Spin-independent RPA calculations

In many papers, such as Takimoto, et. al., PRB 69, 104504 (2004), people use a matrix scheme to calculate spin- and charge-susceptibilites in RPA. In contrast to the generalized susceptibility used in TPRF they don’t use an explicit spin index, which is valid in SU(2) symmetric systems. This allows for more efficient calculations both in memory and time. A mapping between both methods is therefore advantageous and shown in this documentation entry.

With the mapping we can obtain a full generalized RPA susceptibility from spin-independent calculations by decouple a spin-dependent interaction into spin and charge channel.

Matrix RPA

The matrix RPA is based on two equations, one for the spin-susceptibility

(1)\[\hat{\chi}^{(\mathrm{s})} = \big(\mathbb{1} - \hat{\chi}^{(0)} \hat{U}^{(\mathrm{s})}\big)^{-1} \hat{\chi}^{(0)}\,,\]

and the other for the charge susceptibility

(2)\[\hat{\chi}^{(\mathrm{c})} = \big(\mathbb{1} + \hat{\chi}^{(0)} \hat{U}^{(\mathrm{c})}\big)^{-1} \hat{\chi}^{(0)}\,.\]

Note

This is different to the equations in the paper Takimoto, et. al., PRB 69, 104504 (2004) where they have \(\chi\) and \(U\) flipped. But I think that this is the correct way to write it. This difference must be resolved, because it will have difference on the particle-particle vertex, and therefore calculations of the Eliashberg equation in the RPA limit.

Here every quantity with a hat is a matrix and \(\hat{\chi}^{(0)}\) is therefore the matrix representation of the bare particle-hole bubble and the same holds for the spin channel \(\hat{U}^{(\mathrm{s})}\) and charge channel \(\hat{U}^{(\mathrm{c})}\) of the interaction.

The matrices are given as

(3)\[\begin{split}\hat{\chi}^{(0)} = \begin{pmatrix} \chi^{(0)}_{0000} & \chi^{(0)}_{0011} & \chi^{(0)}_{0001} & \chi^{(0)}_{0010}\\ \chi^{(0)}_{1100} & \chi^{(0)}_{1111} & \chi^{(0)}_{1101} & \chi^{(0)}_{1110}\\ \chi^{(0)}_{0100} & \chi^{(0)}_{0111} & \chi^{(0)}_{0101} & \chi^{(0)}_{0110}\\ \chi^{(0)}_{1000} & \chi^{(0)}_{1011} & \chi^{(0)}_{1001} & \chi^{(0)}_{1010}\\ \end{pmatrix}\,,\end{split}\]

and for a Kanamori interaction by

(4)\[\begin{split}\hat{U}^{(\mathrm{s})} = \begin{pmatrix} U & J & 0 & 0\\ J & U & 0 & 0\\ 0 & 0 & U' & J'\\ 0 & 0 & J' & U'\\ \end{pmatrix}\,,\end{split}\]

(5)\[\begin{split}\hat{U}^{(\mathrm{c})} = \begin{pmatrix} U & 2U'-J & 0 & 0\\ 2U'-J & U & 0 & 0\\ 0 & 0 & -U'+2J & J'\\ 0 & 0 & J' & -U'+2J\\ \end{pmatrix}\,.\end{split}\]

The equation for the elements of the bare-particle hole bubbles is given by

(6)\[\chi^{(0)}_{\bar{\alpha}\beta\gamma\bar{\delta}}(q, i\omega) = -\frac{T}{N} \sum_{k, \nu} G^{(0)}_{\gamma\bar{\alpha}}(k+q, i\nu + i\omega) G^{(0)}_{\beta\bar{\delta}}(k, i\nu) \,,\]

were we used greek indices, which we will do exclusively for matrix RPA quantities to highlight that they are spin-independent. The spin and charge channel interaction are given by

