Constrained–random-phase–approximation formalism

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle H = \sum_\sigma \sum_{<ij>} t_{ij}^\sigma c_{i\sigma}^\dagger c_{j\sigma} + \sum_{\sigma\sigma'} \sum_{<ijkl>} U_{ijkl}^{\sigma\sigma'} c_{i\sigma}^\dagger c_{k\sigma'}^\dagger c_{j\sigma} c_{l\sigma'}}

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle i,j,k,l} are site and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \sigma,\sigma'} spin indices, respectively and the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle <\cdots>} indicates summation over nearest neighbors. The hopping matrix elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle t_{ij}^\sigma } describe the hopping of electrons (of same spin) between site Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): i and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): j , while the effective Coulomb matrix elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{ijkl}^{\sigma\sigma'}} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \phi_{n\bf k}^\sigma ({\bf r}) = e^{i{\bf k r}} u_{n\bf k}({r})} of band Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): n at k-point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): k with spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \sigma via

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \bf R=0} and writes instead:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): t 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \epsilon^\sigma_{n\bf k}} denotes these one-electron energies and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \mu^\sigma} is the corresponding Fermi energy, the hopping matrix elements are calculated with following formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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})} }

Similarly, the Coulomb matrix elements are evaluated from the Bloch representation of the effective Coulomb kernel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U_{{\bf G G}'}({\bf q})} via

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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'})} }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle U=U(\omega\to 0)} .

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle {\cal U }^{\sigma\sigma'} = \frac 1 N \sum_{i=1}^N U_{iiii} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle {\cal U' }^{\sigma\sigma'} = \frac{1}{N(N-1)}\sum_{i\neq j}^N U_{ijji} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle {\cal J }^{\sigma\sigma'} = \frac{1}{N(N-1)} \sum_{i\neq j}^N U_{ijij} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): N 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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}) }

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 }

Using Green's functions of the target space

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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] }

the polarizability of the target space reads

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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) }

After a Fourier transform to reciprocal space and imaginary frequency axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle i\omega} one ends up with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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'}} } }

describing the propagation within the target space.

=

Related tags and articles

ALGO, NTARGET_STATES, NCRPA_BANDS LDISENTANGLE LWEIGHTED NUM_WANN WANNIER90_WIN ENCUTGW VCUTOFF

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