Symmetry protected topological phases from decorated domain walls

TL;DR

The article develops a unified decorated-domain-wall framework to realize bosonic SPT phases in d=1–3 by attaching lower-dimensional SPT states to domain walls of a Z_2 subgroup, illuminating edge physics and connections to group cohomology. It provides explicit lattice constructions, edge theories, and effective field theories for 2D Z_2 × Z_2^T and 3D Z_2 × Z_2 SPTs, and analyzes gauging of Z_2, 1D extensions, and time-reversal decorated walls. The work links domain-wall constructions to the Künneth formula, offering a physical interpretation of cohomology classes and outlining how higher-dimensional SPTs emerge from lower-dimensional ingredients. It also discusses soluble Hamiltonians and extensions to time-reversal and other symmetry combinations, suggesting broad applicability and future directions in SPT classification and realization.

Abstract

Symmetry protected topological (SPT) phases with unusual edge excitations can emerge in strongly interacting bosonic systems and are classified in terms of the cohomology of their symmetry groups. Here we provide a physical picture that leads to an intuitive understanding and wavefunctions for several SPT phases in d=1,2,3 dimensions. We consider symmetries which include a Z_2 subgroup, that allows us to define domain walls. While the usual disordered phase is obtained by proliferating domain walls, we show that SPT phases are realized when these proliferated domain walls are `decorated', i.e. are themselves SPT phases in one lower dimension. For example a d=2 SPT phase with Z_2 and time reversal symmetry is realized when the domain walls that proliferate are themselves in a d=1 Haldane/AKLT state. Similarly, d=3 SPT phases with Z_2 * Z_2 symmetry emerges when domain walls in a d=2 SPT with Z_2 symmetry are proliferated. The resulting ground states are shown to be equivalent to that obtained from group cohomology and field theoretical techniques. The result of gauging the Z_2 symmetry in these phases is also discussed. An extension of this construction where time reversal plays the role of Z_2 symmetry allows for a discussion of several d=3 SPT phases. This construction also leads to a new perspective on some well known d=1 SPT phases, from which exactly soluble parent Hamiltonians may be derived.

Figures (8)

• Figure 1: SPT phase in $d=2$ with $Z_2$ and time reversal symmetry ($Z_2\times Z_2^T$). A snapshot of the ground state wave function, where blue and grey are oppositely directed domains of the $Z_2$ symmetry. The ground state preserves the $Z_2$ symmetry since it is a superposition of domain configurations. The domain walls themselves (black lines) are in a $d=1$ SPT phase (the Haldane/AKLT phase) protected by time reversal symmetry. When they end at the edge of the system they create Kramers doublets, leading to a gapless edge state.
• Figure 2: Two-dimensional SPT model with $Z_2 \times Z_2^T$ symmetry. (a) each plaquette hosts a $Z_2$ variable (big black dot) (b) each vertex hosts four spin $1/2$'s (small blue dots) (c) two spin $1/2$'s on the same link form a singlet if they are on a $Z_2$ domain wall, other spin $1/2$'s form singlets within each vertex.
• Figure 3: Two equivalent descriptions of the boundary of 2D SPT model with $Z_2\times Z_2^T$ symmetry. (a) Thick bonds represent $Z_2$ variable in state $\ket{1}$ and thin bonds $\ket{0}$, spin $1/2$ exists on their domain wall. (b) One $Z_2$ variable per bond and one spin variable per vertex. Time reversal acts on spins in a way dependent of neighboring $Z_2$ configurations. Dotted boxes represent local degrees of freedom on the 1D boundary labeled by group elements.
• Figure 4: SPT phase in $d=3$ with $Z_2\times \t Z_2$ symmetry. Red (blue) surfaces represent domain walls of the $Z_2$ ($\t Z_2$) symmetry. They intersect along curves (black lines). The ground state wave function is a superposition of all domain wall configurations which differ by a sign depending on whether there are an even or odd number of intersection curves. This automatically implies a protected edge state when a domain wall intersects the surface of the sample (grey dashed line).
• Figure 5: Three dimensional SPT model with $Z_2 \times \t Z_2$ symmetry. Each cube hosts a $Z_2$ variable (big black dot) and each vertex hosts a $\t Z_2$ variable (small green dot) (a) a cube and the vertices around it (b) a vertex and the cubes around it. Shaded surfaces represent $Z_2$ domain walls.