SCCE/iMF

SCCE/iMF#

SCCE gives access to a variety of cluster embedding schemes, including the inverse mean field based Self-Consistent Cluster Embedding (SCCE/iMF) scheme developed by HQS. In the SCCE/iMF approach, a lattice model is solved iteratively, using a combined system which is divided into a fully-interacting cluster coupled to a non-interacting bath. The simulation proceeds until the result is self-consistent. In this respect, the approach is similar to Dynamical Mean-Field Theory (DMFT) and Density Matrix Embedding Theory (DMET). For a detailed description of these techniques, see section on cluster embedding.

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Lattice divided into a combined cluster-bath system.#

The current version of SCCE beta supports the inverse Hartree-Fock (iHF) based SCCE schemes and CPT with Bloch phase periodization. It is furthermore an easy-to-use backend to a mature DMRG solver for performing simulations of spin and Hubbard-like lattice models.

Basics#

The idea behind SCCE (strictly speaking SCCE/iMF) involves partitioning a lattice model into a "cluster" and its complement, referred to as the "bath." The complete system, consisting of cluster and bath, is then solved for the low-lying states, with a self-consistency condition applied to the full system.

Currently, the self-consistency condition involves using the full many-body description of the cluster to determine a mean-field description for it. This mean-field description is then applied to the bath.

Specifically, the following steps are repeated until convergence:

  • We start with the Hamiltonian \(\mathcal{H}\) of the full system (S).

  • We split the system S into two parts: the cluster (C) and the bath (B).

  • We extract a non-interacting reference system \({\cal H}_0\) which approximates the Hamiltonian \({\cal H}\) of the full system. Currently we simply use the non-interacting part of \({\cal H}\).

    • We then enter the SCCE loop, which proceeds as follows:

    1. Use \({\cal H}_0\) to construct the bath Hamiltonian \({\cal H}_{\mathrm{B}}\) and the cluster-bath coupling \({\cal H}_{\mathrm{CB}}\). Note: certain methods allow for some type of interaction to be retained in \({\cal H}_{\mathrm{B}} / {\cal H}_{\mathrm{CB}}\).

    2. Transform \({\cal H}_{\mathrm{B}}\) into preferred form. Specifically, we perform a Lanczos-type block tridiagonalization, and shuffle the dominantly-occupied bath sites to the left of the cluster and the dominantly empty bath sites to the right of the cluster.

    3. Perform an optional coarse-graining of the bath (this feature is not yet implemented)

    4. Solve the \(({\cal H}_{\mathrm{C}}, {\cal H}_{\mathrm{CB}}, {\cal H}_{\mathrm{B}})\) system with a many-particle wave function method (solver), currently DMRG, that includes, at a minimum, the interaction within the cluster.

    5. Perform any measurements of interest (at a minimum, the observables required for iteration of the SCCE loop).

    6. Extract an effective mean-field Hamiltonian from the cluster and use the lattice vectors to construct a new effective mean-field description \({\cal H}_0\) of the complete system.

    7. Return to step one and iterate until the self-consistency condition is reached.

Effective Hamiltonian Details#

For a detailed description of the inverse mean-field approach, see "Inverse mean field theories", Phys. Chem. Chem. Phys., 20, 27600-27610 (2018), https://doi.org/10.1039/C8CP03763A .

iHF#

The inverse Hartree–Fock (iHF) method calculates the Hartree–Fock Hamiltonian of the cluster by measuring its one-particle reduced density matrix (1RDM). This approach is referred to as the inverse Hartree–Fock method in order to avoid confusion with self-consistent Hartree-Fock.

The Hartree–Fock equations for a term which is quartic in fermionic creation \(\hat{c}^\dagger\) and annihilation \(\hat{c}\) operators is obtained by the mean-field decoupling:

\[\begin{split} \hat{c}^\dagger_i \hat{c}^\dagger_j \hat{c}^{}_k \hat{c}^{}_\ell \rightarrow & \; \hphantom{+} \braket{\hat{c}^\dagger_i \hat{c}^{}_\ell} \,\hat{c}^\dagger_j \hat{c}^{}_k \,+\, \braket{\hat{c}^\dagger_j \hat{c}^{}_k } \, \hat{c}^\dagger_i \hat{c}^{}_\ell \,-\, \braket{ \hat{c}^\dagger_i \hat{c}^{}_\ell} \braket{ \hat{c}^\dagger_j \hat{c}^{}_k} \\ & - \braket{\hat{c}^\dagger_i \hat{c}^{}_k} \,\hat{c}^\dagger_j \hat{c}^{}_\ell \,-\, \braket{\hat{c}^\dagger_j \hat{c}^{}_\ell } \, \hat{c}^\dagger_i \hat{c}^{}_k \,+\, \braket{\hat{c}^\dagger_i \hat{c}^{}_k} \braket{ \hat{c}^\dagger_j \hat{c}^{}_\ell} \\ & +\underbrace{\braket{\hat{c}^\dagger_i \hat{c}^\dagger_j}}_{={\braket{\hat{c}^{}_j \hat{c}^{}_i}}^\dagger} \,\hat{c}^{}_k \hat{c}^{}_\ell \,+\, \braket{\hat{c}^{}_k \hat{c}^{}_\ell } \, \hat{c}^\dagger_i \hat{c}^\dagger_j \,-\, \braket{\hat{c}^\dagger_i \hat{c}^\dagger_j } \braket{ \hat{c}^{}_k \hat{c}^{}_\ell }\end{split}\]

