User:Datzr/Series

The linked cluster series expansion technique is often used in the study of interacting systems.

Ground state series expansions

Series expansions is a perturbative approach to the quantum many-body problem, working directly in the thermodynamic limit. It is a viable technique for dealing with a lattice Hamiltonian ${\mathcal {H}}$ which can be decomposed as

${\mathcal {H}}={\mathcal {H}}_{0}+\lambda V$

where ${\mathcal {H}}_{0}$ is a trivially solved (real-space) Hamiltonian, and $V$ is treated as a perturbation. The dimensionless variable $\lambda$ controls the strength of the perturbation. Quantities of interest such as energy or order parameters are obtained as a power series in $\lambda$.

The basic idea of linked cluster series expansions is that the thermodynamic limit of the ground state expectation value of an observable $\langle {\hat {O}}\rangle$ can be obtained as a series up to a certain power in $\lambda$ by considering and solving only a finite number of linked clusters of lattice sites.

High and low temperature type expansions

While all the series expansions described here focus on ground state (and hence zero temperature) properties, there are two different types of expansions to consider - high- and low-temperature type expansions. Note that these do not refer to actual temperature, but rather the properties of ${\mathcal {H}}_{0}$.

An expansion is high-temperature if ${\mathcal {H}}_{0}$ does not contain any interaction terms between two sites. i.e. if the entire Hamiltonian can be written as a sum of terms each involving only one site. The name comes from the observation that each site is essentially isolated and completely uncorrelated from all the other sites, reminiscent of high temperature states.

An expansion is low-temperature if ${\mathcal {H}}_{0}$ does contain interaction terms between sites (where a basis has been chosen such that these terms are diagonal). Typically the ground state involves a breaking of symmetry, take for example the ferromagnetic Ising model, where one of the two degenerate ground states have to be chosen by fixing all spins up or down.