Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order

TL;DR

This work introduces a foliated field theory to describe abelian foliated fracton orders, where a static foliation structure described by 1-forms $e^k_\mu$ couples a stack of 2+1D $Z_N$ gauge theories to a 3+1D $Z_N$ gauge field. It develops a string-membrane-net lattice model whose ground state is a equal-weight superposition of allowed configurations and shows its low-energy equivalence to the X-cube model via a local unitary circuit, thereby unifying continuum and lattice perspectives. The authors further present a dual coupled-string-net picture and analyze excitations (fractons, lineons, planons) within both pictures, discuss curved foliations, and outline future directions including quantization, generalized foliations, and dynamical foliations with potential gravity connections. Overall, the paper provides a concrete framework for foliated fracton phases, linking field theory, lattice constructions, and dual descriptions to advance the classification and understanding of subdimensional excitations in three dimensions.

Abstract

Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.

Figures (8)

• Figure 1: (a) A foliation structure consisting of three stacks of layers (red, green, and blue), which are often referred to as leaves. In the coarse-grained continuum limit, we will think of the layers as being infinitesimally close to each other. If the layers have a finite separation, then the intersections of the layers results in a cubic lattice. (b) A foliation structure consisting of four stacks of layers (red, green, blue, and yellow), which results in a stack of kagome lattices when the layers have finite separation.
• Figure 2: A 2D slice of 3D space showing a single foliation of 2D layers, which appear as red lines in this cross-section. The foliation field $e_\mu$ points orthogonal to the layers such that $\int_p e^k \equiv \int_p e^k_\mu \mathrm{d} x^\mu$ counts the number of layers that the path $p$ crosses. In the figure, we show two paths and the resulting integrals. A contractible closed path, e.g. the composition of the cyan and purple paths, results in $\oint_p e^k = 0$. In the field theory, the (red) foliating layers are infinitesimally close to each other. A cutoff can be added to allow for a notion of finite layer spacing.
• Figure 3: A 2D slice of 3D space showing two foliations (red and blue). (a) When in isolation, a charge (sourced by$j^\mu$) cannot move through a foliation layer, which results in the mobility constraints tabulated in Tab. \ref{['table:foliation examples']}. However, a dipole $J^{\mu1}$ can move along a red layer and be absorbed by the charge to move the charge to the right, as depicted in (b).
• Figure 4: A graphical representation of the two types terms appearing in the toric code Hamiltonian [Eq. (\ref{['eq:toric']})]. We use a red cross to represent a product of four $Z$ operators on the edges around a vertex and a red zig-zag to represent a product of four $X$ operators around a plaquette.
• Figure 5: (a) A lineon excitation: two different-colored strings that end at a point. (b) The lineon can only move in a straight line since if its path bends, another lineon excitation is left behind at the corner. This occurs because the red string cannot follow the blue string in the $z$-direction as the red string is not allowed on edges in the $z$-direction. The green string is then necessary to avoid excitations of the edge term [Eq. (\ref{['eq:edgeterm']})]. But an excitation remains at the corner since the red and green strings both have endpoints there.