Cluster perturbation theory (CPT)

Within cluster perturbation theory (CPT) one partitions a system into clusters and calculates the cluster Greensfunction (CGF) of the isolated clusters. By partitioning the system into equal clusters one has to calculate one cluster only.

Lattice partitioned into the clusters and the missing bonds.

Once the CGF have been calculated the missing bonds are incorporated by assuming that one can apply a Dyson equation in order to obtain the complete Greensfunction.

We have implemented a Chebyshev expansion of the resolvent [1] combined with a poor man's deconvolution [2].

• [1] Alexander Braun and Peter Schmitteckert, "Numerical evaluation of Greens functions based on the Chebyshev expansion", Phys. Rev. B90, 165112 (2014).

• [2] Peter Schmitteckert, "Calculating Green functions from finite systems", J. Phys.: Conf. Ser. 220, 012022 (2010).

Results

As an example we look at the one-dimensional Hubbard model with an onsite interaction of \(U_0=5t\) and a nearest neighbor hopping of \(t=1\).

We obtain for the spectral function \(A(\omega,k)\) for a \(M=12\) site system using the the sine-transform due to hard wall boundary conditions, see [2],

Spectral function \(A(\omega,k)\) of a \(M=12\) site cluster with \(U_0=5t\) and \(t=1\) using \(P=250\) Chebyshev moments.

Instead of using the Bloch phase boundary condition for periodization we can also perform a real space CPT by repeating the cluster and adding the missing bonds via a Dyson equation. Using 64 repetitions we obtain

Spectral function \(A(\omega,k)\) of a \(M=768 = 64 * 12\) site system consisting of 64 clusters with \(U_0=5t\) and \(t=1\) using \(P=250\) Chebyshev moments.