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Abstract

Tensor network methods are routinely used in approximating various
equilibrium and non-equilibrium scenarios, with the algorithms requiring a
small bond dimension at low enough time or inverse temperature. These
approaches so far lacked a rigorous mathematical justification, since existing
approximations to thermal states and time evolution demand a bond dimension
growing with system size. To address this problem, we construct PEPOs that
approximate, for all local observables, \emph{i)} their thermal expectation
values and \emph{ii)} their Heisenberg time evolution. The bond dimension
required does not depend on system size, but only on the temperature or time.
We also show how these can be used to approximate thermal correlation functions
and expectation values in quantum quenches.