Decomposition of Interaction in two dimensions

In GW-approximation, we calculation self-energy as

\[\Sigma(\mathbf{k},\omega_n)=-T\int \frac{{\rm d}^d \mathbf{q}}{(2\pi)^d} \sum_m G(\mathbf{p},\omega_m)W(\mathbf{k-p},\omega_n-\omega_m) \,, \tag{1} \]

where $G$ is the Green's function and W is the effective interaction. Here, we suppress spin index.

Spherical harmonic representation

We first express the $W(q,\tau)$ function as an expansion in Legendre polynomials $P_\ell(\chi)$

\[\begin{gathered} W(|\mathbf{k}-\mathbf{p}|, \tau)=\sum_{\ell=0}^{\infty} \bar{w}_{\ell}(k, p, \tau) P_{\ell}(\hat{k p}) \,, \\ \bar{w}_{\ell}(k, p, \tau)=\frac{N(d,\ell)}{2} \int_{-1}^{1} d \chi P_{\ell}(\chi) W\left(\sqrt{k^{2}+p^{2}-2 k p \chi} ,\tau\right)\,. \end{gathered}\]

Since the Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using

\[P_{\ell}(\hat{k p})=\frac{\Omega_{d}}{N(d,\ell)} \sum_{m=1}^{N(d,\ell)} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p})\,,\]

where $\Omega_{d}$ is the solid angle in $d$ dimensions, and

\[N(d, \ell)=\frac{2 \ell+d-2}{\ell}\left( \begin{array}{c} \ell+d-3 \\ \ell-1 \end{array}\right)\]

denotes the number of linearly independent homogeneous harmonic polynomials of degree $\ell$ in $d$ dimensions. The spherical harmonics are orthonormal as

\[\int {\rm d}\Omega_{\hat k} Y_{\ell m}(\hat k) Y_{\ell^\prime m^\prime}(\hat k) = \delta_{\ell \ell^\prime} \delta_{mm^\prime} \,.\]

Hence, the $W(q,\tau)$ function can expressed further on as

\[W(|\mathbf{k}-\mathbf{p}|, \tau)=\sum_{\ell} \frac{\Omega_{d}}{N(d,\ell)} \bar{w}_{\ell}(k, p, \tau) \sum_{m} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p})\]


\[W(|\mathbf{k}-\mathbf{p}|, \tau)= \frac{\Omega_{d}}{2}\sum_{\ell} w_{\ell}(k, p, \tau) \sum_{m} Y_{\ell m}(\hat{k}) Y_{\ell m}^{*}(\hat{p} )\,.\]


\[w_{\ell}(k, p, \tau)=\int_{-1}^{1} d \chi P_{\ell}(\chi) W\left(\sqrt{k^{2}+p^{2}-2 k p \chi} ,\tau\right)\]

In addition, the Green's function $G(\mathbf p, \tau)$ has

\[\begin{aligned} G(\mathbf p, \tau)= \sum_{\ell=0}^{\infty}\sum_{m=1}^{N(d,\ell)} G_{\ell m}(p,\tau) Y_{\ell m}(\hat p)\,, \\ G_{\ell m}(p,\tau) = \int {\rm d}\Omega_{\hat k} G(\mathbf p,\tau) Y^*_{\ell m}(\hat p) \end{aligned}\]

Decouple with channels

By the sphereical harmonic expansion of Eq.(1), the self-energy

\[\begin{aligned} \sum_{\ell m} \Sigma_{\ell m }(k,\tau)Y_{\ell m}(\hat k) &= \frac{\Omega_d}{2} \int \frac{{\rm d}\mathbf p}{(2\pi)^d} \sum_{\ell m}G_{\ell m}(p, \tau) Y_{\ell m}(\hat p) \sum_{\ell^\prime m^\prime} w_{\ell^\prime}(k,p,\tau) Y_{\ell^\prime m^\prime}(\hat{k}) Y_{\ell^\prime m^\prime}^{*}(\hat{p} ) \\ &=\frac{\Omega_d}{2} \sum_{\ell m} \int \frac{p^{d-1}dp}{(2\pi)^d} G_{\ell m}(p, \tau) w_{\ell}(k,p,\tau) Y_{\ell m}(\hat k) \end{aligned}\]

Since self-energy is symmertric with $\hat k$, we just need to project on the s-wave channel, namely

\[\Sigma(k,\tau) =\frac{\Omega_d}{2} \int \frac{p^{d-1}dp}{(2\pi)^d} G(p,\tau) w_0(k,p,\tau)\]

Two dimensions

\[\begin{aligned} N(2,\ell) &= 2 \\ Y_{\ell 1}(\hat k) &= \cos(\ell \theta),\; Y_{\ell 2}(\hat k) = \sin(\ell \theta) \\ P_{\ell}(\hat{kp}) &= \pi \cos[\ell(\theta_{\hat k} - \theta_{\hat p})] \end{aligned}\]

Hence, the self-energy is

\[\Sigma(k,\tau) = \int \frac{pdp}{4\pi} G(p,\tau)w_0(k,p,\tau)\]