Fracton phases via exotic higher-form symmetry-breaking

TL;DR

The paper develops a unifying perspective on fracton phases via higher-form symmetries, focusing on $p$-string condensation to connect lattice models with continuum descriptions. It introduces a foliated 1-form symmetry to capture the X-cube fracton order emerging from a coupled-layer toric-code construction and uses a cellular-homology viewpoint to make symmetry actions explicit. The framework is extended to the rank-2 ${\rm U}(1)$ scalar charge theory through a framed 1-form symmetry, with gauging and $p$-string constructions unifying the two viewpoints. Together, these results provide model-independent principles for understanding extended-object condensation, emergent IR symmetries, and the geometric structure underlying fracton phases.

Abstract

We study p-string condensation mechanisms for fracton phases from the viewpoint of higher-form symmetry, focusing on the examples of the X-cube model and the rank-two symmetric-tensor U(1) scalar charge theory. This work is motivated by questions of the relationship between fracton phases and continuum quantum field theories, and also provides general principles to describe p-string condensation independent of specific lattice model constructions. We give a perspective on higher-form symmetry in lattice models in terms of cellular homology. Applying this perspective to the coupled-layer construction of the X-cube model, we identify a foliated 1-form symmetry that is broken in the X-cube phase, but preserved in the phase of decoupled toric code layers. Similar considerations for the scalar charge theory lead to a framed 1-form symmetry. These symmetries are distinct from standard 1-form symmetries that arise, for instance, in relativistic quantum field theory. We also give a general discussion on interpreting p-string condensation, and related constructions involving gauging of symmetry, in terms of higher-form symmetry.

Figures (7)

• Figure 1: Illustration of the direct (black) and dual (dashed red) lattice for the toric code. Two choices of $M^1$ are shown as solid red lines in the dual lattice; the corresponding operators $U(M^1)$ are supported on the blue links $\bar{\ell}$ intersecting $M^1$. The $U(M^1)$ for the top operator corresponds to a cycle with nontrivial homology class, and the $U(M^1)$ for the bottom operator is $A_v$.
• Figure 2: (a) Standard foliation structure and a discretization on $T^3$, shown for $L = 6$. $T^3$ is viewed as a $L \times L \times L$ cube with opposite faces identified, and a $yz$-plane cross section of the cube is shown. The dotted lines (red online) are cross sections of $xy$-plane leaves belonging to one foliation, while the dashed lines (blue online) are cross sections of $xz$-plane leaves belonging to another. The $yz$-plane leaves are parallel to the cross section and are not shown. (b) The twisted foliation structure on $T^3$ discussed in Sec. \ref{['sec:breaking']}. The $xz$ and $yz$-plane leaves are as in (a), but the $xy$-plane leaves are replaced with tilted $\widetilde{xy}$ leaves as indicated by dotted lines (red online). Again $L = 6$, but due to the periodic boundary conditions there are only $L/2 = 3$$\widetilde{xy}$-leaves in the discretization; the thick dotted line indicates a cross-section of a single $\widetilde{xy}$ leaf, which is homeomorphic to $T^2$.
• Figure 3: Illustration of a charged operator ${\cal O}(M^1)$. The black curve and blue dots correspond to the location of $M^1$ and $m$-particles respectively. The operator is formed by taking the product of line operators (dashed lines) within leaves. The choice of ${\cal O}(M^1)$ is not unique, as the line operators can be deformed within each leaf or may include closed $m$-line operators.
• Figure 4: Graphical demonstration that a $\widetilde{xy}$ leaf (solid line, left panel) in the twisted foliation structure on $T^3$ is homologous to a $xz$ leaf (in homology with $\mathbb{Z}_2$ coefficients). In each panel a $yz$-plane cross section of $T^3$ is shown. First, one adds the boundary of a cylinder running along the $x$-direction, shown by the dashed line in the left panel. Joining this with the $\widetilde{xy}$ leaf results in the cycle shown in the middle panel, which can then be slid to the left or right to obtain cycle in the right panel, which is clearly homologous to a $xz$ leaf.
• Figure 5: Illustration of a symmetry operator $U(c^{xy}_2)$ that acts non-trivially in the ground state space of the X-cube phase on $T^3$ with twisted foliation structure, and a charged operator ${\cal O}(M^1)$ that anticommutes with $U(M^{xy}_2)$. A $yz$-plane cross section of $T^3$ is shown, with the dotted (red online) and dashed (blue online) lines indicating $\widetilde{xy}$ and $xz$ leaves as in Fig. \ref{['fig:foliations']}(b). $c_2^{xy}$ is a union of regions from the $f_{\widetilde{xy}}$ and $f_{xz}$ leaves, and the cross section of $c^{xy}_2$ is indicated by the thick dotted and dashed lines. The symmetry operator ${\cal O}(M^1)$ creates a $m$-loop on $M^1$, which lies in a fixed $yz$-plane, and is indicated by the solid thick diagonal line (green online). ${\cal O}(M^1)$ is a product of $Z^f_\ell$ Pauli operators over $\Sigma_{yz}$, which is a surface lying in the same $yz$-plane as $M^1$, shown in the figure by the shaded region (see text for further details).