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Finite-temperature study of the Hubbard model via enhanced exponential tensor renormalization group

Changkai Zhang (张昌凯) and Jan von Delft

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Phys. Rev. B 114, 125114 – Published 18 August, 2026

DOI: https://doi.org/10.1103/7m9y-v22h

Abstract

The two-dimensional (2D) Hubbard model has long attracted interest for its rich phase diagram and its relevance to high-Tc superconductivity. However, reliable finite-temperature studies remain challenging due to the exponential complexity of many-body interactions. Here, we introduce an enhanced 1s+ 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≈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.

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Phys. Rev. B 114, 125114– Published 18 August, 2026
  • Received 15 December 2025
  • Revised 6 July 2026
  • Accepted 13 July 2026
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