Squaring the fermion: The threefold way and the fate of zero modes

Abstract

We investigate topological properties and classification of mean-field theories of stable bosonic systems. Of the three standard classifying symmetries, only time-reversal represents a real symmetry of the many-boson system, while the other two, particle-hole and chiral, are simply constraints that manifest as symmetries of the effective single-particle problem. For gapped systems in arbitrary space dimension we establish three fundamental no-go theorems that prove the absence of: parity switches, symmetry-protected-topological quantum phases, and localized bosonic zero modes under open boundary conditions. We then introduce a squaring, kernel-preserving map connecting noninteracting Hermitian theories of fermions and stable boson systems, which serves as a playground to reveal the role of topology in bosonic phases and their localized midgap boundary modes. Finally, we determine the symmetry classes inherited from the fermionic tenfold-way classification, unveiling an elegant threefold-way topological classification of non-interacting bosons. We illustrate our main findings in one-and two-dimensional bosonic lattice and field-theory models.

Restoring number conservation in quadratic bosonic Hamiltonians with dualities

“…-QBHs can display non-trivial bands characterized in terms of topological invariants. On the one hand, the topological invariants of number-conserving QBHs coincide with the well-established ones for fermions [20], even though the many-body interpretation of these quantities can change drastically [21]. On the other hand, for number-nonconserving QBHs the appropriate topological invariants are defined with respect to the indefinite metric of the Krein space K τ3 on which the effective BdG Hamiltonian acts [22].…”

“…Let G(k) denote an effective BdG Hamiltonian that depends on a vector of parameters k. Suppose further that G(k) is dynamically stable, with a complete basis of eigenstates |n(k) satisfying n(k)|τ 3 |m(k) = λ n δ nm , λ n being either +1 or −1. Following the usual assumptions of adiabatic evolution for the dynamics genereted by G(k) [21], one finds that…”