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

$H=\sum _{\sigma }\sum _{<ij>}t_{ij}^{\sigma }c_{i\sigma }^{\dagger }c_{j\sigma }+\sum _{\sigma \sigma '}\sum _{<ijkl>}U_{ijkl}^{\sigma \sigma '}c_{i\sigma }^{\dagger }c_{k\sigma '}^{\dagger }c_{j\sigma }c_{l\sigma '}$

Here, $i,j,k,l$ are site and $\sigma ,\sigma '$ spin indices, respectively and the symbol $<\cdots >$ indicates summation over nearest neighbors. The hopping matrix elements $t_{ij}^{\sigma }$ describe the hopping of electrons (of same spin) between site $i$ and $j$, while the effective Coulomb matrix elements $U_{ijkl}^{\sigma \sigma '}$ 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 $\phi _{n{\bf {k}}}^{\sigma }({\bf {r}})=e^{i{\bf {kr}}}u_{n{\bf {k}}}({r})$ of band $n$ at k-point $k$ with spin $\sigma$ via

$|w_{i{\bf {R}}}^{\sigma }\rangle ={\frac {1}{N_{k}}}\sum _{n{\bf {k}}}e^{-i{\bf {kR}}}U_{in}^{\sigma ({\bf {k}})}|u_{n{\bf {k}}}^{\sigma }\rangle$

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 ${\bf {R=0}}$ and writes instead:

$|w_{i}^{\sigma }\rangle ={\frac {1}{N_{k}}}\sum _{n{\bf {k}}}U_{in}^{\sigma ({\bf {k}})}|u_{n{\bf {k}}}^{\sigma }\rangle$

In practice one builds a model Hamiltonian only for a 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. Here is an example where the target space contains three states (red bands):

In this case, a minimal Wannier basis containing only target states can be chosen. 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, one cannot represent the band structure, accurately. In the following example (face-centered-cubic Ni), the delocalized s-like band (blue) crosses the five target d-like states (red):

Parameter definitions

Mind: The calculation of the hopping matrix $t$ 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 $\epsilon _{n{\bf {k}}}^{\sigma }$ denotes these one-electron energies and $\mu ^{\sigma }$ is the corresponding Fermi energy, the hopping matrix elements are calculated with following formula

$t_{ij}^{\sigma }={\frac {1}{N_{k}}}\sum _{n{\bf {k}}}U_{in}^{*\sigma ({\bf {k}})}(\epsilon _{n{\bf {k}}}^{\sigma }-\mu ^{\sigma })U_{jn}^{\sigma ({\bf {k}})}$

Similarly, the Coulomb matrix elements are evaluated from the Bloch representation of the effective Coulomb kernel $U_{{\bf {GG}}'}({\bf {q}})$ via

$U_{ijkl}^{\sigma \sigma '}={\frac {1}{N_{k}^{3}}}\sum _{\bf {kkq}}\sum _{n_{1}n_{2}n_{3}n_{4}}U_{in_{1}}^{*({\bf {k}})}U_{jn_{2}}^{({\bf {k-q}})}\langle u_{n_{1}{\bf {k}}}|e^{-i({\bf {q+G}})\cdot {\bf {r}}}|u_{n_{2}{\bf {k-q}}}\rangle U_{{\bf {GG}}'}({\bf {q}})\langle u_{n_{3}{\bf {k'-q}}}|e^{i({\bf {q-G'}})\cdot {\bf {r'}}}|u_{n_{4}{\bf {k'}}}\rangle U_{kn_{3}}^{*({\bf {k'-q}})}U_{ln_{4}}^{({\bf {k'}})}$

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 $U=U(\omega \to 0)$.

