Long Range Action

The long range part of the potential leads to long range action. As a first step, it can be calculated from the primitive approximations as used in (Shumway, 1999). Alternatively, one can use the random phase approximation (RPA) (Pines and Nozieres, 1989) to obtain an improved long range action (for details see (Magro, 1996)), which is constructed in such a way that the local energy is minimized,

$$E_L = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial \beta} + {\mathcal{H}}\rho \right) \quad.$$ (48)

$E_L$ vanishes for a solution of the Bloch equation 2.13. One assumes a given short range density matrix $\rho_s$, which solves the Bloch equation for the Hamiltonian ${\mathcal{H}}_s = - \lambda \nabla^2 + V_s$. For the full Hamiltonian ${\mathcal{H}}= {\mathcal{H}}_s + V_l$ with the additional potential $V_l$, one derives a long range density matrix $\rho_l$ such that the density matrix $\rho_l \rho_s$ leads to $E_L=0$. One writes the long range potential $V_l$ and action $e^{-U_l}$ in the form,

$$V_l({\bf R}) = \sum_{\mathbf{k}} \sum_{\alpha \beta} \; v_k^{\alpha \beta} \sum_... ...rm T(\alpha)} \atop j \in {\rm T(\beta)}} e^{i \mathbf{k}({\bf r}_i-{\bf r}_j)}$$ (49)

$$= \sum_{\mathbf{k}} v_k^{\alpha \beta} \; \rho_\mathbf{k}^\alpha \; \rho_{-\mathbf{k}}^\beta \quad,$$ (50)

$$U_l({\bf R}) = \sum_{\mathbf{k}} u_k^{\alpha \beta} \; \rho_\mathbf{k}^\alpha \; \rho_{-\mathbf{k}}^\beta \quad,$$ (51)

$$\rho_\mathbf{k}^\alpha = \sum_{j \in {\rm T(\alpha)}}e^{i \mathbf{k}{\bf r}_j}\quad,$$ (52)

where $v_k^{\alpha \beta}$ is the Fourier transform of the potential between particle of type $\alpha$ and $\beta$ denoted by T$(\alpha)$ and T$(\beta)$. Setting $E_L=0$, leads to three body terms, which are approximated by the RPA. The resulting first order differential equations, which are integrated numerically in imaginary time from $0$ to $\beta$ with the initial condition $u_k=0$. The calculated coefficients $u^{ij}_k$ then enter PIMC simulations as long range action in form of Eq. 2.42.