where the third term of each r.h.s expression accounts for the double counting of the interaction. Note that we have included the spin degree of freedom in the site index. Due to the fermionic algebra we have \(i\ne j\) and \(k \ne \ell\). However, we may still have \(i = \ell\) and/or \(j = k\)

Restricting ourselves to a density-density interaction, \(\hat{n}_\ell = \hat{c}^\dagger_\ell \hat{c}^{}_\ell\), or setting \(i = \ell\) and \(j = k\) in the HF equations, we obtain

\[\begin{split}\hat{n}_i \hat{n}_j \rightarrow& \;\hphantom{+} \braket{\hat{n}_j}\, \hat{n}_i \,+\, \braket{\hat{n}_i}\, \hat{n}_j \,-\, \braket{\hat{n}_i} \braket{\hat{n}_j} \\& - \braket{ \hat{c}^\dagger_i \hat{c}^{}_j }\, \hat{c}^\dagger_j \hat{c}^{}_i \,-\, \braket{ \hat{c}^\dagger_j \hat{c}^{}_i }\, \hat{c}^\dagger_i \hat{c}^{}_j \,+\, \braket{ \hat{c}^\dagger_i \hat{c}^{}_j } \braket{ \hat{c}^\dagger_j \hat{c}^{}_i } \\&+ \braket{ \hat{c}^\dagger_i \hat{c}^\dagger_j }\, \hat{c}^{}_j \hat{c}^{}_i \,+\, \braket{ \hat{c}^{}_j \hat{c}^{}_i }\, \hat{c}^\dagger_i \hat{c}^\dagger_j \,-\, \braket{ \hat{c}^\dagger_i \hat{c}^\dagger_j } \braket{ \hat{c}^{}_j \hat{c}^{}_i }\end{split}\]

Note that we have extended the standard HF description with a BCS decoupling, so that we are actually performing a HF/BCS decoupling. Here, the first line of the r.h.s denotes the Hartree contribution, the second line the Fock contribution, and the third line the BCS contribution. The BCS contribution is currently not enabled in the SCCE software, however its implementation is planned for the future.

Within the iHF approach we replace the expectation values in the above expression with the measured 1RDM \(\rho\),

\[\rho_{p,q} = \braket{ \hat{c}^\dagger_p \hat{c}^{}_q}\]

iDFT#

The effective Hamiltonian is obtained by using a reverse-engineered inverse density functional theory (iDFT), or more precisely an inverse site occupation function theory (iSOFT). The determination of the the local potentials utilizes a steepest-descent gradient search. In order to avoid problems due to degeneracies a finite inverse temperature \(\beta\) is applied. The default value is \(\beta=10^4\).

iFDFT#

The effective Hamiltonian is obtained by using a reverse-engineered inverse density functional theory (iDFT), or more precisely an inverse site occupation function theory (iSOFT) as in the iDFT. The off diagonal elemts, i.e. the hopping elements are taken from evaluating the Fock terms as in iHF. The determination of the the local potentials utilizes a steepest-descent gradient search. In order to avoid problems due to degeneracies a finite inverse temperature \(\beta\) is applied. The default value is \(\beta=10^4\).

ikDFT#

The effective Hamiltonian is obtained by using a reverse-engineered inverse density functional theory (iDFT), or more precisely an inverse site occupation function theory (iSOFT), combined with an adaptation of the kinetic energy terms. The determination of the effective parameter utilizes a steepest-descent gradient search. The ikDFT is currently not part of the public interface.

Construction of the system mean-field#

Updating the cluster#

Once the mean-field description of the cluster has been obtained, all clusters of \({\cal H}_0\) are updated with the new mean-field description.

The updated \({\cal H}_0\) only contains updated values for bonds and on-site terms within the cluster.

Updating the inter-cluster values#

To update the bonds between clusters, the clusters are shifted by half lattice vectors. Any inter-cluster bonds that appear in this overlay are added to \({\cal H}_0\).

Currently, any interactions or hoppings which are further apart than half of the cluster size are not taken into consideration when updating the cluster.

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