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

${\cal {U}}^{\sigma \sigma '}={\frac {1}{N}}\sum _{i=1}^{N}U_{iiii}$

${\cal {U'}}^{\sigma \sigma '}={\frac {1}{N(N-1)}}\sum _{i\neq j}^{N}U_{ijji}$

${\cal {J}}^{\sigma \sigma '}={\frac {1}{N(N-1)}}\sum _{i\neq j}^{N}U_{ijij}$

Here $N$ 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

$U_{{\bf {G}}{\bf {G}}'}^{\sigma \sigma '}({\bf {q}},\omega )=\left[\delta _{{\bf {G}}{\bf {G}}'}-(\chi _{{\bf {G}}{\bf {G}}'}^{\sigma \sigma '}({\bf {q}},\omega )-{\tilde {\chi }}_{{\bf {G}}{\bf {G}}'}^{\sigma \sigma '}({\bf {q}},\omega ))\cdot V_{{\bf {G}}{\bf {G}}'}({\bf {q}})\right]^{-1}V_{{\bf {G}}{\bf {G}}'}({\bf {q}})$

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:

$|{\tilde {\phi }}_{n{\bf {k}}}\rangle =\sum _{m}\underbrace {P_{mn}^{({\bf {k}})}} _{\sum _{i\in {\cal {T}}}U_{in}^{*({\bf {k}})}U_{im}^{({\bf {k}})}}|\phi _{m{\bf {k}}}\rangle$

Using Green's functions of the target space

${\tilde {G}}^{\sigma }({\bf {r}},{\bf {r}}',i\tau )=-\sum _{n{\bf {k}}}\sum _{ij}U_{in}^{\sigma ({\bf {k}})}\phi _{n{\bf {k}}}^{*\sigma }({\bf {r}})\phi _{n{\bf {k}}}^{\sigma }({\bf {r}}')U_{jn}^{*\sigma ({\bf {k}})}e^{-(\epsilon _{n{\bf {k}}}-\mu )\tau }\left[\Theta (\tau )(1-f_{n{\bf {k}}})-\Theta (-\tau )f_{n{\bf {k}}}\right]$

the polarizability of the target space reads

${\tilde {\chi }}^{\sigma \sigma '}({\bf {r}},{\bf {r'}},i\tau )=-{\tilde {G}}^{\sigma }({\bf {r}},{\bf {r'}},i\tau ){\tilde {G}}^{\sigma '}({\bf {r'}},{\bf {r}},-i\tau )$

After a Fourier transform to reciprocal space and imaginary frequency axis $i\omega$ one ends up with

${\tilde {\chi }}_{{\bf {G,G}}'}^{\sigma }({\bf {q}},i\omega )={\frac {1}{N_{k}^{2}}}\sum _{n{\bf {k}}}\sum _{n'{\bf {k'}}}f_{n{\bf {k}}}(1-f_{n'{\bf {k'}}})\times \int {\rm {d}}{\bf {r}}{\rm {d}}{\bf {r'}}e^{-i{\bf {Gr}}}{\rm {Re}}\left[{\frac {\phi _{n{\bf {k}}}^{*\sigma }({\bf {r}})\phi _{n{\bf {k}}}^{\sigma }({\bf {r'}})\phi _{n'{\bf {k'}}}^{*\sigma '}({\bf {r'}})\phi _{n'{\bf {k'}}}^{\sigma '}({\bf {r}})}{\epsilon _{n{\bf {k}}}-\epsilon _{n'{\bf {k'}}}-i\omega }}\right]e^{-i{\bf {G'r'}}}\times \underbrace {\sum _{ij}U_{in}^{\sigma ({\bf {k}})}U_{jn}^{*\sigma ({\bf {k}})}} _{p_{n{\bf {k}}}^{\sigma }}\underbrace {\sum _{kl}U_{kn'}^{\sigma '({\bf {k'}})}U_{ln'}^{*\sigma '({\bf {k'}})}} _{p_{n'{\bf {k'}}}^{\sigma '}}$

describing the propagation within the target space.

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Related tags and articles

ALGO, NTARGET_STATES, NCRPA_BANDS LDISENTANGLE LWEIGHTED NUM_WANN WANNIER90_WIN ENCUTGW VCUTOFF

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