  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 an
efficient and accurate method for determining the infinite tree tensor network
states on the Bethe lattice. Ground state properties of the quantum transverse
Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The
transverse Ising model is found to undergo a second-order quantum phase
transition with a diverging magnetic susceptibility but a finite correlation
length which is upper-bounded by 1/ln(q-1) even at the transition point (q is
the coordinate number of the Bethe lattice). An intuitive explanation on this
peculiar "critical" phenomenon is given. The XXZ model on the Bethe lattice
undergoes a first-order quantum phase transition at the isotropic point.
Furthermore, the simple update scheme is found to be related with the Bethe
approximation. Finally, by applying the simple update to various tree tensor
clusters, we can obtain rather nice and scalable approximations for
two-dimensional lattices.
