Numerous perturbation theory schemes are based on linked-cluster expansion. They all suffer from the fact that finite graphs have less symmetries than the infinite cluster : for instance, they lack translation symmetry. A collaboration between Sylvain Capponi and Kai Schmidt's group in Dortmund has shown how to go beyond this fundamental difficulty.
In condensed matter theory, linked-cluster expansions are very common since they allow to directly compute thermodynamic or ground-state properties in the thermodynamic limit. However, computing excitations is more involved. For instance, imagine that in the thermodynamic limit, a given finite-momentum excitation is below the continuum, then it will be stable and long-lived. Oppositely, on a finite graph, the absence of translation of symmetry will mix levels so that this excitation becomes unstable ! As shown in the figure, a level crossing becomes an avoided one on a finite cluster. Note that this is a generic situation for any quasi degenerate perturbation theory in quantum mechanics.
Using an approach based on continuous unitary transforms, we have shown how to combine appropriately the eigenstates so that we can obtain reliable properties. This has been confirmed on standard models such as spin 1/2 Heisenberg ladder or computing dispersion relation of the 2d Ising model in transverse field.
This technique looks quite promising in many other cases, and could be applied as well to any scheme using graph expansions.