Exactly solvable model for a 4+1D beyond-cohomology symmetry-protected topological phase

Abstract

We construct an exactly solvable commuting projector model for a 4 + 1 dimensional Z 2 symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with "threefermion" Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective Z 2 symmetry on a 3 + 1 dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a Z 2 domain wall at a trivial boundary) will carry nontrivial chiral central charge 4 mod 8. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of Ref.[1], and define a corresponding anomaly indicator.(⇒) If U acts as a QCA by conjugation, β(X i ) is supported near spin i for a spin Pauli operator X i at i. Hence, since this operator must commute with X j for j far from i, not only is (V z+i ) † V z an operator on the material supported near spin i, it is equal to (V z+i+j ) † V z+j for any j far from i. In words, (V z+i ) † V z depends only on spins near i.Note that U dis acting by conjugation as a QCA is a stronger requirement than just that U mat z acts by conjugation as a QCA as it also imposes locality requirements on the spin degrees of freedom. Heuristically, it means that changing z locally will only change the action of U mat z locally.Following [13], we believe that if one restricts U mat z to just the material degrees of freedom in the domain wall manifold M z , then it is nontrivial, i.e., it cannot be written as a quantum circuit

4D beyond-cohomology topological phase protected by C2 symmetry and its boundary theories

“…This is consistent with the finding in Ref. [47] for the case of on-site Z 2 symmetry. We further show that one of the anomalous signatures of the 3D gauge theory is revealed in the core of the loop excitation, in which there are gapless modes that are adiabatically connected to the edge modes of the E 8 state.…”

“…[52]. This result is consistent with the TQFT classification of 4D SPT phases protected by the on-site Z 2 symmetry [47] as one would expected from the crystalline equivalence principal [32].…”

“…Recently, Ref. [47] constructed an exactly solvable model for this phase and studied its quantized response at the Hamiltonian lattice level. The bulk physical picture of this phase is also revealed by its construction: decorating Z 2 domain walls with 3D Walker-Wang models based on the so-called "3-fermion" topological order, which is a variant of toric-code topological order where all the anyonic excitations are fermionic.…”

Anomalous symmetries end at the boundary

“…Axioms 0, 1, and 2 still hold, but axiom 3 must be modified to account for the possibility that the junctions themselves can host a gravitational anomaly. This has been appreciated in the SPT literature through the discovery of "beyond cohomology" SPT phases [72][73][74][75][76].…”

A review of Quantum Cellular Automata

“…In [130], an exactly solvable 4 + 1 dimensional model with a Z 2 symmetry is introduced which is outside the classification by cohomology. The Walker-Wang model again plays a role here, and we also see the effective realization of the symmetry on a subsystem (in this case a 3 + 1 dimensional boundary) by a QCA.…”