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Simulating quantum systems on the Bethe lattice by translationally invariant infinite-Tree Tensor Network

We construct an algorithm to simulate imaginary time evolution of translationally invariant spin systems with local interactions on an infinite, symmetric tree. We describe the state by symmetric iPEPS and use translation-invariant operators for the updates at each time step. The contraction of this tree tensor network can be computed efficiently by recursion without approximations and one can then truncate all the iPEPS tensors at the same time. The translational symmetry is preserved at each time step that makes the algorithm very well conditioned and stable. The computational cost scales like $O(D^{q+1})$ with the bond dimension $D$ and coordination number $q$, much favourable than that of the iTEBD on trees [D. Nagaj et al. Phys. Rev. B \textbf{77}, 214431 (2008)]. Studying the transverse-field Ising model on the Bethe lattice, we find a second order phase transition with finite correlation lengths.