# Topological Classification of Excitations in Quadratic Bosonic Systems

## TL;DR

This paper develops a topological classification framework for excitations in quadratic bosonic systems, extending symmetry classes and predicting unique topological phases with concrete models.

## Contribution

It introduces a ten-fold symmetry classification for bosonic excitations and derives a topological classification table based on Hamiltonian decomposition.

## Key findings

- Derived a topological classification table for bosonic systems.
- Predicted unique topological phases specific to bosonic excitations.
- Proposed concrete models demonstrating bosonic topological phases.

## Abstract

We investigate the topological classification of excitations in quadratic bosonic systems with an excitation band gap. Time-reversal, charge-conjugation, and parity symmetries in bosonic systems are introduced to realize a ten-fold symmetry classification. We find a specific decomposition of the quadratic bosonic Hamiltonian and use it to prove that each quadratic bosonic system is homotopic to a direct sum of two single-particle subsystems. The topological classification table is thus derived via inheriting from that of Atland-Zirnbauer classes and unique topological phases of bosons are predicted. Finally, concrete topological models are proposed to demonstrate the peculiarity of bosonic excitations.