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We propose the construction of thermodynamic ensembles that minimize theR\'enyi free energy, as an alternative to Gibbs states. For large systems, thelocal properties of these R\'enyi ensembles coincide with those of thermalequilibrium, and they can be used as approximations to thermal states. Weprovide algorithms to find tensor network approximations to the 2-R\'enyiensemble. In particular, a matrix-product-state representation can be found byusing gradient-based optimization on Riemannian manifolds, or via a non-linearevolution which yields the desired state as a fixed point. We analyze theperformance of the algorithms and the properties of the ensembles onone-dimensional spin chains.

R\'enyi free energy and variational approximations to thermal states