Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions

Abstract

Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries. Contents2 This symmetry motivates our choice of notation 2 H (0,1) Z 2 for the Hamiltonian of the plaquette Ising model: the (0, 1) in the superscript indicates that the model has a Z (0,1) 2 (F x , F y ) ∼ symmetry, and the 2 denotes its spatial dimension. We define an infinite family d H (q,k) Z 2 of similar models in §2.1.

Fractionalization of subsystem symmetries in two dimensions

“…In the case of subsystem symmetries, we would also expect that symmetry breaking is required upon condensing the fractionalized anyon. Investigating such condensation transitions and their relations to previously observed forms of sub-dimensional criticality [47,60] remains an interesting open problem.…”

“…In a similar fashion, the model H in the presence of edge perturbations was obtained in Ref. [47] by stacking 1D Ising chains along rows and columns and gauging the resulting global Z 2 symmetry that acts on all chains. This reveals that the phase transition driven by these edge perturbations is equivalent to a stack of 1D critical Ising chains coupled to a 2D Z 2 gauge field.…”

A symmetry principle for gauge theories with fractons

“…We can thus understand the appearance of gapless fractonic phases within the extension of Landau's symmetrybreaking paradigm to higher-form symmetries [28]. There have been other approaches to understanding fractonic phases using a symmetry principle: by considering certain exotic higher-form symmetries [29,30], subsystem symmetries [31], or global symmetries which act on quasiparticles with position-dependent charge [32]. We emphasize that nonuniform higher-form symmetries in fact fall within the standard definition of higher-form symmetries, as the noncommutativity of the charge with translations does not mean that the symmetry generators are not topological.…”

Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries

“…In the fracton phase of matter, the origin of the "Area-law" entropy can be attributed to the so-called subsystem symmetry [9,10,11,12,13,14,15,16,17,18,19]. The Noether charge for the ordinary symmetry (in d dimensional space) is obtained by integrating the charge density over the whole volume: Q 0 = d d x ρ while that for the subsystem symmetry is obtained by integrating the charge density only over codimension q subsystems Q q = d d−q x ρ.…”