Superfluidity in the spin-1/2 XY model with power-law interactions

TL;DR

This work investigates the ground-state superfluid density (spin stiffness) of a one-dimensional spin-1/2 XY model with power-law interactions $1/r^{\alpha}$, as realized in trapped-ion systems. Using an SSE quantum Monte Carlo framework with twisted boundary conditions, the authors measure $\rho_s$ via winding numbers and validate their approach against exact diagonalization in small systems and analytic limits ($\alpha\to\infty$ and $\alpha=0$). They show that long-range interactions enhance superfluidity, with $\rho_s$ diverging in the thermodynamic limit for $\alpha<3$, and they introduce a normalized estimator $\tilde{\rho_s}$ that isolates non-divergent contributions to reveal three distinct regimes: long-range ($\alpha\leq 1$), medium-range ($1<\alpha<\alpha_c$ with $\alpha_c\approx2.7$), and short-range ($\alpha>\alpha_c$). The study combines QMC with mean-field and spin-wave analyses, confirming qualitative and quantitative consistency across methods and providing a practical probe for critical behavior at $\alpha_c$, with relevance to trapped-ion experiments exploring long-range quantum magnets. \nAll mathematical expressions are presented with explicit $\LaTeX$-style formatting, e.g., $\alpha$, $\rho_s$, $E_0$, and the twist $\theta$.

Abstract

In trapped-ion quantum simulators, effective spin-1/2 XY interactions can be engineered via laser-induced coupling between internal atomic states and collective phonon modes. In the simplest one-dimensional ($1d$) traps, these interactions decay as a power-law with distance $1/r^α$, with a tunable exponent $α$. For small $α$, the resulting long-range $1d$ XY model exhibits continuous symmetry breaking, in marked contrast to its nearest neighbor counterpart. In this paper, we examine this model near the phase transition at $α_c$ from the lens of the spin stiffness, or superfluid density. We develop a stochastic series expansion (SSE) quantum Monte Carlo (QMC) simulation and a generalized winding number estimator to measure the superfluid density in the presence of power-law interactions, which we test against exact diagonalization for small lattice sizes. Our results show how conventional superfluidity in the $1d$ XY model is enhanced in the long-range interacting regime. This is observed as a diverging superfluid density as $α\rightarrow 0$ in the thermodynamic limit, which we show is consistent with linear spin-wave theory. Finally, we define a normalized superfluid density estimator that clearly distinguishes the short, medium, and long-range interacting regimes, providing a novel QMC probe of the critical value $α_c$.

Figures (10)

• Figure 1: $1d$ lattice of sites with periodic boundaries. Panel $(a)$ exhibits an infinite chain of spin-1/2 particles or equivalently, the sites between which the bosons hop in the hardcore boson model. The simulation cell is highlighted with the dotted line box surrounding the seven sites in the middle. Panel $(b)$ shows our convention for imposing periodic and twisted boundaries on the simulation cell in $(a)$. Panel $(c)$ displays the SSE operator loop diagram associated with this convention.
• Figure 2: ED results for $\alpha = 50$ (nearest-neighbor limit) compared to the FFS. The black line shows the FSS ($N \rightarrow \infty$) limit and the red line shows the ED results fit. The left panel pertains to $E_0/N$ and the right panel corresponds to $\rho_s$.
• Figure 3: SSE directed loop construction for the XY model. Panel $(a)$ shows the dictionary of matrix elements and their corresponding diagrammatic representations. Each matrix element is associated with four legs. Blue dots denote $|0\rangle$ and red dots denote $|1\rangle$. The blue rectangles represent the matrix element $\langle 10|S_i^- S_{i+r}^+|01\rangle$ and the orange rectangles represent its Hermitian conjugate. Black rectangles correspond to diagonal matrix elements. All four matrix elements shown in panel $(a)$ as well as their Hermitian conjugates are allowed for the XY model. Panel $(b)$ shows the possible transitions for the first and third matrix elements in panel $(a)$. Panel $(c)$ finally exhibits one possible closed loop that can be constructed for the XY model in the off-diagonal SSE update.
• Figure 4: QMC superfluid density and energy per site (insets) as a function of $\beta$ compared to ED for $N=11$. The left panel shows the results for $\alpha = 1$ and the right panel corresponds to $\alpha = 10$.
• Figure 5: Convergence of the QMC estimates of ground state energy (left panel) and superfluid density (right panel) to the mean field results in the limit of $N \rightarrow \infty$. The QMC estimates of energy and superfluid density are denoted with a bar: $\overline{E_0}$ and $\overline{\rho_s}$, and the mean field results are denoted by $E_0^{MF}$ and $\rho_s^{MF}$. The exact finite size effect in $N$ (given by Eq. \ref{['finite_size_mft']}) is also shown for comparison for both $E_0$ and $\rho_s$ by the red lines. The insets show the comparison between QMC estimates and LMG predictions. Here, the denominators $E_0$ and $\rho_s$ correspond to the (exact) LMG results.