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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_v1/":): {\displaystyle | w_{i\bf R}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} e^{-i {\bf k R}} T_{i n}^{\sigma({\bf k})} | \psi_{n\bf k}^\sigma \rangle }

Usually, the basis set is localized such that the interaction between periodic images can be neglected. This allows to work with the Wannier functions 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://wikimedia.org/api/rest_v1/":): {\displaystyle \bf R=0} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | w_{i}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} T_{i n}^{\sigma({\bf k})} | \psi_{n\bf k}^\sigma \rangle }

Mind: Here, T instead of the widely used U notation for the transformation matrix is used to avoid confusion with the Hubbard parameter.

In practice one builds a model Hamiltonian only for a small subset of Bloch functions. These target states are typically centered around the chemical potential (or Fermi energy) and are strongly localized around ions. The model Hamiltonian can be solved successfully, only if the target states are well represented by the Wannier basis. As a measure of the Wannier representation one usually compares the original band structure with the Wannier interpolated one.

In the following example (SrVO3) the target space consists of three Bloch bands (red bands) that may be represented well by three Wannier states:

In this optimal case, the Wannier basis contains only target states and no additional Wannier states are required.

More often, however, one has delocalized states that mix with the target space of the model. Without including these crossing states in the Wannier basis, a good representation of the band structure is not possible. Below is an example (face-centered-cubic Ni), where the delocalized s-band (blue) crosses the five target d-states (red):

This system requires at least six Wannier states to represent the electronic structure of five target states well. That is, a one-to-one mapping between Bloch and Wannier space requires a six-by-six unitary matrix.

If a modification of the band structure is acceptable within an energy window, these five target states might be disentangled from the remaining ones and one arrives at following picture:

Here the original Bloch bands (gray lines) are projected to five non-crossing Wannier states.

In the following Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cal T} denotes the subset of Wannier states in the Wannier basis that belong to the target space.

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://wikimedia.org/api/rest_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 Kohn-Sham energies, while in GW+DMFT 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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ij}^\sigma = \frac{1}{N_k}\sum_{n\bf k}T_{in}^{*\sigma({\bf k})} (\epsilon^\sigma_{n{\bf k}} - \mu^\sigma) T_{jn}^{\sigma({\bf k})} }

Similarly, 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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_v1/":): {\displaystyle U_{ijkl}^{\sigma\sigma'} = \frac{1}{N^3_k}\sum_{{\bf k k q}}\sum_{n_1n_2n_3n_4} T_{in_1}^{*({\bf k})} T_{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 T_{kn_3}^{*({\bf k'-q})} T_{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://wikimedia.org/api/rest_v1/":): {\displaystyle U=U(\omega\to 0)} .

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\cal U }^{\sigma\sigma'} = \frac 1 N \sum_{i\in \cal T} U_{iiii}^{\sigma\sigma'} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\cal U' }^{\sigma\sigma'} = \frac{1}{N(N-1)}\sum_{i,j \in{\cal T}, i\neq j}^N U_{ijji}^{\sigma\sigma'} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\cal J }^{\sigma\sigma'} = \frac{1}{N(N-1)} \sum_{i,j \in{\cal T}, i\neq j}^N U_{ijij}^{\sigma\sigma'} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): N specifies the number of Wannier functions in 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://wikimedia.org/api/rest_v1/":): {\displaystyle \cal T} .

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://wikimedia.org/api/rest_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 are described by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde \chi} and can be separated best in Wannier space. To this end, we work with the target projectors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{mn}^{\sigma({\bf k})} = \sum_{i\in \cal T} T_{i m}^{*\sigma({\bf k})} T_{i n}^{\sigma({\bf k})} }

that filter out target space contributions to each Bloch state. Using these projectors, one can write the 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://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde G^\sigma({\bf r},{\bf r}',i\tau)=-\frac 1{N_k}\sum_{n{\bf k}} e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right] \sum_{mm'} P_{n m}^{*\sigma ({\bf k}) } \psi_{m{\bf k}}^{*\sigma }({\bf r}) \psi_{m'{\bf k}}^{\sigma }({\bf r}') P_{m' n}^{\sigma ({\bf k}) } }