(7)\[\begin{split}U^{(\mathrm{s})}(\alpha\bar{\beta}\bar{\gamma}\delta) = \begin{cases} U, & \mathrm{if}\;\alpha=\bar{\beta}=\bar{\gamma}=\delta \\ U', & \mathrm{if}\;\alpha=\bar{\gamma}\neq \bar{\beta}=\delta \\ J, & \mathrm{if}\;\alpha=\bar{\beta}\neq \bar{\gamma}=\delta \\ J', & \mathrm{if}\;\alpha=\delta\neq \bar{\beta}=\bar{\gamma} \\ 0, & \mathrm{else} \end{cases}\,,\end{split}\]

(8)\[\begin{split}U^{(\mathrm{c})}(\alpha\bar{\beta}\bar{\gamma}\delta) = \begin{cases} U, & \mathrm{if}\;\alpha=\bar{\beta}=\bar{\gamma}=\delta \\ -U'+2J, & \mathrm{if}\;\alpha=\bar{\gamma}\neq \bar{\beta}=\delta \\ 2U'-J, & \mathrm{if}\;\alpha=\bar{\beta}\neq \bar{\gamma}=\delta \\ J', & \mathrm{if}\;\alpha=\delta\neq \bar{\beta}=\bar{\gamma} \\ 0, & \mathrm{else} \end{cases}\,.\end{split}\]

Here we have to note that operator order used in the matrix RPA differs from the one we use in TPRF. While in TPRF the susceptibilites are ordered as \(c^\dagger cc^\dagger c\) and the vertices as \(cc^\dagger cc^\dagger\), in matrix RPA the last two indices are flipped. This means \(c^\dagger ccc^\dagger\) for susceptibilites and \(cc^\dagger c^\dagger c\) for vertices. This flipping of the last two indices corresponds to the particle-hole product, see Particle-Hole channel (), which the matrix RPA explicitly keeps in the notation. Therefore, when comparing matrix RPA susceptibilites to TPRF ones this flip of indices has to be taken into account.

Mapping between spin-dependent and independent quantities

While the greek indices only carry orbital information the latin indices used for the quantities in TPRF carry orbital and spin information. To map the spin-dependent to independent quantities and vice versa, we introduce following notation

(9)\[a = \mathrm{orb}(a)_{\sigma(a)} = \alpha_{\sigma(a)}\,,\]

where the \(\mathrm{orb}\) function extracts the orbital information, mapping to the greek letters, and the \(\sigma\) extracts the spin.

With this we can state the mapping between the susceptibilites as

(10)\[\chi^{(s)}(\bar{\alpha}\beta\bar{\gamma}\delta) = \chi^{\mathrm{(RPA)}}(\bar{\alpha}_\uparrow \beta_\uparrow \bar{\gamma}_\uparrow \delta_\uparrow)- \chi^{\mathrm{(RPA)}}(\bar{\alpha}_\uparrow \beta_\uparrow \bar{\gamma}_\downarrow \delta_\downarrow)\,,\]

(11)\[\chi^{(c)}(\bar{\alpha}\beta\bar{\gamma}\delta) = \chi^{\mathrm{RPA}}(\bar{\alpha}_\uparrow \beta_\uparrow \bar{\gamma}_\uparrow \delta_\uparrow)+ \chi^{\mathrm{RPA}}(\bar{\alpha}_\uparrow \beta_\uparrow \bar{\gamma}_\downarrow \delta_\downarrow)\,,\]

and

(12)\[\begin{split}\chi^{(\mathrm{RPA})}(\bar{a}b\bar{c}d) = \begin{cases} \frac{1}{2}\big(\chi^{(\mathrm{c})} + \chi^{(\mathrm{s})}\big)(\bar{\alpha}\beta\bar{\gamma}\delta),& \mathrm{if}\; \sigma(\bar{a}) = \sigma(b) = \sigma(\bar{c}) = \sigma(d)\\ \frac{1}{2}\big(\chi^{(\mathrm{c})} - \chi^{(\mathrm{s})}\big)(\bar{\alpha}\beta\bar{\gamma}\delta),& \mathrm{if}\; \sigma(\bar{a}) = \sigma(b) \neq \sigma(\bar{c}) = \sigma(d)\\ \chi^{(\mathrm{s})}(\bar{\alpha}\beta\bar{\gamma}\delta),& \mathrm{if}\; \sigma(\bar{a}) = \sigma(d) \neq \sigma(b) = \sigma(\bar{c})\\ 0, & \mathrm{else} \end{cases}\,.\end{split}\]

