User:EverettYou/Books/Physics/CMFT/BCS

Introduction
See BCS theory.

Green's Function: Equations and Solutions
Consider the electron band structure modeled by the following action
 * $$S_0=\sum_{k}c_k^\dagger(-i\omega+\xi(\boldsymbol{k}))c_k$$,

where $$c_k$$ is the electron field containing spin and orbital degrees of freedom, and $$\xi(\boldsymbol{k})$$ is the matrix of band Hamiltonian. The bare Green's function is defined as the correlation between the field $$c_k$$ and its conjugate field $$G_0(k)\equiv -\langle c_k c_k^\dagger\rangle_0$$, which can be read out from the action $$S_0$$ as
 * $$G_0(k)=-(i\omega-\xi(\boldsymbol{k}))^{-1}$$.

Turn on the pairing interaction $$S=S_0+S_\text{int}$$, the Fermi surface instability drives the system into the superconducting phase at low temperature. The superconducting off-diagonal long range order indicates the correlation between the field $$c_k$$ and $$c_{-k}$$ as well, which is know as the abnormal Green's function $$F(k)\equiv-\langle c_k c_{-k}\rangle$$. In contrast to the normal Green's function $$G(k)\equiv-\langle c_k c_k^\dagger\rangle$$. These Green's functions are subject to the equations of motion known as the Gor'kov equations, which can be understood as the special case of the Dynson equations for self-energy correction, taking the following form
 * $$G(k)=G_0(k)+G_0(k)\Delta(k)F^\dagger(k)$$,
 * $$F^\dagger(k)=-G_0^\mathsf{T}(-k)\Delta^\dagger(k)G(k)$$,

where $$\Delta(k)$$ is basically a self-energy function, conventionally called the gap function, as it determines the energy gap of pairing. The gap function is related to the abnormal Green's function such that the above equations can be closed. Their relation depends on the details of pairing interaction, and can not be formulated in general. However a simple assumption may be $$\Delta(k)\propto F(k)$$, which can be justified in the limit of short-range and instantaneous interaction. A more sophisticated model is to consider the pairing interaction mediated by some bosonic field $$b_q$$ (like phonon or magnon) with the correlation $$D_0(q)\equiv -\langle b_q b_q^\dagger\rangle_0$$, then the gap function takes the form of the exchange self-energy
 * $$\Delta(k)=-\sum_q \mathrm{Tr} D_0(q)vF(k-q)(-v^\mathsf{T})$$,

where $$v$$ is the vertex operator coupling the electron field to the bosonic field, and Tr trace out the bosonic degrees of freedom.

In general, the Gor'kov equations could only be solved numerically by iterative approach. Sometimes, assumptions can be made to simplify the equations so as to obtain some analytical result. First of all, for a time reversal symmetric electron band structure, $$G^\mathsf{T}(k) = G(k)$$. Secondly, superconductivity has U(1) gauge structure that can be fixed by enforcing $$F(k)$$ to be real, so that $$F^\dagger(k)=F^\mathsf{T}(k)=-F(-k)$$. With these assumptions, the Gor'kov equations can be expressed in terms of G and F only, and can be further merged into a single equation
 * $$G(k)=G_0(k)-G_0(k)\Delta(k)G_0(-k)\Delta^\dagger(k)G(k)$$,

which has a formal solution
 * $$G(k)=\left(G_0^{-1}(k)+\Delta(k)G_0(-k)\Delta^\dagger(k)\right)^{-1}$$.

In the weak coupling limit, the gap function can be considered as a perturbation. Then $$G_0^{-1}(k)$$ always dominates the denominator unless k is on the mass-shell where $$G_0^{-1}(k)=0$$. Therefore the value of $$\Delta(k)$$ is only desired on the mass-shell. In the low temperature limit, only zero-frequency component is important. If the pairing symmetry is know, a form factor $$f(\boldsymbol{k})$$ (including the matrix part describing the spin and orbital degrees of freedom) can be assigned, then the gap function is only controlled by a single parameter Δ,
 * $$\Delta(k) = \Delta f(\boldsymbol{k})$$ for $$i\omega=0$$.

In this case, it is possible to obtain an analytic expression of the normal Green's function G, and hence the abnormal Green's function F. Then the other physical properties can be calculated based on the knowledge of G and F.

Physical Properties
Let Π be the (minus) correlation function between two Fermion bilinear vertex operators v1 and v2, i.e. $$\Pi=-\langle v_1 v_2\rangle$$. It can be calculated from Green's functions G and F as
 * $$\Pi(q)=\sum_k \mathrm{Tr}\left(v_1G(k)v_2G(k+q)+ v_1F(k)(-v_2^\mathsf{T})F^\dagger(k+q)\right)$$.

Here Tr trace out the fermionic degrees of freedom. In particular, if vi is the velocity operator (or density-current operator), then Π stands for the polarizability function (the self-energy correction to the propagator of electromagnetic field); if vi is the spin operator, then Π stands for the susceptibility function.

Spin Orbital Coupling Case
Suppose
 * $$\xi(\boldsymbol{k})=\xi_0(\boldsymbol{k})\sigma_0 + \xi_i(\boldsymbol{k})\sigma_i$$,

where σ0 is the identity matrix, and σi (i=1,2,3) stand for the Pauli matrices in the spin space. The spin orbital coupling (SOC) strength is reflected by $$|\boldsymbol{\xi}|=(\xi_1^2+\xi_2^2+\xi_3^2)^{1/2}$$. In the weak SOC limit ($$|\boldsymbol{\xi}|/\xi_0\rightarrow 0$$), the solution of Green's functions are
 * $$G(k)=G_+(k)\sigma_0+\hat{\xi}_i(\boldsymbol{k})\sigma_i G_-(k)$$,
 * $$F(k)=F_+(k)\sigma_0+\hat{\xi}_i(\boldsymbol{k})\sigma_i F_-(k)$$,

where
 * $$G_\pm(k)=-\frac{1}{2}\left(\frac{i\omega+\xi_+(\boldsymbol{k})}{\omega^2+\xi_+^2(\boldsymbol{k})+|\Delta|^2} \pm \frac{i\omega+\xi_-(\boldsymbol{k})}{\omega^2+\xi_-^2(\boldsymbol{k})+|\Delta|^2}\right)$$,
 * $$F_\pm(k)=-\frac{1}{2}\left(\frac{1}{\omega^2+\xi_+^2(\boldsymbol{k})+|\Delta|^2} \pm \frac{1}{\omega^2+\xi_-^2(\boldsymbol{k})+|\Delta|^2}\right)\Delta f(\boldsymbol{k})$$,

with $$\hat{\xi}_i(\boldsymbol{k})=\xi_i(\boldsymbol{k})/|\boldsymbol{\xi}(\boldsymbol{k})|$$, $$\xi_\pm(\boldsymbol{k})=\xi_0(\boldsymbol{k})\pm|\boldsymbol{\xi}(\boldsymbol{k})|$$, and $$f(\boldsymbol{k})=i\sigma_2(f_s(\boldsymbol{k}) \sigma_0 + f_p(\boldsymbol{k}) \hat{\xi}_i(\boldsymbol{k})\sigma_i)$$. The parameter Δ controls the overall gap, while fs and fp specifies the s-wave and p-wave form factors respectively. The gap Δ and form factors should be determined from a self-consistent mean field equation numerically, or analytically with certain approximations.