Tensor network simulation of the (1+1)-dimensional $O(3)$ nonlinear $\sigma$-model with $\theta=\pi$ term

TL;DR

This paper uses tensor network methods to simulate the (1+1)-dimensional $O(3)$ nonlinear sigma model with a $ heta=\pi$ term, revealing its massless nature and extracting the central charge to compare with theoretical predictions.

Contribution

The authors develop a tensor network approach for the $O(3)$ sigma model with a $ heta=\pi$ term, employing monopole harmonics and continuous matrix product operators, to study its finite-temperature properties.

Findings

The model exhibits a massless phase at finite temperature.

The central charge matches field theory predictions.

Tensor network methods effectively simulate topologically nontrivial field theories.

Abstract

We perform a tensor network simulation of the (1+1)-dimensional $O(3)$ nonlinear $\sigma$-model with $\theta=\pi$ term. Within the Hamiltonian formulation, this field theory emerges as the finite-temperature partition function of a modified quantum rotor model decorated with magnetic monopoles. Using the monopole harmonics basis, we derive the matrix representation for this modified quantum rotor model, which enables tensor network simulations. We employ our recently developed continuous matrix product operator method [Tang et al., Phys. Rev. Lett. 125, 170604 (2020)] to study the finite-temperature properties of this model and reveal its massless nature. The central charge as a function of the coupling constant is directly extracted in our calculations and compared with field theory predictions.