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A central problem in many-body quantum physics is the determination of theground state of a thermodynamically large physical system. We construct acluster expansion for ground states of local Hamiltonians, which naturallyincorporates physical requirements inherited by locality as conditions on itscluster amplitudes. Applying a diagrammatic technique we derive the relation ofthese amplitudes to thermodynamic quantities and local observables. Moreover wederive a set of functional equations that determine the cluster amplitudes fora general Hamiltonian, verify the consistency with perturbation theory anddiscuss non-perturbative approaches. Lastly we verify the persistence oflocality features of the cluster expansion under unitary evolution with a localHamiltonian and provide applications to out-of-equilibrium problems: asimplified proof of equilibration to the GGE and a cumulant expansion for thestatistics of work, for an interacting-to-free quench.

Cluster expansion for ground states of local Hamiltonians