Finite-Temperature Study of the Hubbard Model via Enhanced Exponential Tensor Renormalization Group

TL;DR

This paper introduces an enhanced tensor renormalization group algorithm that enables efficient finite-temperature simulations of the 2D Hubbard model, allowing detailed exploration of its phase diagram and superconducting properties.

Contribution

The authors develop an improved exponential tensor renormalization group method that achieves higher accuracy and speed for finite-temperature studies of the 2D Hubbard model.

Findings

Achieves up to 50% acceleration in simulations.

Enables exploration of superconducting order down to very low temperatures.

Provides a comprehensive dataset for AI analysis and experimental comparison.

Abstract

The two-dimensional (2D) Hubbard model has long attracted interest for its rich phase diagram and its relevance to high-$T_c$ superconductivity. However, reliable finite-temperature studies remain challenging due to the exponential complexity of many-body interactions. Here, we introduce an enhanced $1\text{s}^+$ eXponential Tensor Renormalization Group algorithm that enables efficient finite-temperature simulations of the 2D Hubbard model. By exploring an expanded space, our approach achieves two-site update accuracy at the computational cost of a one-site update, and delivers up to 50% acceleration for Hubbard-like systems, which enables simulations down to $T\!\approx\!0.004t$. This advance permits a direct investigation of superconducting order over a wide temperature range and facilitates a comparison with zero-temperature infinite Projected Entangled Pair State simulations. Finally, we compile a comprehensive dataset of snapshots spanning the relevant region of the phase diagram, providing a valuable reference for Artificial Intelligence-driven analyses of the Hubbard model and a comparison with cold-atom experiments.