The constrained random-phase approximation (CRPA) is a method that allows to calculate the effective interaction parameter U, J and J' for model Hamiltonians. The main idea is to neglect screening effects of specific target states in the screened Coulomb interaction W of the GW method. The resulting partially screened Coulomb interaction is usually evaluated in a localized basis that spans the target space and is described by the model Hamiltonian. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as dynamical mean field theory (DMFT).

Model Hamiltonians

A model Hamiltonian describes a small subset of electrons around the chemical potential and has, in second quantization, following form

[math]\displaystyle{ H = \sum_\sigma \sum_{\lt ij\gt } t_{ij}^\sigma c_{i\sigma}^\dagger c_{j\sigma} + \sum_{\sigma\sigma'} \sum_{\lt ijkl\gt } U_{ijkl}^{\sigma\sigma'} c_{i\sigma}^\dagger c_{k\sigma'}^\dagger c_{j\sigma} c_{l\sigma'} }[/math]

Here, [math]\displaystyle{ i,j,k,l }[/math] are site and [math]\displaystyle{ \sigma,\sigma' }[/math] spin indices, respectively and the symbol [math]\displaystyle{ \lt \cdots\gt }[/math] indicates summation over nearest neighbors. The hopping matrix elements [math]\displaystyle{ t_{ij}^\sigma }[/math] describe the hopping of electrons (of same spin) between site [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math], while the effective Coulomb matrix elements [math]\displaystyle{ U_{ijkl}^{\sigma\sigma'} }[/math] describe the interaction of electrons between sites.

Wannier basis and target space

To use model Hamiltonians successfully a localized basis set is chosen. In most applications this basis set consists of Wannier states that are connected with the Bloch functions [math]\displaystyle{ \phi_{n\bf k}^\sigma ({\bf r}) = e^{i{\bf k r}} u_{n\bf k}({r}) }[/math] of band [math]\displaystyle{ n }[/math] at k-point [math]\displaystyle{ k }[/math] with spin [math]\displaystyle{ \sigma }[/math] via

[math]\displaystyle{ | w_{i\bf R}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} e^{-i {\bf k R}} U_{i n}^{\sigma({\bf k})} | u_{n\bf k}^\sigma \rangle }[/math]

Usually, the basis set is localized such that the interaction between periodic images can be neglected. Hence, in practice one works with the Wannier function in the unit cell at [math]\displaystyle{ \bf R=0 }[/math] and writes instead:

[math]\displaystyle{ | w_{i}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} U_{i n}^{\sigma({\bf k})} | u_{n\bf k}^\sigma \rangle }[/math]

In practice one selects only a subset of Bloch functions, typically around the chemical potential (i.e.

Parameter definitions

Mind: The calculation of the hopping matrix [math]\displaystyle{ t }[/math] depends on the theory that is used to solve the effective model.

For instance, in DFT+DMFT (often termed LDA+DMFT) one calculates the hopping matrix from the Kohn-Sham energies, while in GW+DMFT the GW quasi-particle energies are used. If [math]\displaystyle{ \epsilon^\sigma_{n\bf k} }[/math] denotes these one-electron energies and [math]\displaystyle{ \mu^\sigma }[/math] is the corresponding Fermi energy, the hopping matrix elements are calculated with following formula

[math]\displaystyle{ t_{ij}^\sigma = \frac{1}{N_k}\sum_{n\bf k}U_{in}^{*\sigma({\bf k})} (\epsilon^\sigma_{n{\bf k}} - \mu^\sigma) U_{jn}^{\sigma({\bf k})} }[/math]

Similarly, the Coulomb matrix elements are evaluated from the Bloch representation of the effective Coulomb kernel [math]\displaystyle{ U_{{\bf G G}'}({\bf q}) }[/math] via

[math]\displaystyle{ U_{ijkl}^{\sigma\sigma'} = \frac{1}{N^3_k}\sum_{{\bf k k q}}\sum_{n_1n_2n_3n_4} U_{in_1}^{*({\bf k})} U_{jn_2}^{({\bf k-q})} \langle u_{n_1\bf k}| e^{-i({\bf q + G})\cdot {\bf r}} |u_{n_2\bf k-q}\rangle U_{{\bf G G}'}({\bf q}) \langle u_{n_3\bf k'-q}| e^{i({\bf q - G'})\cdot {\bf r'} }|u_{n_4\bf k'}\rangle U_{kn_3}^{*({\bf k'-q})} U_{ln_4}^{({\bf k'})} }[/math]

Mind: The effective Coulomb kernel is frequency dependent and thus the effective interaction in the model as well.

