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Universality of mean-field antiferromagnetic order in an anisotropic 3D Hubbard model at half-filling

E. Langmann, J. Lenells

TL;DR

The paper analyzes the universality of mean-field antiferromagnetic order in the anisotropic 3D Hubbard model at half-filling, focusing on how the interlayer hopping $t_z$ governs the density of states and the Néel temperature. Using Hartree--Fock theory and all-order density-of-states expansions, the authors derive precise small-$U$ asymptotics for $T_N$ and the AF gap, and show that the gap ratio $\hat{m}$ adheres to a universal BCS-like function $f_{BCS}$ for any $t_z>0$, while universality breaks down in the strictly 2D limit $t_z\to0$ with a distinct $\sqrt{U}$ correction. They also establish all-orders expansions for $N_0(\varepsilon)$ and $N_{t_z}(\varepsilon)$, analyze the $t_z\to0$ behavior of $N_{t_z}(0)$, and provide an exponentially small subleading term refinement for the mean-field description. The results clarify which universal features persist when moving from 3D to quasi-2D Hubbard systems and quantify the singular nature of the $t_z\to0$ limit, offering insight for modeling layered materials and cold-atom realizations.

Abstract

We study the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter $t$ in the $x$- and $y$-directions and a possibly different hopping parameter $t_z$ in the $z$-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases $t_z=0$ and $t_z=t$, respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that $t=1$, we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, $U$, and on the hopping parameter $t_z$. We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as $t_z \to 0$. It is found that the asymptotic formulas are qualitatively different for $t_z = 0$ (the two-dimensional case) and $t_z > 0$ (the case of nonzero hopping in the $z$-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit $t_z \to 0$ in which the three-dimensional model reduces to the two-dimensional model.

Universality of mean-field antiferromagnetic order in an anisotropic 3D Hubbard model at half-filling

TL;DR

The paper analyzes the universality of mean-field antiferromagnetic order in the anisotropic 3D Hubbard model at half-filling, focusing on how the interlayer hopping governs the density of states and the Néel temperature. Using Hartree--Fock theory and all-order density-of-states expansions, the authors derive precise small- asymptotics for and the AF gap, and show that the gap ratio adheres to a universal BCS-like function for any , while universality breaks down in the strictly 2D limit with a distinct correction. They also establish all-orders expansions for and , analyze the behavior of , and provide an exponentially small subleading term refinement for the mean-field description. The results clarify which universal features persist when moving from 3D to quasi-2D Hubbard systems and quantify the singular nature of the limit, offering insight for modeling layered materials and cold-atom realizations.

Abstract

We study the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter in the - and -directions and a possibly different hopping parameter in the -direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases and , respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that , we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, , and on the hopping parameter . We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as . It is found that the asymptotic formulas are qualitatively different for (the two-dimensional case) and (the case of nonzero hopping in the -direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit in which the three-dimensional model reduces to the two-dimensional model.
Paper Structure (33 sections, 29 theorems, 239 equations, 9 figures)

This paper contains 33 sections, 29 theorems, 239 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Mean-field equation for antiferromagnetic order
  3. Néel temperature
  4. Scaling properties
  5. Density of states
  6. Brief description of main results
  7. Universality
  8. Related other works
  9. Organization of the paper
  10. Notation
  11. Main results
  12. Asymptotic formulas for the density of states
  13. The small $U$ behavior of the Néel temperature
  14. The small $U$ behavior of the mean field
  15. An exponentially small subleading term
  16. ...and 18 more sections

Key Result

Theorem 2.1

The two-dimensional density of states $N_0(\epsilon)$ given by (2Ddensityofstates) obeys the following asymptotic expansion to all orders as $\epsilon \downarrow 0$: The functions $A_{k,l}$ in (N0expansion) are defined in terms of the Gamma function $\Gamma(z)$ and the digamma function $\psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}$ by where the principal branch is used for the logarithms. In particula

Figures (9)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 4 more figures

Theorems & Definitions (54)

  • Theorem 2.1: Behavior of $N_0(\epsilon)$ as $\epsilon \to 0$
  • Theorem 2.2: Behavior of $N_{t_z}(\epsilon)$ as $\epsilon \to 0$ for $t_z > 0$
  • Theorem 2.3: Behavior of $N_{t_z}(0)$ as $t_z \to 0$
  • Theorem 2.4: Small $U$ behavior of the 2D Néel temperature
  • proof
  • Theorem 2.5: Small $U$ behavior of the 3D Néel temperature
  • proof
  • Theorem 2.6: Small $U$ behavior of 2D mean field
  • proof
  • Theorem 2.7: Small $U$ behavior of 3D mean field
  • ...and 44 more