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

TL;DR

This paper introduces an efficient algorithm for simulating imaginary time evolution of infinite, translationally invariant spin systems on a Bethe lattice using symmetric iPEPS, preserving symmetry and enabling stable, scalable computations.

Contribution

The authors develop a novel, stable, and efficient tensor network algorithm for simulating quantum spin systems on Bethe lattices, improving computational scaling over previous methods.

Findings

Identified a second order phase transition in the transverse-field Ising model on the Bethe lattice.

Demonstrated the algorithm's stability and efficiency in simulating infinite systems.

Achieved finite correlation lengths near the phase transition.

Abstract

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.