Hidden charge-conjugation, parity, and time-reversal symmetries and massive Goldstone (Higgs) modes in superconductors

A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode, is a collective excitation that arises in a system involving spontaneous breaking of a continuous symmetry, along with a gapless Nambu-Goldstone mode. It has been known in the previous studies that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions exhibits the particle-hole (p-h) symmetry. However, clear understanding of the relation between the symmetry of the Hamiltonian and the MG modes has not been reached. Here we reveal the fundamental connection between the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes. To this end, we introduce nontrivial charge-conjugation ($mathcal C$), parity ($mathcal P$), and time-reversal ($mathcal T$) operations that involve the swapping of pairs of wave vectors symmetrical with respect to the Fermi surface. The product of $mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous to the CPT theorem in the relativistic field theory. It is shown that a fermionic Hamiltonian with a p-h symmetric dispersion exhibits the discrete symmetries under $mathcal C$, $mathcal P$, $mathcal T$, and $mathcal{CPT}$. We find that in the superconducting ground state, $mathcal T$ and $mathcal P$ are spontaneously broken simultaneously with the U(1) symmetry. Moreover, we rigorously show that amplitude and phase fluctuations of the gap function are uncoupled due to the unbroken $mathcal C$. In the normal phase, the MG and NG modes become degenerate, and they have opposite parity under $mathcal T$. Therefore, we conclude that the lifting of the degeneracy in the superconducting phase and the resulting emergence of the pure amplitude MG mode can be identified as a consequence of the the spontaneous breaking of $mathcal T$ symmetry but not of $mathcal P$ or U(1).