  We study the Higgs mode in a Bardeen-Cooper-Schrieffer (BCS) superconductor
with a particle-hole symmetric fermion energy dispersion. We show that the
emergence of the Higgs mode is due to spontaneous breakdown of $\mathbb{Z}_2$
symmetry under the time-reversal operation $\mathcal T$ in the pseudospin
space. Motivated by the observation that the full U(1) symmetry of the BCS
Hamiltonian is not essential for the Higgs mode, we study the Ising-type
Hamiltonian in the pseudospin representation. We demonstrate that the Higgs
mode arises as an excited state of the Ising-type Hamiltonian above the ground
state that involves spontaneous breakdown of $\mathbb{Z}_2$ under $\mathcal T$.
We further predict the existence of multiple Higgs modes that have quantized
energy $2(n+1)\Delta_0$ ($1\le n\le N_{k_F}$), where $\Delta_0$ is the
superconducting gap, $n$ is an integer, and $N_{k_F}$ is the number of states
on the Fermi surface.