In most applications, however, one considers the static limit [math]\displaystyle{ U=U(\omega\to 0) }[/math].

In practice one often, simplifies the model Hamiltonian further and works with the Hubbard-Kanamori parameters:^[1]

[math]\displaystyle{ {\cal U }^{\sigma\sigma'} = \frac 1 N \sum_{i=1}^N U_{iiii} }[/math]

[math]\displaystyle{ {\cal U' }^{\sigma\sigma'} = \frac{1}{N(N-1)}\sum_{i\neq j}^N U_{ijji} }[/math]

[math]\displaystyle{ {\cal J }^{\sigma\sigma'} = \frac{1}{N(N-1)} \sum_{i\neq j}^N U_{ijij} }[/math]

Here [math]\displaystyle{ N }[/math] specifies the number of Wannier functions in the basis set.

Effective Coulomb kernel in constrained random-phase approximation

In analogy to the screened Coulomb kernel in GW, the effective coulomb kernel is calculated as

[math]\displaystyle{ U^{\sigma\sigma'}_{{\bf G}{\bf G}'}({\bf q},\omega)=\left[\delta_{{\bf G}{\bf G}'}-(\chi^{\sigma\sigma'}_{{\bf G}{\bf G}'}({\bf q},\omega) - \tilde\chi^{\sigma\sigma'}_{{\bf G}{\bf G}'}({\bf q},\omega) ) \cdot V_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}V_{{\bf G}{\bf G}'}({\bf q}) }[/math]

In contrast to the GW method, however, the polarizability contains all RPA screening effects, except those from the target space. These effects can be obtained with the target Bloch states:

[math]\displaystyle{ |\tilde \phi_{n\bf k}\rangle = \sum_{ m } \underbrace{ P_{mn}^{({\bf k})} }_{ \sum_{i\in \cal T} U_{i n}^{*({\bf k})} U_{i m}^{({\bf k})} } | \phi_{m\bf k}\rangle }[/math]

Using Green's functions of the target space

[math]\displaystyle{ \tilde G^\sigma({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}} \sum_{ij} U_{i n}^{\sigma ({\bf k}) } \phi_{n{\bf k}}^{*\sigma }({\bf r}) \phi_{n{\bf k}}^{\sigma }({\bf r}') U_{j n}^{*\sigma ({\bf k}) } e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right] }[/math]

the polarizability of the target space reads

[math]\displaystyle{ \tilde \chi^{\sigma \sigma'}({\bf r},{\bf r'},i\tau) = -\tilde G^\sigma({\bf r},{\bf r'},i\tau)\tilde G^{\sigma'}({\bf r'},{\bf r},-i\tau) }[/math]

After a Fourier transform to reciprocal space and imaginary frequency axis [math]\displaystyle{ i\omega }[/math] one ends up with

[math]\displaystyle{ \tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)= \frac 1{N^2_k}\sum_{n{\bf k}}\sum_{n'{\bf k'}} f_{n\bf k}(1-f_{n'\bf k'}) \times \int {\rm d}{\bf r }{\rm d}{\bf r'} e^{-i \bf G r} {\rm Re }\left[\frac{ \phi_{n {\bf k }}^{*\sigma }({\bf r }) \phi_{n {\bf k }}^{ \sigma }({\bf r'}) \phi_{n'{\bf k'}}^{*\sigma' }({\bf r'}) \phi_{n'{\bf k'}}^{ \sigma' }({\bf r }) }{ \epsilon_{n{\bf k}} - \epsilon_{n'\bf k'} - i \omega } \right]e^{-i \bf G' r'} \times \underbrace{ \sum_{ij } U_{i n }^{ \sigma ({\bf k }) } U_{j n }^{*\sigma ({\bf k }) }}_{ p^\sigma_{n {\bf k} } } \underbrace{ \sum_{ kl } U_{k n'}^{ \sigma' ({\bf k'}) } U_{l n'}^{*\sigma' ({\bf k'}) } }_{p^{\sigma'}_{ n'{\bf k'}} } }[/math]

describing the propagation within the target space.

=

References

↑ L. Vaugier, H. Jiang, and S. Biermann, Phys. Rev. B 86, 165105 (2012).

[vaugier:prb:86-1] L. Vaugier, H. Jiang, and S. Biermann, Phys. Rev. B 86, 165105 (2012).

[1]

Constrained–random-phase–approximation formalism