Emergence of multiple Higgs modes in a superconductor due to spontaneous breakdown of a Z₂ symmetry

Higgs modes and Nambu-Goldstone (NG) modes are ubiquitous collective excitations that arise in systems with spontaneous symmetry breaking. Although Higgs modes are believed to emerge with NG modes when continuous symmetries are spontaneously broken, Higgs modes do not necessarily appear in systems exhibiting spontaneous breakdown of continuous symmetries. We study the Higgs mode in a Bardeen-Cooper-Schrieffer (BCS) superconductor. Motivated by the observation that U(1) symmetry of the BCS Hamiltonian is not essential for the Higgs mode, we study the Ising-like Hamiltonian in the pseudospin representation. We show that the Higgs mode emerges as the lowest excited state of the Ising-like Hamiltonian due to spontaneous breakdown of Z₂ symmetry under the time-reversal operation T in the pseudospin space. We further predict the existence of multiple Higgs modes that have quantized energy 2(n+1)∆₀ (0 ≤ n ≤ N_kF ), where ∆₀ is the superconducting gap, n is an integer, and N_kF is the number of states on the Fermi surface.