Correlation function diagnostics for type-I fracton phases

TL;DR

The paper introduces correlation-function diagnostics for Type I fracton phases by formulating a generalized plaquette Ising gauge theory (PGT) that maps to the X-cube fracton order. It derives a 4D Euclidean path integral for the PGT, defines Wilson-loop–type observables and horseshoe generalizations to probe partial deconfinement, and analyzes an anisotropic version to connect X-cube physics with layered 2D Ising models. The authors validate the approach with SSE Monte Carlo, detailing update schemes and presenting a phase diagram that reveals first-order confinement transitions and the partial deconfinement of fracton-pair excitations. The work provides a practical framework for diagnosing fracton deconfinement in Type I models and suggests extensions to other Type I systems, while outlining open questions for Type II fracton theories.

Abstract

Fracton phases are recent entrants to the roster of topological phases in three dimensions. They are characterized by subextensively divergent topological degeneracy and excitations that are constrained to move along lower dimensional subspaces, including the eponymous fractons that are immobile in isolation. We develop correlation function diagnostics to characterize Type I fracton phases which build on their exhibiting {\it partial deconfinement}. These are inspired by similar diagnostics from standard gauge theories and utilize a generalized gauging procedure that links fracton phases to classical Ising models with subsystem symmetries. En route, we explicitly construct the spacetime partition function for the plaquette Ising model which, under such gauging, maps into the X-cube fracton topological phase. We numerically verify our results for this model via Monte Carlo calculations.

Figures (4)

• Figure 1: The Euclidean time representation of the Wilson loop and horseshoe generalizations for the PGT and its dual, which realize the X-cube topological phase. Blue circles represents $\tau$ (which lie on vertices), red represent $\sigma$ (which lie on the spatial plaquettes in the PGT, but on spatial links in its dual), and green lines represent the auxiliary spin $\lambda$ (which lie on the links along the imaginary time direction). Non-equal time operators are shown projected to a 2+1D subspace, with the time direction pointing "up" in the page. The three possible cut orientations are labeled by $a$,$b$, and $c$.
• Figure 2: The behavior of the spatial cut (Fig \ref{['fig:loops']}a) ratio $\mathcal{R}(L)$ (Eq \ref{['eq:ratio']}) for large $L$ across the (first-order) confinement transition as $J/\Gamma_M$ is increased with fixed $\Gamma/K=0.8$. Inset is a sketch of the zero-temperature phase diagram, where lines indicate first order transitions, as obtained by quantum Monte Carlo (see supplementary material supmat for details).
• Figure 3: Plots of the energy $\langle E \rangle$ as a function of $J$ and $\Gamma$, for a $10\times 10\times 10$ system at $\beta=8$ with $K=\Gamma_M=1$, with the QMC constants (Eq \ref{['eq:app_beginops']}-\ref{['eq:app_endops']}) subtracted out. We only show data until the QMC state becomes unstable and transitions into a lower-energy state. (left) Sweeping $J$ at various values of $\Gamma$, sweeping right from the X-cube limit and left from the trivial $\sigma^z=1$ large-$J$ limit, showing strong first order transitions at $J/\Gamma_M\approx0.3$. (center) Sweeping of $\Gamma$ with $J=0.2$, showing a first-order transition at around $\Gamma/K\approx0.93$ (other values of $J<0.3$ look very similar). (right) Sweeping $\Gamma$ with $J=0.5$ (which is confining), showing a first-order transition between the two confined phases.
• Figure 4: (left) The expectation value of the diagnostic $\mathcal{R}(L)$ defined in the main text, which approaches zero (a constant) in the deconfined (confined) phase as $L\rightarrow\infty$. Here, the loop is taken to be an $L\times L$ square, and the horseshoe has dimensions $L/2\times L$. We look at the transition induced by increasing $J$ at fixed $\Gamma/K=0.8$. The correlation lengths are very short near the first order transition and already $L=2$ is indistinguishable from $L=4$, thus we are already in the large-$L$ limit and $\mathcal{R}(L)$ shows the expected behavior. Note that we only show the lower-energy state at the first order transition. (right) A schematic phase diagram summarizing the sweep results from Figure \ref{['fig:app_transition']}. All transitions are first-order.