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Physical Review Letters

Incommensurate antiferromagnetism in the two-dimensional Hubbard model

Heinz J. Schulz

Published Mar 19, 1990• 405 citations
Publisher

Abstract

The magnetic states occurring in the two-dimensional Hubbard model in the vicinity of half-filling of the conduction band are studied in the case of weak correlations, using Hartree-Fock theory. It is shown that at zero temperature the commensurate antiferromagnetic state is unstable against domain-wall formation for arbitrarily small deviations from half-filling. The resulting incommensurate antiferromagnet remains initially insulating. The predictions of the model for the polarization and modulation wave vector of the incommensurate antiferromagnet are discussed.

DOI
10.1103/physrevlett.64.1445

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The magnetic states occurring in the two-dimensional Hubbard model in the vicinity of half-filling of the conduction band are studied in the case of weak correlations, using Hartree-Fock theory. It is shown that at zero temperature the commensurate antiferromagnetic state is unstable against domain-wall formation for arbitrarily small deviations from half-filling. The resulting incommensurate antiferromagnet remains initially insulating. The predictions of the model for the polarization and modulation wave vector of the incommensurate antiferromagnet are discussed.
We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard model on a square lattice. We study the stability of the usual commensurate antiferromagnetic phase against the formation of domain walls for various system sizes, band filling, and Hubbard repulsion. An instability towards diagonal domain walls is found for the values of the Hubbard repulsion studied (U=4-10t). We compute the condensation energy of the holes in walls. Such an incommensurate antiferromagnetic phase is much stabler than any other solution. Whether paramagnetic or purely superconducting, but superconductivity is found still to coexist with incommensurate antiferromagnetism. We carefully compare our results with those of the Hartree-Fock model and shed some light on the limitations of the Hartree-Fock solutions.
Commensurate and incommensurate antiferromagnetic fluctuations in the two-dimensional repulsive $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$-Hubbard model are investigated using functional renormalization group equations. For a sufficient deviation from half-filling we establish the existence of local incommensurate order below a pseudocritical temperature Tpc{T}_{pc}Tpc​. Fluctuations not accounted for in the mean-field approximation are important---they lower Tpc{T}_{pc}Tpc​ by a factor $\ensuremath{\approx}2.5$.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
A scheme of symmetry breaking is proposed for the two-dimensional Hubbard model at less than half filling. The self-induced mean field is a random spin-dependent potential. The energy of the spin-disordered state is shown to be lower than that of commensurate or incommensurate antiferromagnetic states. Long-wavelength charge and spin fluctuations obey the diffusion equation, leading to deviations from Fermi-liquid behavior. At wave vector q=(\ensuremath{\pi},\ensuremath{\pi}) the transverse spin-excitation spectrum has a gap.
Abstract We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, Δ\Delta Δ Δ , vanishes like exp⁡(−2πt/U)\exp (-2\pi \sqrt{t/U})exp(−2πt/U​) exp ( - 2 π t / U ) in the weak-coupling limit U/t↓0U/t\downarrow 0U/t↓0 U / t ↓ 0 ( U>0U>0U>0 U > 0 and t>0t>0t>0 t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.
Antiferromagnetic and metal to insulator transitions in the ground state of a two dimensional half-filled Hubbard model with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Hartree-Fock approximation. The results indicate that the phase diagram depends on the band structure or t ′ / t ratio; a discontinuous transition between the paramagnetic metallic (PM) and the antiferromagnetic insulator (AI) phases takes place for t ′ / t =0.2, while continuous transitions from PM to AM (antiferromagnetic metal) and then to AI phases take place for t ′ / t =0.5. The results are compared with those of the Gutzwiller approximation and discussed in the light of the other existing theories. The general importance of considering antiferromagnetism in this problem is emphasized.