# Overcoming the Sign Problem at Finite Temperature: Quantum Tensor Network for the Orbital $e_g$ Model on an Infinite Square Lattice

## TL;DR

This paper demonstrates that tensor network methods can effectively overcome the quantum Monte Carlo sign problem at finite temperature for the orbital $e_g$ model, accurately capturing phase transition properties and critical exponents.

## Contribution

It introduces a tensor network approach to study a sign-problematic 2D quantum model at finite temperature, providing precise estimates of critical temperature and exponents.

## Key findings

- Confirmed finite order parameter below $T_c$
- Estimated critical temperature $T_c$ accurately
- Critical exponents within 1% of 2D Ising universality

## Abstract

The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied for the first time to a model suffering the notorious quantum Monte Carlo sign problem --- the orbital $e_g$ model with spatially highly anisotropic orbital interactions. Coarse-graining of the tensor network along the inverse temperature $\beta$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension $D$ --- limiting the amount of entanglement --- is a natural refinement parameter. Increasing $D$ we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature $T_c$, provide a numerically exact estimate of~$T_c$, and give the critical exponents within $1\%$ of the 2D Ising universality class.