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On the Hartree-Fock phase diagram for the two-dimensional Hubbard model

Christophe Charlier, Edwin Langmann, Jonatan Lenells

TL;DR

The paper develops an analytic, asymptotic framework to construct Hartree–Fock phase diagrams for Hubbard-like models, focusing on the zero-temperature 2D Hubbard model on the square lattice. By analyzing the paramagnetic, ferromagnetic, and antiferromagnetic mean-field equations in three asymptotic sectors (I–III) at the diagram edges, the authors derive explicit boundary curves and dopings, showing that four critical boundaries in the doped phase diagram can be obtained with high precision and that mixed phases must occur. The main contributions are three theorems for the asymptotic sectors, including precise expansions for the phase-boundary curves and the free-energy landscapes, and a rigorous demonstration of mixed HF phases in this canonical strongly correlated system. The results provide deep analytic control over HF phase structure, refine prior numerical boundaries, and suggest that mixed or exotic states may underlie high-temperature superconductivity phenomena, with potential extensions to finite temperature and other lattice geometries.

Abstract

We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.

On the Hartree-Fock phase diagram for the two-dimensional Hubbard model

TL;DR

The paper develops an analytic, asymptotic framework to construct Hartree–Fock phase diagrams for Hubbard-like models, focusing on the zero-temperature 2D Hubbard model on the square lattice. By analyzing the paramagnetic, ferromagnetic, and antiferromagnetic mean-field equations in three asymptotic sectors (I–III) at the diagram edges, the authors derive explicit boundary curves and dopings, showing that four critical boundaries in the doped phase diagram can be obtained with high precision and that mixed phases must occur. The main contributions are three theorems for the asymptotic sectors, including precise expansions for the phase-boundary curves and the free-energy landscapes, and a rigorous demonstration of mixed HF phases in this canonical strongly correlated system. The results provide deep analytic control over HF phase structure, refine prior numerical boundaries, and suggest that mixed or exotic states may underlie high-temperature superconductivity phenomena, with potential extensions to finite temperature and other lattice geometries.

Abstract

We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.
Paper Structure (54 sections, 45 theorems, 318 equations, 4 figures)

This paper contains 54 sections, 45 theorems, 318 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Construction of Hartree--Fock phase diagrams
  3. Mixed phases
  4. The method
  5. The case of the 2D Hubbard model
  6. Organization of the paper
  7. Theorems for the 2D Hubbard model
  8. The P, F, and AF free energies
  9. Physical interpretation
  10. The three asymptotic sectors
  11. First theorem: Sector I
  12. Second theorem: Sector II
  13. Third theorem: Sector III
  14. The paramagnetic Hartree--Fock function
  15. P mean-field equation
  16. ...and 39 more sections

Key Result

Theorem 2.1

Let $\delta >0$. There exist a $U_0>0$ and a smooth real-valued function $\mu_{\text{\upshape I}}(U)$ of $U\in[U_0, +\infty)$ such that if $(U, \mu) \in \text{\upshape I}_{U_0, \delta}$, then the following hold: Moreover, it holds that:

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (90)

  • Theorem 2.1: The AF-Mixed-F interface
  • proof
  • Remark : Nagaoka's theorem
  • Theorem 2.2: The F-Mixed-P interface
  • proof
  • Theorem 2.3: The AF-Mixed-P interface
  • proof
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • ...and 80 more