8.1. 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:
(8.1)\[\begin{split}\begin{aligned}
U_{mn}({\bf R},\omega)&=&\int_V d{\bf r} \int_V d{\bf r'}
w_{m{\bf 0}}^*({\bf r}) w_{m{\bf 0}}({\bf r})
W({\bf r,r'},\omega)
w_{n{\bf R}}^*({\bf r'}) w_{n{\bf R}}({\bf r'}),\nonumber\\
J_{mn}({\bf R},\omega)&=&\int_V d{\bf r}\int_V d{\bf r'}
w_{m{\bf 0}}^*({\bf r}) w_{n{\bf R}}({\bf r})
W({\bf r,r'},\omega)
w_{n{\bf R}}^*({\bf r'}) w_{m{\bf 0}}({\bf r'}).
\end{aligned}\end{split}\]
Here, \(V\) is the volume of the crystal, \(w_ {i {\bf R}}({\bf r})\) is the \(i\) -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:
(8.2)\[\begin{split}\begin{aligned}
{\cal H}_{\rm int} &= \frac{1}{2}\sum_{\sigma\rho }\sum_{ij}\sum_{nm} \Bigl[ U_{mn}({\bf R}_{ij})c_{im, \sigma}^{\dagger}c_{jn, \rho}^{\dagger}c_{jn, \rho}c_{im, \sigma}\nonumber\\
&+ J_{mn}({\bf R}_{ij})(c_{im, \sigma}^{\dagger}c_{jn,\rho}^{\dagger}c_{im,\rho}c_{jn,\sigma} + c_{im, \sigma}^{\dagger}c_{im,\rho}^{\dagger}c_{jn,\rho}c_{jn,\sigma} )\Bigr],
\end{aligned}\end{split}\]
where \({\bf R}_{ij} \equiv {\bf R}_i-{\bf R}_j\) . 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:
(8.3)\[\begin{split}\begin{aligned}
{\cal H}_{\rm int} &= \sum_{i,m} U_{mm}({\bf 0})n_{im,\uparrow} n_{im, \downarrow} +\sum_{(i,m)<(j,n)}U_{mn}({\bf R}_{ij})n_{im}n_{jn}\nonumber\\
& - \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij})(n_{im, \uparrow}n_{jn,\uparrow}+n_{im, \downarrow}n_{jn,\downarrow}) \nonumber\\
& + \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij})(c_{im, \uparrow}^{\dagger}c_{jn,\downarrow}^{\dagger}c_{im,\downarrow}c_{jn,\uparrow}+{\rm h.c.}) \nonumber\\
& + \sum_{(i,m)<(j,n)}J_{mn}({\bf R}_{ij}) (c_{im, \uparrow}^{\dagger}c_{im,\downarrow}^{\dagger}c_{jn,\downarrow}c_{jn,\uparrow} + {\rm h.c.} ).
\end{aligned}\end{split}\]
The lattice model is defined by the following Hamiltonian:
(8.4)\[\begin{aligned}
{\cal H} &=
\sum_{m,n, i, j,\sigma}
\left[t_{mn}({\bf R}_{ij}) - t_{mn}^{\rm DC}({\bf R}_{ij})\right] c_{im \sigma}^{\dagger} c_{jn \sigma}
+ {\cal H}_{int},
\end{aligned}\]
where \(t_{mn}^{\rm DC}({\bf R}_{ij})\) is the one-body correction term given as:
(8.5)\[\begin{split}\begin{aligned}
t_{mm}^{\rm DC}({\bf 0}) &\equiv \alpha U_{mm}({\bf 0}) D_{mm}({\bf 0})
+ \sum_{({\bf R}, n) \neq ({\bf 0}, m)} U_{m n} ({\bf R})D_{nn}({\bf 0})\\
& - (1-\alpha) \sum_{({\bf R}, n) \neq ({\bf 0}, 0)} J_{m n}({\bf R}) D_{nn}({\bf R}),\\
t_{mn}^{\rm DC}({\bf R}_{ij}) &\equiv \frac{1}{2} J_{mn}({\bf R}_{ij}) \left(D_{nm}({\bf R}_{ji}) + 2 {\rm Re} [D_{nm}({\bf R}_{ji})]\right)\\
&-\frac{1}{2} U_{mn}({\bf R}_{ij}) D_{nm}({\bf R}_{ji}),
\quad ({\bf R}_{ij}, m) \neq ({\bf 0}, n),
\\
D_{mn}({\bf R}_{ij}) &\equiv \sum_{\sigma}
\left\langle c_{im \sigma}^{\dagger} c_{jn \sigma}\right\rangle_{\rm KS},
\end{aligned}\end{split}\]
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
. These terms are introduced to avoid double counting in analyzing the lattice model. To adopt theses 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.
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 \({\mathcal H}\Phi\).
Therefore, these programs must be available in our machine.
