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Progress in describing finitetemperature phase transitions in strongly correlated quantum systems is obtained by noticing that the Gibbs operator for an infinite 2D lattice system can be represented by a 3D tensor network, the third dimension being the inverse temperature \beta. Coarsegraining the network along \beta results in a 2D projected entangledpair operator (PEPO) with a finite bond dimension D. The coarsegraining is performed by a tree tensor network of isometries. The isometries are optimized variationally â€” taking into account full tensor environment â€” to maximize the accuracy of the PEPO. I will present benchmark results in the 2D quantum compass model, the Hubbard model at high temperature, the e_g orbital model, and the Hubbard model for spinpolarized fermions on a hexagonal lattice. Our results have similar accuracy as the best results obtained by conventional methods like quantum Monte Carlo or dynamical cluster approximation but, like all tensor networks, our unbiased method is also applicable where the sing problem makes the QMC fail. Host: Lukasz Cincio 