And for the interaction they are given by

(13)\[U^{(s)}(\alpha\bar{\beta}\gamma\bar{\delta}) = U(\alpha_\uparrow \bar{\beta}_\uparrow \gamma_\uparrow \bar{\delta}_\uparrow)- U(\alpha_\uparrow \bar{\beta}_\uparrow \gamma_\downarrow \bar{\delta}_\downarrow)\,,\]

(14)\[U^{(c)}(\alpha\bar{\beta}\gamma\bar{\delta}) = -U(\alpha_\uparrow \bar{\beta}_\uparrow \gamma_\uparrow \bar{\delta}_\uparrow)- U(\alpha_\uparrow \bar{\beta}_\uparrow \gamma_\downarrow \bar{\delta}_\downarrow)\,,\]

(15)\[\begin{split}U(a\bar{b}c\bar{d}) = \begin{cases} \frac{1}{2}\big(U^{(\mathrm{s})} - U^{(\mathrm{c})}\big)(\alpha\bar{\beta}\gamma\bar{\delta}),& \mathrm{if}\; \sigma(a) = \sigma(\bar{b}) = \sigma(c) = \sigma(\bar{d})\\ \frac{1}{2}\big(-U^{(\mathrm{c})} - U^{(\mathrm{s})}\big)(\alpha\bar{\beta}\gamma\bar{\delta}),& \mathrm{if}\; \sigma(a) = \sigma(\bar{b}) \neq \sigma(c) = \sigma(\bar{d})\\ U^{(\mathrm{s})}(\alpha\bar{\beta}\gamma\bar{\delta}),& \mathrm{if}\; \sigma(a) = \sigma(\bar{d}) \neq \sigma(\bar{b}) = \sigma(c)\\ 0, & \mathrm{else} \end{cases}\,.\end{split}\]

Example

If you have a spin-dependent bare particle-hole bubble chi00_wk and a spin-dependent vertex U_abcd, you could use the following code snippet to produce the corresponding spin-dependent general RPA susceptibility chi_wk, without doing a spin-dependent calculation.

from triqs_tprf.rpa_tensor import lose_spin_degree_of_freedom
chi00_wk_wo_spin = lose_spin_degree_of_freedom(chi00_wk, spin_fast=False)

from triqs_tprf.rpa_tensor import lose_spin_degree_of_freedom
U_c, U_s = split_quartic_tensor_in_charge_and_spin(U_abcd)

from triqs_tprf.lattice import solve_rpa_PH
chi_s = solve_rpa_PH(chi00_wk_wo_spin, U_s)
chi_c = solve_rpa_PH(chi00_wk_wo_spin, -U_c) # Minus for correct charge rpa equation

from triqs_tprf.rpa_tensor import general_susceptibility_from_charge_and_spin
chi_wk = general_susceptibility_from_charge_and_spin(chi_c, chi_s, spin_fast=False)

Or you could already start at the spin-dependent Green’s function g0_wk to construct a spin-independent bare particle-hole bubble.

from triqs_tprf.rpa_tensor import lose_spin_degree_of_freedom
g0_wk_wo_spin = lose_spin_degree_of_freedom(g0_wk, spin_fast=False)

from triqs_tprf.lattice_utils import imtime_bubble_chi0_wk
chi00_wk_wo_spin = imtime_bubble_chi0_wk(g0_wk_wo_spin, nw=1)