# Thermal Tensor Network Simulations of the Heisenberg Model on the Bethe Lattice

## TL;DR

This paper extends the tensor network method to finite temperatures to study the phase diagram and magnetic properties of the Heisenberg XXZ model on the Bethe lattice, revealing various magnetic phases and excitation spectra.

## Contribution

The authors develop a finite-temperature tensor network approach for the Bethe lattice, enabling detailed phase diagram and thermodynamic property calculations for the Heisenberg model.

## Key findings

- Identified three magnetic phases: ferromagnetic, XY, and antiferromagnetic.
- Discovered continuous and first-order phase transitions at specific coupling points.
- Found that magnetic excitations are gapped, with Goldstone-like modes in ordered phases.

## Abstract

We have extended the canonical tree tensor network (TTN) method, which was initially introduced to simulate the zero-temperature properties of quantum lattice models on the Bethe lattice, to finite temperature simulations. By representing the thermal density matrix with a canonicalized tree tensor product operator, we optimize the TTN and accurately evaluate the thermodynamic quantities, including the internal energy, specific heat, and the spontaneous magnetization, etc, at various temperatures. By varying the anisotropic coupling constant $\Delta$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three kinds of magnetic ordered phases, namely the ferromagnetic, XY and antiferromagnetic ordered phases, are found in low temperatures and separated from the high-$T$ paramagnetic phase by a continuous thermal phase transition at $T_c$. The XY phase is separated from the other two phases by two first-order phase transition lines at the symmetric coupling points $ \Delta=\pm 1$. We have also carried out a linear spin wave calculation on the Bethe lattice, showing that the low-energy magnetic excitations are always gapped, and find the obtained magnon gaps in very good agreement with those estimated from the TTN simulations. Despite the gapped excitation spectrum, Goldstone-like transverse fluctuation modes, as a manifestation of spontaneous continuous symmetry breaking, are observed in the ordered magnetic phases with $|\Delta|\le 1$. One remarkable feature there is that the prominent transverse correlation length reaches $\xi_c=1/\ln{(z-1)}$ for $T\leq T_c$, the maximal value allowed on a $z$-coordinated Bethe lattice.