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

Abstract

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 $1/(gN(0))^{2}\cdot (2{\Delta })^{2}/(\omega ^{2}-{{c}_{s}^{2}}\mathbf {q}^{2})$ with the coupling constant g and $c_{s}=v_{F}/\sqrt {3}$ 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: $\omega \sim 2{\Delta }$ . The Green function H(ω, q) of the Higgs mode is given by $H(\omega ,\mathbf {q})\propto (2{\Delta })^{2}/((2{\Delta })^{2}-\frac {1}{3}\omega ^{2} +\frac {1}{3}{c_{s}^{2}}\mathbf {q}^{2})$ for small ω and q, and H(ω, q) is proportional to $1/(gN(0))^{2}\cdot {\Delta }/\sqrt { (2{\Delta })^{2}+{c_{s}^{2}}\mathbf {q}^{2}-\omega ^{2}}$ for $\omega \sim 2{\Delta }$ with the cut in the interval − ω(q) ≤ z ≡ q0 ≤ ω(q) where $\omega (\mathbf {q})=\sqrt { (2{\Delta })^{2}+{c_{s}^{2}}\mathbf {q}^{2}}$ . 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 Hc2 may be realized in a superconductor with multiple order parameters.