Abstract
Cluster perturbation theory (CPT) is a simple approximation scheme that applies to lattice models with local interactions, like the Hubbard model, or models where the local interaction is predominant. It proceeds by tiling the lattice into identical, finite-size clusters, solving these clusters exactly and treating the inter-cluster hopping terms at first order in strong-coupling perturbation theory. This review will focus on the kinematical aspects of CPT, in particular the periodization procedure, and on the practical implementation of CPT using an exact diagonalization solver for the cluster. Applications of CPT will be briefly reviewed.
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Notes
- 1.
In the VCA, \(\vec{V }\) is the difference between the lattice Hamiltonian H and the reference Hamiltonian H′, and as such may also contain intra-cluster terms.
- 2.
For simplicity, we will suppress the spin and band indices \(\sigma \) in this section, but the whole discussion is trivially generalized to the case where there are many electron states per lattice site.
- 3.
Dependence on quasi-continuous indices, like \(\vec{k}\) and \(\tilde{\vec{k}}\), will be indicated by parentheses instead of subscripts. This notation may rightfully be deemed capricious, since the labels i and \(\vec{k}\) take the same number N of values, but we adopt it nonetheless as it helps reminding us that the values of the labels are closely separated.
- 4.
We use the term Brillouin zone in a rather liberal manner, as a complete and irreducible set of wavevectors, and not as the Wigner–Seitz cell of the reciprocal lattice.
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Acknowledgments
The author would like to thank the following people for discussions which, over the years, have strengthened and widened his understanding of quantum cluster methods: M. Civelli, G. Kotliar, B. Kyung, M. Jarrell, Th. Maier, S. Okamoto, D. Plouffe, M. Potthoff, A-M. Tremblay, and C. Weber. Computational resources for this review were provided by RQCHP and Compute Canada.
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Sénéchal, D. (2012). Cluster Perturbation Theory. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21831-6_8
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