an define the screening effects within the target space as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_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://wikimedia.org/api/rest_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://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)= \frac 1{N_k}\sum_{nn'{\bf k}} f_{n\bf k}(1-f_{n'\bf k-q}) \times \sum_{m_1m_1'm_2m_2' } \int {\rm d}{\bf r }{\rm d}{\bf r'} e^{-i \bf G r} {\rm Re }\left[\frac{ \psi_{m_1 {\bf k }}^{*\sigma }({\bf r }) \psi_{m_1'{\bf k }}^{ \sigma }({\bf r'}) \psi_{m_2 {\bf k-q}}^{*\sigma' }({\bf r'}) \psi_{m_2'{\bf k-q}}^{ \sigma' }({\bf r }) }{ \epsilon_{n{\bf k}} - \epsilon_{n'\bf k-q} - i \omega } \right]e^{-i \bf G' r'} \times P_{m_1 n }^{*\sigma ({\bf k })} P_{m_1' n }^{ \sigma ({\bf k })} P_{m_2 n'}^{*\sigma' ({\bf k-q})} P_{m_2' n'}^{ \sigma' ({\bf k-q})} }

Evaluating the Fourier transform and rearraning one arrives at following expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)= \frac 1{N_k}\sum_{nn'{\bf k}} \frac{ f_{n\bf k}-f_{n'\bf k-q} }{ \epsilon_{n{\bf k}} - \epsilon_{n'\bf k-q} - i \omega } \times \sum_{m_1m_2' } P_{m_1 n }^{*\sigma ({\bf k })} \langle u_{m_1 {\bf k }}^{*\sigma } |e^{-i \bf (G+q) r}| u_{m_2'{\bf k-q}}^{ \sigma' } \rangle P_{m_2' n'}^{ \sigma' ({\bf k-q})} \times \sum_{m_1'm_2 } P_{m_2 n'}^{*\sigma' ({\bf k-q})} \langle u_{m_2 {\bf k-q}}^{*\sigma' } |e^{-i \bf (G'-q)r'} | u_{m_1'{\bf k }}^{ \sigma } \rangle P_{m_1' n }^{ \sigma ({\bf k })} }

Note, if the target space forms an isolated set of bands (like in the first example above), the projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} are the identity operators. In this case the expression above is essentially the Adler and Wiser formula for the target space. ^[2]^[3]

Weighted CRPA

Sasioglu, Friedrich and Blügel propose an alternative approach,^[4] where the screening effects within the target space are approximated as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)\approx \frac 1{N_k}\sum_{nn'{\bf k}} \frac{ f_{n\bf k}-f_{n'\bf k-q} }{ \epsilon_{n{\bf k}} - \epsilon_{n'\bf k-q} - i \omega } p_{n\bf k }^{\sigma} p_{n'\bf k-p }^{\sigma'} \langle u_{n {\bf k }}^{*\sigma } |e^{-i \bf (G+q) r}| u_{n'{\bf k-q}}^{ \sigma' } \rangle \langle u_{n' {\bf k-q}}^{*\sigma' } |e^{-i \bf (G'-q)r'} | u_{n'{\bf k }}^{ \sigma } \rangle }

Here the weighting factors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n\bf k}^\sigma = \sum_{i\in\cal T} |T_{i n}^{\sigma({\bf k})}|^2 , \quad 0 \le p_{n\bf k}^\sigma \le 1 }

measure the probability for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_{n\bf k}^\sigma\rangle} being in the target space.

References

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

[adler:1962-2] S. L. Adler, Phys. Rev. 126, 413 (1962)

[wiser:1963-3] N. Wiser, Phys. Rev. 129, 62 (1963)

[sasioglu:prb:83-4] E. Sasioglu, C. Friedrich, and S. Blügel, Phys. Rev. B 83, 121101 (2011).

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