Walker Classes#

Several types of Slater determinant random walkers are implemented based on how spin-symmetry is handled. Each type is described below. The following list includes all walker types.

  1. RHF/Closed Walkers

  2. UHF/Collinear Walkers

  3. GHF/Noncollinear Walkers

  4. Fully Spin-polarized Walkers

The walker type can be set in the AFQMC input file using the walker_type keyword within a “walker_set” input block. The walker type can be set to one of the following values:

RHF/Closed Walkers#

RHF/Closed walkers are treated as having identical spin-up and spin-down channels. Formally,

\[|\Phi_k\rangle = |\Phi_{k \uparrow}\rangle \otimes |\Phi_{k \tilde{\uparrow}}\rangle,\]

where \(|\Phi_{k \uparrow}\rangle\) is the spin-up channel walker with walker index \(k\), and \(|\Phi_{k \tilde{\uparrow}}\rangle\) is the spin-down channel walker with walker index \(k\). The notation \(\tilde{\uparrow}\) is used to denote that the spin-down channel Slater matrix is identical to the spin-up channel Slater matrix.

Note

Closed walkers can only be used with Hamiltonians that have a “Closed” representation.

Warning

Closed walkers are not allowed in calculations using Lattice Model Hamiltonians.

UHF/Collinear Walkers#

UHF/Collinear walkers are treated as having independent spin-up and spin-down channels. Formally,

\[|\Phi_k\rangle = |\Phi_{k \uparrow}\rangle \otimes |\Phi_{k \downarrow}\rangle,\]

where \(|\Phi_{k \uparrow}\rangle\) is the spin-up channel walker with walker index \(k\), and \(|\Phi_{k \downarrow}\rangle\) is the spin-down channel walker with walker index \(k\). Both spin channels have independent Slater matrices which are separately updated during the random walk.

Note

Collinear walkers can only be used with Hamiltonians that have a “Collinear” or “Closed” representation.

GHF/Noncollinear Walkers#

GHF/Noncollinear walkers include both spin-up, spin-down channels as well as spin-flip channels within a single Slater matrix. Formally,

\[\begin{split}|\Phi_k\rangle = \begin{bmatrix} \Phi^{\uparrow \uparrow}_k & \Phi^{\uparrow \downarrow}_k \\ \Phi^{\downarrow \uparrow}_k & \Phi^{\downarrow \downarrow}_k \end{bmatrix}\end{split}\]

where $Phi^{sigma sigma’}_k$ represent each respective spin channel. In practice, the entire Slater matrix is stored and propagated during the random walk with no partitioning between spin channels.

Note

Noncollinear walkers can be used with Hamiltonians that have any spin symmetry representation (“Closed”, “Collinear”, “Noncollinear”).

Fully Spin-polarized Walkers#

Fully Spin-polarized walkers are treated as having a single spin channel. This can be thought of as a special case of the UHF/Collinear walker in which there are no walkers in the spin-down channel.

Formally,

\[|\Phi_k\rangle = |\Phi_{k \uparrow}\rangle,\]

where \(|\Phi_{k \uparrow}\rangle\) is the spin-up channel walker with walker index \(k\).

Note

Fully spin-polarized walkers can only be used with Hamiltonians that have a “Collinear” or “Closed” representation.