[Paper Review] The Weak, the Strong and the Long Correlation Regimes of the Two-Dimensional Hubbard Model at Finite Temperature

\v{S}imkovic IV, Fedor, Riccardo Rossi+1 more|arXiv (Cornell University)|Oct 12, 2021

Physics of Superconductivity and Magnetism112 references4 citations

TL;DR

This paper introduces a novel connected determinant diagrammatic Monte Carlo algorithm to study the two-dimensional Hubbard model at finite temperature, enabling numerically exact results at unprecedentedly low temperatures (T ≥ 0.067) and large system sizes. It identifies three distinct correlation regimes—weak, strong, and long-range magnetic correlations—revealing a spin-channel crossover from commensurate to incommensurate correlations, while charge correlations remain short-ranged, indicating spin-charge decoupling.

ABSTRACT

We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a novel connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at unprecedentedly low temperatures $T\geq 0.067$ for interactions up to $U\leq 8$, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

Motivation & Objective

• To resolve the open question of whether long magnetic correlation lengths are necessary for high-temperature superconductivity in the 2D Hubbard model.
• To clarify the relationship between spin and charge correlation lengths and their dependence on doping, temperature, and interaction strength.
• To investigate the nature of the crossover from weak to strong coupling and the emergence of long-range magnetic correlations.
• To provide numerically exact results at low temperatures and large system sizes, minimizing finite-size effects.
• To connect perturbative series analytic structure to physical insights in the strongly correlated regime.

Proposed method

• Development of a new connected determinant diagrammatic Monte Carlo algorithm for double expansions in interaction and temperature.
• Use of determinant Monte Carlo techniques to compute momentum-resolved spin and charge susceptibilities, chemical potential, and double occupancy.
• Implementation of a novel algorithm that enables numerically exact results at T ≥ 0.067 and U ≤ 8 on arbitrarily large lattices.
• Application of Padé approximants and singular value decomposition (SVD) to analyze the analytic structure of perturbative series.
• Use of finite-size scaling and comparison with smaller lattices to quantify finite-size effects.
• Incorporation of advanced diagrammatic techniques, including connected diagrams and irreducible vertex expansions, to improve convergence and accuracy.

Experimental results

Research questions

• RQ1 What are the distinct correlation regimes in the 2D Hubbard model at finite temperature, and how do they depend on doping, temperature, and U?
• RQ2 Does a crossover from commensurate to incommensurate spin correlations occur, and what drives this transition?
• RQ3 Are spin and charge correlations coupled, or do they exhibit decoupled behavior in the thermodynamic limit?
• RQ4 How do long magnetic correlation lengths emerge, and is their presence necessary for unconventional superconductivity?
• RQ5 To what extent do finite-size effects distort the physical picture in previous studies of the Hubbard model?

Key findings

• Three distinct correlation regimes are identified: weak-coupling, strong-coupling with short-range correlations, and intermediate-coupling with long magnetic correlation lengths.
• A clear crossover from commensurate to incommensurate spin correlations is observed in the spin channel as a function of doping and temperature.
• Charge correlations remain short-ranged across all studied temperatures and interaction strengths, indicating a decoupling of spin and charge responses.
• The method achieves numerically exact results at T ≥ 0.067 and U ≤ 8 on large lattices, significantly reducing finite-size effects compared to prior studies.
• The analytic structure of perturbative series, analyzed via Padé approximants, provides physical insights into the onset of strong correlations.
• Finite-size effects are substantial in smaller lattices, particularly in the intermediate-coupling regime, underscoring the importance of large system sizes for accurate phase diagram determination.