Foliated Quantum Field Theory of Fracton Order

TL;DR

The paper develops a continuum description of foliated fracton orders by introducing foliated quantum field theory (FQFT) on manifolds with layered foliations, where mobility constraints arise from gauge invariance and excitations are bound to intersections of leaves. The approach defines foliated gauge fields $A^k$ with the constraint $A^k \wedge e^k=0$ and quantized levels $M_k$, $n_k$, $N$, with $m_k=\frac{n_k M_k}{N}\in\mathbb{Z}$, yielding a stack-of-leaves structure that reproduces X-cube-like mobility. A central result is the exfoliation duality, a finite-depth IR transformation that maps portions of the FQFT to decoupled stacks of 2+1D BF theories via spatially varying coefficients $\tilde{n}_1(z)$, illustrating how foliated fracton order can emerge and be dialed. The work connects to lattice fracton models and provides a flexible framework for studying boundaries, curvature, and higher-dimensional generalizations, with potential implications for robust quantum information storage in foliated topological phases.

Abstract

We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and exhibit a duality that spatially transforms the coefficients. This duality occurs because the FQFT is a foliated fracton order. That is, the duality can decouple 2+1D gauge theories from the FQFT through a process we dub exfoliation.

Figures (4)

• Figure 1: A depiction of some leaves (colored surfaces) for three different foliations. A foliation consists of an infinite number of infinitesimally-spaced layers, which are called leaves.
• Figure 2: Spatial pictures of: (a) Three leaves intersecting at a point. (b) A 1-dimensional manifold $\mathcal{M}_1^\text{L}$ (blue) at the intersection of two leaves (red and green). (c) A 2-dimensional manifold $\mathcal{M}_2^\text{P}$ (blue) with boundaries supported on leaves (red).
• Figure 3: An XY planar slice of spacetime showing a depiction of $U_1$, $U_2$, $U_3$, and $U_4$ from Eqs. \ref{['eq:3cover']} and \ref{["eq:3cover'"]}.
• Figure 4: Examples of how the duality Eq. \ref{['eq:aB duality']} acts on field configurations that satisfy the equations of motion $da + \sum_k m_k A^k = dA^k = db = (dB^k + n_k b) \wedge e^k = 0$. Some leaves are shown in green. (a) A membrane of $a \neq 0$ (light red) that ends on a loop of $A^1 \neq 0$ (red) gets mapped to (b) the same fields but with the membrane extended vertically (from the $A^1 \neq 0$ loop) to $z_2$, ending on a new loop of $\tilde{A}^1 \neq 0$. $A^k = 0$ and $a = 0$ elsewhere. (c) A membrane of $B^1 \neq 0$ (light blue) with a boundary on a loop of $b \neq 0$ (purple). (c) is mapped to (d), which adds an additional membrane of $\tilde{B}^1 \neq 0$ within $z_1 < z < z_2$. Importantly, note that $da + \sum_k m_k A^k = (dB^1 + n_1 b) \wedge e^1 =0$ is satisfied everywhere in (a) and (c) since membranes of $a \neq 0$ and $B^1 \neq 0$ end on loops of $A^1 \neq 0$ and $B^1 \neq 0$; while $d\tilde{a} = d\tilde{B}^1 \wedge e^1 = 0$ is satisfied for $z_1 < z < z_2$ in (b) and (d) since membranes of $\tilde{a} \neq 0$ and $\tilde{B}^1 \neq 0$ do not have boundaries in this region.