed by the data point at $\alpha = 3.05$.