Linearized Eliashberg Equation
Note
References:
- [A.A. Abrikosov, L.P. Gor’kov, et.al., Pergamon, Oxford (1965)]
- [Takimoto, et. al., PRB 69, 104504 (2004)]
- [Yanase, et. al., Physics Reports 387, 1-149 (2003)]
Note
All indices on this page only represent orbital degrees of freedom.
Spin is not treated explicitly and therefore only spin-independent Hamiltonians can be used for calculations.
We asssume a homogenous system with some arbitrary effective pairing interaction \(\Gamma\), which leads to the formation of Cooper pairs.
Anomalous Green’s Functions
Note
Explain what happens with all spin quantum numbers in the single-particle Green’s function. Do we work with a particular combination of spins? \(G_{a\bar{b}} = G_{\alpha \uparrow \bar{b} \downarrow}\)?
With the arise of Cooper pairs we need in addition to the normal single-particle Green’s function
(1)\[G_{a\bar{b}}(\tau - \tau')
\equiv
- \langle \mathcal{T} c_{a}(\tau) c^\dagger_{\bar{b}}(\tau') \rangle
=
- \langle \mathcal{T} a(\tau) \bar{b}(\tau') \rangle\,,\]
and its backwards propagating counterpart
(2)\[\bar{G}_{\bar{a}b}(\tau - \tau')
\equiv
- \langle \mathcal{T} c^\dagger_{\bar{a}}(\tau) c_{b}(\tau') \rangle
=
- \langle \mathcal{T} \bar{a}(\tau) b(\tau') \rangle\,,\]
the single-particle anomalous Green’s functions \(F\) and \(\bar{F}\) to describe a superconducting state.
These are defined as
(3)\[F_{ab}(\tau - \tau')
\equiv
\langle \mathcal{T} c_{a}(\tau) c_{b}(\tau') \rangle
=
\langle \mathcal{T} a(\tau) b(\tau') \rangle
\,,\]
(4)\[\bar{F}_{\bar{a}\bar{b}}(\tau - \tau')
\equiv
\langle \mathcal{T} c^\dagger_{a}(\tau) c^\dagger_{\bar{b}}(\tau') \rangle
=
\langle \mathcal{T} \bar{a}(\tau) \bar{b}(\tau') \rangle\,.\]
Fourier transforming to Matsubara frequency space then gives that
(5)\[\begin{split}\bar{G}_{\bar{a}b}(\mathbf{k}, i\nu_n) = [ G_{b\bar{a}}(-\mathbf{k}, -i\nu_n) ]^{*}
\\
\bar{F}_{\bar{a}\bar{b}}(\mathbf{k}, i\nu_n) = [ F_{ba}(-\mathbf{k}, -i\nu_n) ]^{*}\end{split}\]
Dyson-Gorkov Equations
The former properties of a superconductor are given by the Dyson-Gorkov equations
(6)\[\mathbf{G}(\mathbf{k}, i\nu_n)
=
\mathbf{G}^{(0)}(\mathbf{k}, i\nu_n)
+ \mathbf{G}^{(0)}(\mathbf{k}, i\nu_n)
\ast \mathbf{\Sigma}(\mathbf{k}, i\nu_n)
\ast \mathbf{G}(\mathbf{k}, i\nu_n)\]
(7)\[\begin{split}\mathbf{G} \equiv
\left[ \begin{array}{cc}
G_{a\bar{b}} & F_{ab} \\
\bar{F}_{\bar{a}\bar{b}} & \bar{G}_{\bar{a}b} \\
\end{array} \right]
\quad
\mathbf{G}^{(0)}
\equiv
\left[ \begin{array}{cc}
G^{(0)}_{a\bar{b}} & 0 \\
0 & \bar{G}^{(0)}_{\bar{a}b} \\
\end{array} \right]
\quad
\mathbf{\Sigma}
\equiv
\left[ \begin{array}{cc}
\Sigma_{a\bar{b}} & \Delta_{ab} \\
\bar{\Delta}_{\bar{a}\bar{b}} & \bar{\Sigma}_{\bar{a}b} \\
\end{array} \right]\end{split}\]
In component form this becomes,
(8)\[G(a\bar{b}) = G^{(0)}(a\bar{b}) + G^{(0)}(a\bar{c})\Sigma(\bar{c}d)G(d\bar{b}) +
