SWAP algorithm for lattice spin models

TL;DR

The paper introduces a Δ-model that extends SWAP dynamics to finite-dimensional lattice spin systems by assigning a random length to each spin and interleaving non-local swaps with local flips under Metropolis acceptance. In the 2D Edwards–Anderson spin glass, SWAP with partial annealing of spin lengths dramatically accelerates relaxation at low temperature and enables rapid ground-state discovery, outperforming standard methods in several regimes. The work clarifies that non-local exchanges soften rugged energy landscapes created by quenched disorder, linking SWAP efficiency to the structure of free-energy barriers, and suggests directions for extending the approach to 3D and optimizing annealing schedules. Overall, SWAP provides a practical framework to study slow spin-glass dynamics and ground-state search by enhancing exploration of low-energy configurations.

Abstract

We adapted the SWAP molecular dynamics algorithm for use in lattice Ising spin models. We dressed the spins with a randomly distributed length and we alternated long-range spin exchanges with conventional single spin flip Monte Carlo updates, both accepted with a stochastic rule which respects detailed balance. We show that this algorithm, when applied to the bidimensional Edwards-Anderson model, speeds up significantly the relaxation at low temperatures and manages to find ground states with high efficiency and little computational cost. The exploration of spin models should help in understanding why SWAP accelerates the evolution of particle systems and shed light on relations between dynamics and free-energy landscapes.

Figures (36)

• Figure 1: Sketch of a spin configuration of the modified 2DEA model. Two spins are singled out for analysis (surrounded by green bubbles). The neighboring up and down spins are colored red and blue, respectively. The solid (red) and dashed (blue) links represent $J_{ij}>0$ and $J_{ij}<0$, respectively. The length of the arrows are proportional to the length of the spins, that is, the local $\tau_i$ values. They are here chosen to take only two values, for simplicity.
• Figure 2: The two-time correlations of the couplings ${\mathcal{J}}_{ij}$ in the $\Delta$-model with $\Delta = 1.5$, $L=32$ quenched to $T=0$ evolved with SWAP ($p_{\rm swap} = 0.1$). The waiting times $t_w$ are given in the key. Inset: the cumulative probability of local frustrations $f_P$ at three times after the quench.
• Figure 3: (a) Asymptotic probability of reaching a ground state after a $T=0$ quench and a quadratic annealing, starting from $T_0 = 1.0$ during $t_f \approx 10^7$ MCs. $L=32$. Inset (b) Probability distributions of the ground state energy density differences found after $T=0$ quenches and annealing protocols, Eq. (\ref{['eq:annealing_protocol-def']}), of models with different $\Delta$. In all cases, the $J_{ij}$ and initial lengths $\{\tau_i(t = 0)\}$ are the same, and the data are sampled over $10^3$ initial Ising spin conditions $\{\sigma_i(t = 0) =\pm 1\}$. (c) The overlap of an early (left panel) and final (right panel) $s_i$ configuration with the ground state of the model with couplings ${\mathcal{J}}^*_{ij} ={\mathcal{J}}_{ij}(t_{\rm max})$, for a quench to $T=0$ with $\Delta = 1.5$. The light bullets and triangles are located at frustrated plaquettes with local frustration $f_P$ being greater or smaller than one-half in magnitude, respectively.
• Figure 4: The two-time Ising spin self-correlation at three waiting-times. Data for four kinds of $T=0$ evolution of a system with $L=32$ and $\Delta = 1.5$ starting from random initial conditions: (i) SWAP with non-local moves, (iv) local spin exchanges and spin flips, and solely single spin flips of the $\Delta$ model with (ii) random ${\mathcal{J}}_{ij}$ and (iii) optimized ${\mathcal{J}}^*_{ij}$ couplings.
• Figure 5: Characteristic relaxation time $\tau_\alpha$ extracted from the decay of the $\sigma$ self-correlation after quenches to the target temperatures, evolved with (i) SWAP (i.e. non-local moves), (iv) local spin exchanges, both with the same $p_{\rm swap} =0.1$, and (iii) pure single-spin-flip dynamics ($p_{\rm swap} =0$) with optimized bonds ($\mathcal{J}^*_{ij}$), for a model with $\Delta = 1.5$. In the inset, comparison of the two relaxation times, for local exchanges and pure single-spin-flips with respect to SWAP, as a function of temperature for two $\Delta$-models.