Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions

Abstract

We build exactly solvable lattice Hamiltonians for fermionic symmetry-protected topological (SPT) phases in (3 + 1)D classified by group supercohomology. A central benefit of our construction is that it produces an explicit finite-depth quantum circuit (FDQC) that prepares the ground state from an unentangled symmetric state. The FDQC allows us to clearly demonstrate the characteristic properties of supercohomology phases -namely, symmetry fractionalization on fermion parity flux loops -predicted by continuum formulations. By composing the corresponding FDQCs, we also recover the stacking relations of supercohomology phases. Furthermore, we derive topologically ordered gapped boundaries for the supercohomology models by extending the protecting symmetries, analogous to the construction of topologically ordered boundaries for bosonic SPT phases. Our approach relies heavily on dualities that relate certain bosonic 2-group SPT phases with supercohomology SPT phases. We develop physical motivation for the dualities in terms of explicit lattice prescriptions for gauging a 1-form symmetry and for condensing emergent fermions. We also comment on generalizations to supercohomology phases in higher dimensions and to fermionic SPT phases outside of the supercohomology framework.

Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

“…𝑓 ! 𝑓 " 𝑓 # Figure 4: It can be checked that for any set of faces f 1 , f 2 , f 3 on a tetrahedron, there is always a minus sign in [41]. Therefore, we conclude that U f is the hopping operator of a fermionic particle.…”

“…The discussion follows the approach in Refs. [9,23,[38][39][40][41]. We introduce qubits on each pn i `1q-simplex, acted by Pauli matrices…”

“…Our strategy follows the auxiliary vertex approach in Ref. [41], which is reviewed below. To apply our bulk formalism to the case with boundary, we construct a closed manifold by introducing auxiliary simplices: these simplices are given by first introducing a vertex v 0 , and then introducing auxiliary simplices by connecting v 0 to all vertices on the boundary N 3 of M 4 , forming auxiliary edges x0iy @i P N 3 , auxiliary faces x0ijy @xijy P N 3 , etc...…”

Symmetry-protected sign problem and magic in quantum phases of matter

“…the 2D cluster state) [63,64] or protected by higher-form symmetries (e.g. the 3D cluster state) [23,[65][66][67][68][69]. In our argument, we assumed that the protecting symmetry is a 0-form symmetry, i.e., it is supported on a codimension-0 manifold.…”

Equivalence between fermion-to-qubit mappings in two spatial dimensions

“…In the last two decades, there have been many proposals of fermionto-qubit mappings for two dimensions [1][2][3][4][5][6][7][8][9] and three or arbitrary dimensions [10][11][12]. These fermion-to-qubit mappings play important roles in various topics of modern physics, such as exactly solvable models for topological phases [3,[13][14][15], fermionic quantum simulations [2,4,5,8,10], and quantum error correction [16][17][18][19][20][21]. In particular, the exact bosonizations in Refs.…”