Theory of Green’s Functions of Nambu-Goldstone and Higgs Modes in Superconductors

We examine fundamental properties of Green’s functions of Nambu-Goldstone and Higgs modes in superconductors with multiple order parameters. Nambu-Goldstone and Higgs modes are determined once the symmetry of the system and that of the order parameters are specified. Multiple Nambu-Goldstone modes and Higgs modes exist when we have multiple order parameters. The Nambu-Goldstone Green function D ( ω , q ) has the form 1 / ( g N ( 0 ) ) 2 ⋅ ( 2 Δ ) 2 / ( ω 2 − c s 2 q 2 ) with the coupling constant g and c s = v F / 3 for small ω and q , with a pole at ω = 0 and q = 0 indicating the existence of a massless mode. It is shown, based on the Ward-Takahashi identity, that the massless mode remains massless in the presence of intraband scattering due to non-magnetic and magnetic impurities. The pole of D ( ω , q ), however, disappears as ω increases as large as 2Δ: ω ∼ 2 Δ . The Green function H ( ω , q ) of the Higgs mode is given by H ( ω , q ) ∝ ( 2 Δ ) 2 / ( ( 2 Δ ) 2 − 1 3 ω 2 + 1 3 c s 2 q 2 ) for small ω and q , and H ( ω , q ) is proportional to 1 / ( g N ( 0 ) ) 2 ⋅ Δ / ( 2 Δ ) 2 + c s 2 q 2 − ω 2 for ω ∼ 2 Δ with the cut in the interval − ω ( q ) ≤ z ≡ q 0 ≤ ω ( q ) where ω ( q ) = ( 2 Δ ) 2 + c s 2 q 2 . The constant part of the action for the Higgs modes is important since it determines the coherence length of a superconductor. There is the case that it has a large eigenvalue, indicating that the large upper critical field H c 2 may be realized in a superconductor with multiple order parameters.