Green's functions of Nambu-Goldstone modes 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 $D(\omega ,\mathbf{q})$ has the form $1/(gN(0))^2\cdot (2\Delta)^2/(\omega^2-c_s^2{\bf q}^2)$withthecouplingconstant$g$and$c_s=v_F/\sqrt{3}$forsmall$\omega$and${\bf q}$,withapoleat$\omega=0$and${\bf q}=0$ 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 nonmagnetic and magnetic impurities. The pole of $D(\omega ,\mathbf{q})$, however, disappears as $\omega$ increases as large as $2\mathrm{\Delta }$: $\omega \sim 2\mathrm{\Delta }$. The Green function $H(\omega,{\bf q})$oftheHiggsmodeisgivenby$H(\omega,{\bf q})\propto (2\Delta)^2/((2\Delta)^2-\frac{1}{3}\omega^2+\frac{1}{3}c_s^2{\bf q}^2)$ for small $\omega$ and $\mathbf{q}$. $H(\omega ,\mathbf{q})$ is proportional to $1\mathrm{/}(gN(0){)}^2\cdot \mathrm{\Delta }\mathrm{/}\sqrt{(2\mathrm{\Delta }{)}^2+c_s^2{\mathbf{q}}^2-{\omega }^2}$ for $\omega \sim 2\mathrm{\Delta }$ and $\omega <\omega (\mathbf{q})$. This behavior is similar to that of the $\sigma$-particle Green function in the Gross-Neveu model. That is, the Higgs Green function $H(\omega ,\mathbf{q})$ has the same singularity as the Green function of the $\sigma$ boson of the Gross-Neveu model. 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 $H_{c2}$ may be realized in a superconductor with multiple order parameters.