G^{(0)}(a\bar{c})\bar{\Delta}(\bar{c}\bar{d})\bar{F}(\bar{d}\bar{b})\]
(9)\[\bar{G}(\bar{a}b) = \bar{G}^{(0)}(\bar{a}b) + \bar{G}^{(0)}(\bar{a}c)\bar{\Sigma}(c\bar{d})\bar{G}(\bar{d}b) +
\bar{G}^{(0)}(\bar{a}c)\Delta(cd)F(db)\]
(10)\[F(ab) = G^{(0)}(a\bar{c}) \Sigma(\bar{c}d) F(db)+
G^{(0)}(a\bar{c}) \bar{\Delta}(\bar{c}\bar{d}) \bar{G}(\bar{d}b)\]
(11)\[\bar{F}(\bar{a}\bar{b}) = \bar{G}^{(0)}(\bar{a}c) \bar{\Sigma}(c\bar{d}) \bar{F}(\bar{d}\bar{b})+
\bar{G}^{(0)}(\bar{a}c) \Delta(cd) G(d\bar{b})\]
Here \(\Sigma\) is the normal self-energy and \(\Delta\) and \(\bar{\Delta}\) the anomalous self-energies, which are equal in the absence of a magnetic field and will be treated as from now on.
Anomalous self-energy and particle-particle vertex
Note
Define \(\Gamma\). It should be the particle-particle vertex \(\Gamma^{(pp)}\) related to the generalized susceptibility \(\chi\) through the Bethe-Salpeter equation in the particle-particle channel. This would give the definition of the four orbital(spin) indices and their order.
The anomalous self-energy can be expressed with the effective pairing interaction \(\Gamma\) and the anomalous Green’s function \(F\) as
(12)\[\Delta_{\bar{a}\bar{b}}(\mathbf{k},i\nu) = -\frac{1}{N_k \beta}\sum_{\mathbf{q}} \sum_{i\nu'}
\Gamma_{A\bar{a}\bar{b}B}(\mathbf{k}-\mathbf{q}, i\nu - i\nu') F_{AB}(\mathbf{q}, i\nu')\,.\]
Linearization in \(\Delta\)
Around the transition point to the superconducting state the anomalous self-energy \(\Delta\) is approximately zero, and, because we are only interested in the transition point, we linearize \(F\) in the Dyson-Gorkov equations with respect to \(\Delta\). This yields
(13)\[\begin{split}F & = g \Sigma F + g \Delta \bar{G} \\
G & = g + g \Sigma G + g \Delta \bar{F}\end{split}\]
(14)\[\begin{split}F & = (g^{-1} - \Sigma)^{-1} \Delta \bar{G} \\
\bar{G} & = (\bar{g}^{-1} - \Sigma)^{-1} + \bar{\Delta} F\end{split}\]
(15)\[F = (g^{-1} - \Sigma)^{-1} \Delta (\bar{g}^{-1} - \bar{\Sigma})^{-1} + \mathcal{O}(\Delta^2)\]
We then insert (15) into (12) and obtain the linearized Eliashberg equation
(16)\[\begin{split}\Delta_{\bar{a}\bar{b}}(\mathbf{k},i\nu) = -\frac{1}{N_k \beta}\sum_{\mathbf{q}} \sum_{i\nu'}
\Gamma_{A\bar{a}\bar{b}B}(\mathbf{k}-\mathbf{q}, i\nu - i\nu')
\\ \times
\big({G^{(0)}}^{-1}(\mathbf{q}, i\nu') - \Sigma(\mathbf{q}, i\nu') \big)^{-1}_{A\bar{c}}
\Delta_{\bar{c}\bar{d}}(\mathbf{q}, i\nu')
\big({G^{(0)}}^{-1}_{}(-\mathbf{q}, -i\nu') - \Sigma_{}(-\mathbf{q}, -i\nu') \big)^{-1}_{B\bar{d}}\,.\end{split}\]
To make use of this equations it is usually interpreted as an eigenvalue equation
(17)\[\lambda \Delta = \Lambda \Delta\,,\]
where the eigenvalue \(\lambda\) is seen as a measurement for the strength of superconducting ordering and a phase transition occurs when it reaches unity.
