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| 1 | +# Decomposition of Interaction in two dimensions |
| 2 | + |
| 3 | +In GW-approximation, we calculation self-energy as |
| 4 | +```math |
| 5 | +\Sigma(\mathbf{k},\omega_n)=-T\int \frac{{\rm d}^d \mathbf{q}}{(2\pi)^d} \sum_m G(\mathbf{p},\omega_m)W(\mathbf{k-p},\omega_n-\omega_m) \,, |
| 6 | +\tag{1} |
| 7 | +``` |
| 8 | +where ``G`` is the Green's function and W is the effective interaction. Here, we suppress spin index. |
| 9 | + |
| 10 | +## Spherical harmonic representation |
| 11 | +We first express the ``W(q,\tau)`` function as an expansion in Legendre polynomials ``P_\ell(\chi)`` |
| 12 | +```math |
| 13 | +\begin{gathered} |
| 14 | +W(|\mathbf{k}-\mathbf{p}|, \tau)=\sum_{\ell=0}^{\infty} \bar{w}_{\ell}(k, p, \tau) P_{\ell}(\hat{k p}) \,, \\ |
| 15 | +\bar{w}_{\ell}(k, p, \tau)=\frac{N(d,\ell)}{2} \int_{-1}^{1} d \chi P_{\ell}(\chi) W\left(\sqrt{k^{2}+p^{2}-2 k p \chi} ,\tau\right)\,. |
| 16 | +\end{gathered} |
| 17 | +``` |
| 18 | +Since the Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using |
| 19 | +```math |
| 20 | +P_{\ell}(\hat{k p})=\frac{\Omega_{d}}{N(d,\ell)} \sum_{m=1}^{N(d,\ell)} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p})\,, |
| 21 | +``` |
| 22 | +where ``\Omega_{d}`` is the solid angle in ``d`` dimensions, and |
| 23 | +```math |
| 24 | +N(d, \ell)=\frac{2 \ell+d-2}{\ell}\left( |
| 25 | +\begin{array}{c} |
| 26 | +\ell+d-3 \\ |
| 27 | +\ell-1 |
| 28 | +\end{array}\right) |
| 29 | +``` |
| 30 | +denotes the number of linearly independent homogeneous harmonic polynomials of degree ``\ell`` in ``d`` dimensions. The spherical harmonics are orthonormal as |
| 31 | +```math |
| 32 | +\int {\rm d}\Omega_{\hat k} Y_{\ell m}(\hat k) Y_{\ell^\prime m^\prime}(\hat k) = \delta_{\ell \ell^\prime} \delta_{mm^\prime} \,. |
| 33 | +``` |
| 34 | +Hence, the ``W(q,\tau)`` function can expressed further on as |
| 35 | +```math |
| 36 | +W(|\mathbf{k}-\mathbf{p}|, \tau)=\sum_{\ell} \frac{\Omega_{d}}{N(d,\ell)} \bar{w}_{\ell}(k, p, \tau) \sum_{m} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p}) |
| 37 | +``` |
| 38 | + |
| 39 | +or |
| 40 | +```math |
| 41 | +W(|\mathbf{k}-\mathbf{p}|, \tau)= \frac{\Omega_{d}}{2}\sum_{\ell} w_{\ell}(k, p, \tau) \sum_{m} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p} )\,. |
| 42 | +``` |
| 43 | +with |
| 44 | +```math |
| 45 | +w_{\ell}(k, p, \tau)=\int_{-1}^{1} d \chi P_{\ell}(\chi) W\left(\sqrt{k^{2}+p^{2}-2 k p \chi} ,\tau\right) |
| 46 | +``` |
| 47 | +In addition, the Green's function ``G(\mathbf p, \tau)`` has |
| 48 | +```math |
| 49 | +\begin{aligned} |
| 50 | + G(\mathbf p, \tau)= \sum_{\ell=0}^{\infty}\sum_{m=1}^{N(d,\ell)} G_{\ell m}(p,\tau) Y_{\ell m}(\hat p)\,, \\ |
| 51 | + G_{\ell m}(p,\tau) = \int {\rm d}\Omega_{\hat k} G(\mathbf p,\tau) Y^*_{\ell m}(\hat p) |
| 52 | +\end{aligned} |
| 53 | +``` |
| 54 | +### Decouple with channels |
| 55 | +By the sphereical harmonic expansion of Eq.(1), the self-energy |
| 56 | + |
| 57 | +```math |
| 58 | +\begin{aligned} |
| 59 | +\sum_{\ell m} \Sigma_{\ell m }(k,\tau)Y_{\ell m}(\hat k) &= \frac{\Omega_d}{2} \int \frac{{\rm d}\mathbf p}{(2\pi)^d} \sum_{\ell m}G_{\ell m}(p, \tau) Y_{\ell m}(\hat p) \sum_{\ell^\prime m^\prime} w_{\ell^\prime}(k,p,\tau) Y_{\ell^\prime m^\prime}(\hat{k}) Y_{\ell^\prime m^\prime}^{*}(\hat{p} ) \\ |
| 60 | +&=\frac{\Omega_d}{2} \sum_{\ell m} \int \frac{p^{d-1}dp}{(2\pi)^d} G_{\ell m}(p, \tau) w_{\ell}(k,p,\tau) Y_{\ell m}(\hat k) |
| 61 | +\end{aligned} |
| 62 | +``` |
| 63 | +Since self-energy is symmertric with ``\hat k``, we just need to project on the s-wave channel, namely |
| 64 | +```math |
| 65 | +\Sigma(k,\tau) =\frac{\Omega_d}{2} \int \frac{p^{d-1}dp}{(2\pi)^d} G(p,\tau) w_0(k,p,\tau) |
| 66 | +``` |
| 67 | + |
| 68 | +## Two dimensions |
| 69 | +```math |
| 70 | +\begin{aligned} |
| 71 | +N(2,\ell) &= 2 \\ |
| 72 | +Y_{\ell 1}(\hat k) &= \cos(\ell \theta),\; Y_{\ell 2}(\hat k) = \sin(\ell \theta) \\ |
| 73 | +P_{\ell}(\hat{kp}) &= \pi \cos[\ell(\theta_{\hat k} - \theta_{\hat p})] |
| 74 | +\end{aligned} |
| 75 | +``` |
| 76 | +Hence, the self-energy is |
| 77 | +```math |
| 78 | +\Sigma(k,\tau) = \int \frac{pdp}{4\pi} G(p,\tau)w_0(k,p,\tau) |
| 79 | +``` |
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