Subsystem symmetry protected topological order

TL;DR

Subsystem symmetry protected topological (SSPT) order extends SPT concepts to symmetries acting on lower-dimensional subsystems, yielding gapless edge modes and nonlocal order while resisting symmetric perturbations. The authors construct exactly solvable bosonic and fermionic models in $2D$ and $3D$ that realize $d=1$ subsystem symmetries, analyze edge-state projective representations, flux/gauge responses, and dualities to gauge theories, and generalize to $Z_n^{sub}$ structures. They distinguish weak versus strong SSPTs via edge and bulk response (including flux insertions and nonlocal order parameters), and provide concrete instances with $Z_2^{sub}$ and $ ext{T}^{sub}$ symmetries, as well as a fermionic SSPT based on Fidkowski–Kitaev interactions. The work connects SSPTs to decorated-domain constructions, membrane/volume order parameters, and potential fracton physics upon gauging, offering a framework for future classifications and experimental signatures.

Abstract

In this work, we introduce a new type of topological order which is protected by subsystem symmetries which act on lower dimensional subsets of lattice many-body system, e.g. along lines or planes in a three dimensional system. The symmetry groups for such systems exhibit a macroscopic number of generators in the infinite volume limit. We construct a set of exactly solvable models in $2d$ and $3d$ which exhibit such subsystem SPT (SSPT) phases with one dimensional subsystem symmetries. These phases exhibit analogs of phenomena seen in SPTs protected by global symmetries: gapless edge modes, projective realizations of the symmetries at the edge and non-local order parameters. Such SSPT phases are proximate, in theory space, to previously studied phases that break the subsystem symmetries and phases with fracton order which result upon gauging them.

Figures (15)

• Figure 1: The terms in the TPIM Hamiltonian. The Pauli spins $\tau,\sigma$ live on the red/blue sites. The interaction $\sigma^z_i \sigma^z_j \sigma^z_k \sigma^z_l \tau^x$ involves the four $\sigma_z$ spins on the blue plaquette and the $\tau_x$ in the middle. The interaction $\tau^z_i \tau^z_j \tau^z_k \tau^z_l \sigma^x$ involves the four $\tau_z$ spins on the red plaquette and the $\sigma_x$ in the middle.
• Figure 2: Ground state of the TPIM. The blue lines denote domain walls for $\sigma$ spins, where $\sigma_z= +1 (-1)$ outside/inside a domain. The corners of these domains are decorated by a $\tau^x=-1$ spin, indicated by the red arrows. The ground state is an equal superposition of all such configurations.
• Figure 3: The dark blue line gives an example of a subsystem: a single row on sublattice $A$ (where we call the spin operator $\sigma$). The green square indicates the boundary of the membrane order parameter, which involves product of $\sigma_z$(blue) at the corner of the membrane and product of $\tau_x$(red) inside the membrane.
• Figure 4: Red ovals show the physical spins that take part in the edge operators $\pi^\alpha_i$, and form a spin-$1/2$ degree of freedom at the edge. The action of the subsystem symmetries (green lines) on the ground state manifold may be expressed in terms of such $\pi^\alpha_i$ operators. Near a corner of the type shown here, the symmetry becomes a local symmetry, and the corresponding boundary modes can be gapped out.
• Figure 5: The spins in the blue rectangle are involved in the edge operators $\pi^\alpha_i$, in the case of a $45^\circ$ edge.