.. _eliashberg: 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 :math:`\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? :math:`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 .. math:: 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 .. math:: \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 :math:`F` and :math:`\bar{F}` to describe a superconducting state. These are defined as .. math:: F_{ab}(\tau - \tau') \equiv \langle \mathcal{T} c_{a}(\tau) c_{b}(\tau') \rangle = \langle \mathcal{T} a(\tau) b(\tau') \rangle \,, .. math:: \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 .. math:: \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) ]^{*} Dyson-Gorkov Equations ---------------------- The former properties of a superconductor are given by the Dyson-Gorkov equations .. math:: \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) .. math:: \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] In component form this becomes, .. math:: 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}) .. math:: \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) .. math:: 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) .. math:: \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 :math:`\Sigma` is the normal self-energy and :math:`\Delta` and :math:`\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 :math:`\Gamma`. It should be the particle-particle vertex :math:`\Gamma^{(pp)}` related to the generalized susceptibility :math:`\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 :math:`\Gamma` and the anomalous Green's function :math:`F` as .. math:: \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')\,. :label: anom_self_energy Linearization in :math:`\Delta` ------------------------------- Around the transition point to the superconducting state the anomalous self-energy :math:`\Delta` is approximately zero, and, because we are only interested in the transition point, we linearize :math:`F` in the Dyson-Gorkov equations with respect to :math:`\Delta`. This yields .. math:: F & = g \Sigma F + g \Delta \bar{G} \\ G & = g + g \Sigma G + g \Delta \bar{F} .. math:: F & = (g^{-1} - \Sigma)^{-1} \Delta \bar{G} \\ \bar{G} & = (\bar{g}^{-1} - \Sigma)^{-1} + \bar{\Delta} F .. math:: F = (g^{-1} - \Sigma)^{-1} \Delta (\bar{g}^{-1} - \bar{\Sigma})^{-1} + \mathcal{O}(\Delta^2) :label: lin_anom_gf We then insert :eq:`lin_anom_gf` into :eq:`anom_self_energy` and obtain the linearized Eliashberg equation .. math:: \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}}\,. To make use of this equations it is usually interpreted as an eigenvalue equation .. math:: \lambda \Delta = \Lambda \Delta\,, where the eigenvalue :math:`\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 :math:`\Gamma` for a singlet Cooper pairs is given by .. math:: \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)\,, and for a triplet by .. math:: \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)\,. Here :math:`\chi^{(\mathrm{s})}` is the spin-susceptibility tensor defined by .. math:: \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 :math:`\chi^{(\mathrm{c})}` is the charge-susceptibility tensor defined by .. math:: \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 :math:`\chi^{(0)}` is the non-interacting particle-hole bubble. The spin and charge interaction tensors are given by .. math:: 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} .. math:: 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} where :math:`U`, :math:`U'`, :math:`J` and :math:`J'` are the usual Kanamori interaction parameters.