About projectors¶

See also

Sometimes one might want to project onto a correlated subspace \(\mathcal{C}_i = \{ |\chi_{Rm} \rangle_i \}\) from a larger set of states \(\mathcal{H} = \{ |\Psi_{k \nu} \rangle \}\), where \(R\) labels a reference unit cell, \(i\) labels inequivalent subspaces in the reference cell, \(m\) labels one of the \(M_i\) correlated orbitals in \(\mathcal{C}_i\), \(k\) labels a reciprocal lattice vector \(\vec{k}\), and \(\nu\) labels one of the \(N\) orbitals in \(\mathcal{H}\). It is not a requirement that \(N\) is the same for each \(k\).

At each \(k\), the projection onto \(\mathcal{C}_i\) can be encoded into \(M_i \times N\) rectangular matrices \(\mathbf{P}_i(k)\). Any single-particle quantity, represented as an \(N \times N\) matrix \(\mathbf{A}^{\mathcal{H}}(k)\), can be projected from \(\mathcal{H}\) into the correlated subspace \(\mathcal{C}_i\) as

\[ \mathbf{A}_i^{\mathcal{C}} = \frac{1}{\mathcal{N}} \sum_k \mathbf{P}_i(k) \mathbf{A}^{\mathcal{H}}(k) \mathbf{P}^{\dagger}_i(k), \]

where \(\mathcal{N}\) is the number of unit cells on the lattice. The above is called downfolding in DMFT[1]. Assuming homogeneity such that \(\mathbf{A}_i^{\mathcal{C}}\) is equivalent in all unit cells, the reverse process from all correlated subspaces \(\mathcal{C}_i\) to \(\mathcal{H}\) is given by

\[ \mathbf{A}^{\mathcal{H}}(k) = \sum_i \sum_k \mathbf{P}^{\dagger}_i(k) \mathbf{A}_i^{\mathcal{C}} \mathbf{P}_i(k). \]

The above is called upfolding in DMFT[1].

In RISB the above projectors are used to upfold the renormalization matrix \(\mathbf{\mathcal{R}}_i\) in each correlated subspace \(\mathcal{C}_i\) as

\[ \mathbf{\mathcal{R}}(k) = \sum_i \sum_k \mathbf{P}^{\dagger}_i(k) \mathbf{\mathcal{R}}_i \mathbf{P}^{}_i(k), \]

and similarly for the correlation potential matrix \(\mathbf{\lambda}_i\) to obtain \(\mathbf{\lambda}(k)\).

The reverse process is used to downfold the quasiparticle density matrix \(\Delta^{\mathrm{qp}}\) of \(\hat{H}^{\mathrm{qp}}\) into each correlated subspace \(\mathcal{C}_i\) as

\[ \Delta^{\mathrm{qp}}_i = \frac{1}{\mathcal{N}} \sum_k \mathbf{P}_i(k)^{} f(\mathbf{H}^{\mathrm{qp}}(k)) \mathbf{P}^{\dagger}_i(k), \]

where \(f(\xi)\) is the Fermi-Dirac function,

\[ \mathbf{H}^{\mathrm{qp}}(k) = \mathbf{\mathcal{R}}(k) \mathbf{H}_0^{\mathrm{kin}}(k) \mathbf{\mathcal{R}}^{\dagger}(k) + \mathbf{\lambda}(k), \]

is the matrix representation of \(\hat{H}^{\mathrm{qp}}\) at each \(k\)-point, and \(\mathbf{H}_{0}^{\mathrm{kin}}(k)\) is the non-interacting dispersion matrix on the lattice that does not include the non-interacting quadratic terms in the subspaces \(\mathcal{C}_i\).

Similarly, downfolding is used to obtain the lopsided kinetic energy of the quasiparticles as

\[ E^{c,\mathrm{qp}}_i = \frac{1}{\mathcal{N}} \sum_k \mathbf{P}^{}_i(k) \mathbf{H}_0^{\mathrm{kin}}(k) \mathbf{\mathcal{R}}^{\dagger}(k) f(\mathbf{H}^{\mathrm{qp}}(k)) \mathbf{P}^{\dagger}_i(k). \]