RPA Approach
Note
Explain what happens with momenta
The linearized Eliashberg equation can be studied in the RPA limit.
In this case the normal self-energy is set to zero and the effective pairing interaction \(\Gamma\) for a singlet Cooper pairs is given by
(18)\[\begin{split}\Gamma^{(\mathrm{singlet})}(a\bar{b}c\bar{d}) =
\frac{3}{2} U^{(\mathrm{s})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{s})}(\bar{B}A\bar{C}D)
U^{(\mathrm{s})}(D\bar{C}c\bar{d}) \\
-\frac{1}{2} U^{(\mathrm{c})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{c})}(\bar{B}A\bar{C}D)
U^{(\mathrm{c})}(D\bar{C}c\bar{d}) \\
+ \frac{1}{2}\big(U^{(\mathrm{s})}(a\bar{b}c\bar{d})+
U^{(\mathrm{c})}(a\bar{b}c\bar{d})\big)\,,\end{split}\]
and for a triplet by
(19)\[\begin{split}\Gamma^{(\mathrm{triplet})}(a\bar{b}c\bar{d}) =
-\frac{1}{2} U^{(\mathrm{s})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{s})}(\bar{B}A\bar{C}D)
U^{(\mathrm{s})}(D\bar{C}c\bar{d}) \\
-\frac{1}{2} U^{(\mathrm{c})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{c})}(\bar{B}A\bar{C}D)
U^{(\mathrm{c})}(D\bar{C}c\bar{d}) \\
+ \frac{1}{2}\big(U^{(\mathrm{s})}(a\bar{b}c\bar{d})+
U^{(\mathrm{c})}(a\bar{b}c\bar{d})\big)\,.\end{split}\]
Here \(\chi^{(\mathrm{s})}\) is the spin-susceptibility tensor defined by
(20)\[\chi^{(\mathrm{s})}(\bar{a}b\bar{c}d) = \big(\mathbb{1} - \chi^{(0)}(\bar{a}b\bar{A}B)
U^{(\mathrm{s})}(B\bar{A}C\bar{D})\big)^{-1} \chi^{(0)}(\bar{D}C\bar{c}d)\,,\]
and \(\chi^{(\mathrm{c})}\) is the charge-susceptibility tensor defined by
(21)\[\chi^{(\mathrm{c})}(\bar{a}b\bar{c}d) = \big(\mathbb{1} + \chi^{(0)}(\bar{a}b\bar{A}B)
U^{(\mathrm{c})}(B\bar{A}C\bar{D})\big)^{-1} \chi^{(0)}(\bar{D}C\bar{c}d)\,,\]
here \(\chi^{(0)}\) is the non-interacting particle-hole bubble.
The spin and charge interaction tensors are given by
(22)\[\begin{split}U^{(\mathrm{s})}(a\bar{a}b\bar{b}) =
\begin{cases}
U, & \mathrm{if}\;a=\bar{a}=b=\bar{b} \\
U', & \mathrm{if}\;a=\bar{b}\neq \bar{a}=b \\
J, & \mathrm{if}\;a=\bar{a}\neq b=\bar{b} \\
J', & \mathrm{if}\;a=b\neq \bar{a}=\bar{b} \\
0, & \mathrm{else}
\end{cases}\end{split}\]
(23)\[\begin{split}U^{(\mathrm{c})}(a\bar{a}b\bar{b}) =
\begin{cases}
U, & \mathrm{if}\;a=\bar{a}=b=\bar{b} \\
-U'+2J, & \mathrm{if}\;a=\bar{b}\neq \bar{a}=b \\
2U'-J, & \mathrm{if}\;a=\bar{a}\neq b=\bar{b} \\
J', & \mathrm{if}\;a=b\neq \bar{a}=\bar{b} \\
0, & \mathrm{else}
\end{cases}\end{split}\]
where \(U\), \(U'\), \(J\) and \(J'\) are the usual Kanamori interaction parameters.