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We show that the simple update approach proposed by Jiang et. al. [H.C.Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is anefficient and accurate method for determining the infinite tree tensor networkstates on the Bethe lattice. Ground state properties of the quantum transverseIsing model and the Heisenberg XXZ model on the Bethe lattice are studied. Thetransverse Ising model is found to undergo a second-order quantum phasetransition with a diverging magnetic susceptibility but a finite correlationlength which is upper-bounded by 1/ln(q-1) even at the transition point (q isthe coordinate number of the Bethe lattice). An intuitive explanation on thispeculiar "critical" phenomenon is given. The XXZ model on the Bethe latticeundergoes a first-order quantum phase transition at the isotropic point.Furthermore, the simple update scheme is found to be related with the Betheapproximation. Finally, by applying the simple update to various tree tensorclusters, we can obtain rather nice and scalable approximations fortwo-dimensional lattices.

Efficient simulation of infinite tree tensor network states on the Bethe lattice