8.1. Overview

In this document, we introduce how we compute downfolded models with mVMC or HPhi++ in conjunction to RESPACK.

\[\begin{split}\begin{aligned} {\cal H} &= \sum_{R, R', i, j, \sigma} \left(t_{(R'-R) i j} - t_{(R'-R) i j}^{\rm DC}\right) c_{R' j \sigma}^{\dagger} c_{R i \sigma} \nonumber \\ &+ \sum_{R, i} U_{0 i j} n_{R i \uparrow} n_{R i \downarrow} + \sum_{(R, i) < (R', j)} U_{(R'-R) i j} n_{R i} n_{R' j} - \sum_{(R, i) < (R', j)} J_{(R'-R) i j} (n_{R i \uparrow} n_{R' j \uparrow} + n_{R i \downarrow} n_{R' j \downarrow}) \nonumber \\ &+ \sum_{(R, i) < (R', j)} J_{(R'-R) i j} ( c_{R i \uparrow}^{\dagger} c_{R' j \downarrow}^{\dagger} c_{R i \downarrow} c_{R' j \uparrow} + c_{R' j \uparrow}^{\dagger} c_{R i \downarrow}^{\dagger} c_{R' j \downarrow} c_{R i \uparrow} ) \nonumber \\ &+ \sum_{(R, i) < (R', j)} J_{(R'-R) i j} ( c_{R i \uparrow}^{\dagger} c_{R i \downarrow}^{\dagger} c_{R' j \downarrow} c_{R' j \uparrow} + c_{R' j \uparrow}^{\dagger} c_{R' j \downarrow}^{\dagger} c_{R i \downarrow} c_{R i \uparrow} ), \\ t_{0 i i}^{\rm DC} &\equiv \frac{1}{2}U_{0 i i} D_{0 i i} + \sum_{(R, j) (\neq 0, i)} U_{R i j} D_{0 j j} - \frac{1}{2} \sum_{(R, j) (\neq 0, i)} J_{R i j} D_{0 j j}, \\ t_{R i j}^{\rm DC} &\equiv \frac{1}{2} J_{R i j} (D_{R i j} + 2 {\rm Re} [D_{R i j}]) -\frac{1}{2} U_{R i j} D_{R i j}, \quad (R, j) \neq (0, i), \\ D_{R i j} &\equiv \sum_{\sigma} \left\langle c_{R j \sigma}^{\dagger} c_{0 i \sigma}\right\rangle_{\rm KS}. \end{aligned}\end{split}\]

8.1.1. Prerequisite

We compute the Kohn-Sham orbitals with QuantumESPRESSO or xTAPP, and obtain the Wannier function, the dielectric function, the effective interaction with RESPACK, and simulate quantum lattice models with mVMC or HPhi++. Therefore, these programs must be available in our machine.