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

TL;DR

This paper tackles the challenging finite-temperature study of the 2D Hubbard model by introducing an enhanced 1s+ eXponential Tensor Renormalization Group (XTRG) algorithm that leverages Controlled Bond Expansion to enlarge the variational space. The method attains near 2-site accuracy at approximately 1-site cost, delivering up to a 50% speedup and enabling cooling to $T/t\approx0.004$, which permits direct comparison with zero-temperature iPEPS results and exploration of superconducting correlations. Key findings include pairing enhancement at larger doping and positive $t'/t$, a pseudogap onset temperature $T^*$ that decreases with doping (especially for $t'/t>0$), and a possible Nagaoka polaron signature; a comprehensive density-matrix snapshot dataset is also generated for AI-driven analyses and cold-atom comparisons. Overall, the approach provides a scalable, symmetry-preserving framework for finite-temperature investigations of the Hubbard model with practical implications for quantum simulations and experimental benchmarking.

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.

Figures (5)

• Figure 1: The error of the free energy $F$ of the free-fermion model relative to the exact value as a function of the inverse temperature $\beta$, obtained via the 1s+ XTRG algorithm with bond dimension $D = 400, 600, 800$, respectively.
• Figure 2: The ground state energy per site (red for uniform and orange for striped states) obtained from iPEPS simulations, and the energy at $T/t = 1/256$ (blue) obtained from the 1s+ XTRG algorithm. The iPEPS ground states are acquired on an infinite lattice with 4×2 (uniform) or 8×2 (striped) supercell and the XTRG density matrices are generated on an 8×8 lattice with PBC on the $y$ direction. The light blue curves mark the projected energies for the XTRG density matrices with PBC on both directions. The insets show the difference $\Delta e$ between the projected XTRG energies and the striped iPEPS ground state (GS) energies with respect to temperature for two representative doping levels.
• Figure 3: The pair correlation indicator as a function of doping $\delta$ and temperature $T/t$, obtained from the 1s+ XTRG for $t'/t = -0.25$ (left) and $t'/t = +0.25$ (right). Red circles mark the positions with exact data, and the colored contour map displays the interpolation between the data points. The pairing correlation is found strengthened at large doping, low temperature and a positive $t'/t$ ratio.
• Figure 4: The spin susceptibility (a,b) as a function of doping $\delta$ and temperature $T/t$, and (c,d) as a function of temperature $T/t$ for three representative doping levels (via interpolation), obtained from the 1s+ XTRG for (a,c) $t'/t = -0.25$ and (b,d) $t'/t = +0.25$. Red circles mark the positions with exact data, and the colored contour map displays the interpolation between the data points. The thick dashed line indicates the locus of the susceptibility peak, signaling the onset temperature $T^*$ where pseudogap starts to develop.
• Figure 5: Convergence behavior of hole density $n_h$ and double occupancy $n_d$ as a function of sample size at $\delta\simeq 0.1694$ and $T/t = 1/16$. Green dashed lines indicate reference values extracted directly from the density matrix.