Overview

In this document, we introduce how we compute downfolded models with mVMC or \({\mathcal H}\Phi\) in conjunction to RESPACK. In RESPACK, the screened direct integrals \(U_{mn}({\bf R},\omega)\) and the screened exchanged integrals \(J_{mn}({\bf R},\omega)\) are given as follows:

\[\begin{split}\begin{aligned} U_{mn}({\bf R},\omega)&=&\int_V d{\bf r} \int_V d{\bf r'} w_{m{\bf 0}}^*({\bf r}) w_{m{\bf 0}}({\bf r}) W({\bf r,r'},\omega) w_{n{\bf R}}^*({\bf r'}) w_{n{\bf R}}({\bf r'}),\nonumber\\ J_{mn}({\bf R},\omega)&=&\int_V d{\bf r}\int_V d{\bf r'} w_{m{\bf 0}}^*({\bf r}) w_{n{\bf R}}({\bf r}) W({\bf r,r'},\omega) w_{n{\bf R}}^*({\bf r'}) w_{m{\bf 0}}({\bf r'}). \end{aligned}\end{split}\]

Here, \(V\) is the volume of the crystal, \(w_ {n {\bf R}}({\bf r})\) is the \(n\)-th wannier function at \(\bf R\)-th cell, \(W({\bf r,r'}, \omega)\) is the screened Coulomb interactions, respectively. In the following, the components at \(\omega=0\) are only considered. Then, the Hamiltonian of the two-body interactions are given as follows:

\[\begin{split}\begin{aligned} {\cal H}_{\rm int} &= \frac{1}{2}\sum_{\sigma\rho }\sum_{ij}\sum_{nm} \Bigl[ U_{mn}({\bf R}_{ij})c_{im, \sigma}^{\dagger}c_{jn, \rho}^{\dagger}c_{jn, \rho}c_{im, \sigma}\nonumber\\ &+ J_{mn}({\bf R}_{ij})(c_{im, \sigma}^{\dagger}c_{jn,\rho}^{\dagger}c_{im,\rho}c_{jn,\sigma} + c_{im, \sigma}^{\dagger}c_{im,\rho}^{\dagger}c_{jn,\rho}c_{jn,\sigma} )\Bigr], \end{aligned}\end{split}\]

where \({\bf R}_{ij} \equiv {\bf R}_i-{\bf R}_j\) . \({\bf R}_i\) is the position vector of the \(i\)-th cell. Since mVMC and \({\mathcal H}\Phi\) cannot directly treat the following type of interactions \({c_{i, \sigma}^{\dagger}c_{j, \rho}^{\dagger}c_{k, \rho'}c_{l, \sigma'}}\) , the Hamiltonian must be rewritten as follows:

\[\begin{split}\begin{aligned} {\cal H}_{\rm int} &= \sum_{i,m} U_{mm}({\bf 0})n_{im,\uparrow} n_{im, \downarrow} +\sum_{(i,m)<(j,n)}U_{mn}({\bf R}_{ij})n_{im}n_{jn}\nonumber\\ & - \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij})(n_{im, \uparrow}n_{jn,\uparrow}+n_{im, \downarrow}n_{jn,\downarrow}) \nonumber\\ & + \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij})(c_{im, \uparrow}^{\dagger}c_{jn,\downarrow}^{\dagger}c_{im,\downarrow}c_{jn,\uparrow}+{\rm h.c.}) \nonumber\\ & + \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij}) (c_{im, \uparrow}^{\dagger}c_{im,\downarrow}^{\dagger}c_{jn,\downarrow}c_{jn,\uparrow} + {\rm h.c.} ). \end{aligned}\end{split}\]

The lattice model is defined by the following Hamiltonian:

\[\begin{aligned} {\cal H} &= \sum_{m,n, i, j,\sigma} \left[t_{mn}({\bf R}_{ij}) - t_{mn}^{\rm DC}({\bf R}_{ij})\right] c_{im \sigma}^{\dagger} c_{jn \sigma} + {\cal H}_{int}, \end{aligned}\]

where \(t_{mn}^{\rm DC}({\bf R}_{ij})\) is the one-body correction term given as:

\[\begin{split}\begin{aligned} t_{mm}^{\rm DC}({\bf 0}) &\equiv \alpha U_{mm}({\bf 0}) D_{mm}({\bf 0}) + \sum_{({\bf R}, n) \neq ({\bf 0}, m)} U_{m n} ({\bf R})D_{nn}({\bf 0})\\ & - (1-\alpha) \sum_{({\bf R}, n) \neq ({\bf 0}, 0)} J_{m n}({\bf R}) D_{nn}({\bf R}),\\ t_{mn}^{\rm DC}({\bf R}_{ij}) &\equiv \frac{1}{2} J_{mn}({\bf R}_{ij}) \left(D_{nm}({\bf R}_{ji}) + 2 {\rm Re} [D_{nm}({\bf R}_{ji})]\right)\\ &-\frac{1}{2} U_{mn}({\bf R}_{ij}) D_{nm}({\bf R}_{ji}), \quad ({\bf R}_{ij}, m) \neq ({\bf 0}, n), \\ D_{mn}({\bf R}_{ij}) &\equiv \sum_{\sigma} \left\langle c_{im \sigma}^{\dagger} c_{jn \sigma}\right\rangle_{\rm KS}, \end{aligned}\end{split}\]

Here, \(t_{mm}^{\rm DC}({\bf 0})\) is the term to correct the chemical potntial, \(t_{mn}^{\rm DC}({\bf R}_{ij})\) is term to correct transfer integrals . \(\alpha\) is the tuning parameter for one-body correction from the on-site Coulomb interactions. These terms are introduced to avoid double counting in analyzing the lattice model. To adopt these corrections or not can be selected by the option doublecounting in the input file. The strength of \(U_{Rij}\) and \(J_{Rij}\) can be controled by multiplying tuning parameters \(\lambda_U, \lambda_J\). For details, see Input parameters for Standard mode.

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 \({\mathcal H}\Phi\). Therefore, these programs must be available in our machine.