Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Abstract

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension $d \geq 2$ of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological emergent $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

Figures (26)

• Figure 1: The schematic phase diagram of a symmetry $G$ (left) in the Hamiltonian $H(\lambda_i,\lambda_j,\dots)$'s coupling space $\{\lambda_i,\lambda_j,\dots\}$, and the schematic phase diagram of a symmetry $H$ (right) in the Hamiltonian $H(\lambda_i',\lambda_j',\dots)$'s coupling space $\{\lambda_i',\lambda_j',\dots\}$. Many $G$-SPT states may be trivialized to an $H$-trivial vacuum/insulator, by pulling $G$ back to $H$. Some SPT states in $G$ (here G-SPTs$_1$ and G-SPTs$_2$) may be symmetry-enforced gapless on the physical boundary (that may have either perturbative anomalies or non-perturbative global anomalies). Other SPT states in $G$ (here G-SPTs$_2$ and G-SPTs$_4$) may have symmetry-preserving gapped boundaries (that must have non-perturbative global anomalies) by pulling $G$ back to a larger symmetry group (say $H_2$ and $H_4$, where $H_2 \neq H_4$ in general). We could consider the effective schematic phase diagram in a certain larger $H$. (We may choose $H \supseteq H_2$ and $H \supseteq H_4$.) The effective Hilbert space for the whole systems in the $H$-symmetry may be different/larger than that in the $G$-symmetry. The "$\dots \dots$" in the schematic $H$ phase diagram, implies other possible new phases that occur in a larger $H$-symmetry but do not occur in a $G$-symmetry. The phase boundaries shown are schematically only, which could be the first order, second order or any continuous higher order phase transitions.
• Figure 2: The CZX model. Each site (a large disc) contains four qubits or objects of spin 1/2 (shown as small black dots). The squares, formed by red links, are plaquettes, introduced later.
• Figure 3: A pair of adjacent spins: To preserve the symmetry $U_{CZX}$, we choose a Hamiltonian that only flips the spins in a plaquette if pairs of adjacent spins in neighboring plaquettes are equal. Thus the spins shown here at the top of this plaquette are only flipped if the two spins just above them are equal. Both the spins in the plaquette and the ones just above them are in different sites, as shown.
• Figure 4: Each plaquette Hamiltonian $H_p$ acts on the spins contained in an octagon, as depicted in dashed gray line in the left subfigure (1) and also in the lower left of the right subfigure (2). In the subfigure (2), the octagon in the lower left contains the four spins in plaquette $p$ and four adjacent pairs of spins. In the case of a finite sample made of complete sites, as depicted here, most of the spins can be grouped in plaquettes, but there is a row of spins on the boundary -- shown here on the right of the figure -- that are not contained in any plaquette. However, the Hamiltonian acts on these boundary spins through the projection operators $P_p^\alpha$ from a neighboring plaquette.
• Figure 5: By omitting the right row of spins from the boundary of Fig. \ref{['bdryA1']}(2), we get an alternative boundary of the CZX model. Now all spins are contained in plaquettes, but on the boundary there are "incomplete sites," shown as semicircles on the right of the figure, that contain only two spins instead of four. The "upper" and "lower" spins of the $i^{th}$ boundary site have been labeled $\upsigma_{i+}$ and $\upsigma_{i-}$.