Infinite Stability in Disordered Systems

TL;DR

The paper demonstrates that disorder can fail to destroy finite-temperature ordering in XY models by introducing a bilayer setup with random interlayer phase shifts. Through a strong-disorder reduction to an effective Hamiltonian $\mathcal{H}^{\mathrm{s}}$, coupled with site-percolation and Ginibre inequalities, it proves infinite stability of LRO in $d\ge3$ under a percolation condition, and shows a 2D non-planar-graph analogue yields infinite quasi-stability. It also analyzes weak-disorder behavior via a decoupled RFO(2)-like regime, and provides a rigorous comparison between disordered and clean models, plus detailed bounds in 2D that highlight the delicate role of dimensionality. The results have implications for disorder-induced ordering in layered superconductors and for understanding percolation-driven mechanisms of robust order in disordered systems.

Abstract

In quenched disordered systems, the existence of ordering is generally believed to be only possible in the weak disorder regime (disregarding models of spin-glass type). In particular, sufficiently large random fields is expected to prohibit any finite temperature ordering. Here, we prove that this is not necessarily true, and show rigorously that for physically relevant systems in $\mathbb{Z}^d$ with $d\ge 3$, disorder can induce ordering that is \textit{infinitely stable}, in the sense that (1) there exists ordering at arbitrarily large disorder strength and (2) the transition temperature is asymptotically nonzero in the limit of infinite disorder. Analogous results can hold in 2 dimensions provided that the underlying graph is non-planar (e.g., $\mathbb{Z}^2$ sites with nearest and next-nearest neighbor interactions).

Figures (9)

• Figure 1: Schematic Phase Diagram. The green line sketches the transition temperature $T_c(h)$ of single layer models with disorder strength $h$, e.g., the random-field Ising and XY models aharony1978tricriticalaharony1978spinbray1985scaling, the RFO(2) model dotsenko19812ddotsenko1982spincrawford2024randomcrawford2013randomcrawford2014random. If $h \to 0$, then $T_c(h)\sim 1$ is on the scale of the XY coupling strength $\kappa=1$ (in 2D, there is debate on whether LRO or QLRO occurs below $T_c$crawford2024random). For these models, there exists a critical threshold $h_c \sim 1$, beyond which there is no finite temperature ordering. In comparison, this manuscript introduces a bilayer model with $T_c(h)$ sketched by the red line. If $h \to \infty$, then $T_c(h)$ is asymptotically nonzero (dashed red line). For completeness, the dashed orange line denotes an intermediate scenario where there is finite temperature ordering at large disorder, but $T_c(h)$ is asymptotically zero.
• Figure 2: The strong disorder change of variable. In each subfigure, the purple, orange arrows denote the $\sigma^\pm =e^{i\theta^\pm}$ spins, respectively. Subplots (a), (b) denote the change of variables described by Eq. \ref{['eq:COV']}, $\alpha_x=0$, where the grey arrow indicates $\zeta$ pointing in the middle of the smaller region. Note the discontinuity near $\sigma^+ =-\sigma^-$. Subplots (d), (e) denote that described by Eq. \ref{['eq:COV']}, $\alpha_x=\pi$, where the grey arrow indicates $\zeta$ pointing in the middle of the region characterized by the counter-clockwise orientation from $\sigma^+$ to $\sigma^-$. Note the discontinuity near $\sigma^+ = \sigma^-$. The grey box in subplots (c), (f) denote the conventional $(-\pi,\pi)^2$ box. If $\alpha_x=0$ and $\theta^\pm$ are restricted within the green box in subplot (c), then $\zeta = e^{i\theta},w=e^{i\phi}$ where $\theta = (\theta^++\theta^-)/2,\phi = \theta^+ -\theta^-$ and $(\theta,\phi)\in (-\pi,\pi)^2$. The green dashed line denotes $\phi =0$. Similarly, if $\alpha_x=\pi$ and $\theta^\pm$ are restricted within the red box in subplot (f), then $\zeta = e^{i\theta},w=e^{i(\phi-\pi)}$ where $\theta = (\theta^++\theta^-)/2,\phi = \theta^+ -\theta^-$ and $(\theta,\phi-\pi)\in (-\pi,\pi)^2$. The red dashed line denotes $\phi = \pi$.
• Figure 3: GS Schematic $e^{i\phi} \sim^\text{s} +i$. In the strong disorder regime, (a) shows a typical disorder realization of clusters with random phase $\alpha=0,\pi$ where the green and red dashed lines denote the cluster boundary. By Eq. \ref{['eq:GS']} (with the assumption that $\nabla \theta =0$), the slanting $\tau_x=0$ vanishes within the interiors of the clusters (e.g., the colored dots deep within each cluster). However, at the boundary of each cluster, $\tau_x > 0$, since there is a nontrivial jump in neighboring phase gradients $\nabla \alpha = \pm \pi$. (b) shows the corresponding GS $\phi$ in spin space $\dS^1$, given by the heuristics $e^{i\phi} = e^{i\alpha} +i\tau$ in Eq. \ref{['eq:heuristic']}. At the boundary (dashed lines), there is uniform slanting in the $+i$ direction. (c) shows the GS in the original spins $\sigma^\pm$ denoted by the purple, orange arrows, respectively.
• Figure 4: Largest cluster. For a given disorder realization $\alpha$ (with periodic boundary conditions), (a) plots the largest $\alpha=0$ cluster (green) and $\alpha =\pi$ cluster (red) on $\tilde{\dZ}^2$, while (b) plots those on $\dZ^2$. All other lattice sites are colored white. The difference between subplots (a), (b) is due to $p_c^\text{site}(\tilde{\dZ}^2) < 1/2 <p_c^\text{site} (\dZ^2)$
• Figure 5: Numerics in $\dZ^3$ with $p=1/2$. The red/blue depict the critical transition temperature of the bilayer/effective Hamiltonian $\sH,\sH^\mathrm{s}$ defined in Eq. \ref{['eq:H-alpha']} and \ref{['eq:H-strong']}, respectively. Error bars are also provided based on the Binder cumulant.

Theorems & Definitions (63)

• Theorem 1.1: Infinite Stability
• Theorem 1.2: Infinite Quasi-Stability
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 1
• Theorem 2.3: XY ordering on site percolation clusters dario2023phase
• Theorem 2.4: Weak disorder crawford2013randomcrawford2014randomcrawford2024random
• Definition 3.1