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arXiv:cond-mat/9407036 · Physical review. B, Condensed matter

Scaling properties of the ferromagnetic state in the Hubbard model

Koichi Kusakabe, Hideo Aoki

Published Nov 1, 1994• 11 citations
arXivHTMLPDF

Abstract

A numerical scaling analysis is used to show that Nagaoka's ferromagnetic state in the two-dimensional Hubbard model with one hole is superseded by an antiferromagnetic (AF) state with a discontinuous jump in the total spin due to the AF coupling as the Hubbard U is made finite. The same applies to the two-hole system, which has a spiral spin structure. We can show, via the scaling, that the crossover to an AF state is a precursor of a pathological coalescence of states having the minimum spin and Nagaoka's state at U=\ensuremath{\infty} in the thermodynamic limit.

DOI
10.1103/physrevb.50.12991

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A numerical scaling analysis is used to show that Nagaoka's ferromagnetic state in the two-dimensional Hubbard model with one hole is superseded by an antiferromagnetic (AF) state with a discontinuous jump in the total spin due to the AF coupling as the Hubbard U is made finite. The same applies to the two-hole system, which has a spiral spin structure. We can show, via the scaling, that the crossover to an AF state is a precursor of a pathological coalescence of states having the minimum spin and Nagaoka's state at U=\ensuremath{\infty} in the thermodynamic limit.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.
We investigate magnetic properties in the ground state of the one-dimensional flat-band Hubbard model in high electron filling. Until now, we found the new saturated ferromagnetism for odd numbers of holes, and its properties were investigated in detail. However, it was remained as a problem how we understand the complete alternation of magnetism between saturated ferromagnetism and singlet for odd and even numbers of holes, respectively. This alternation causes a difficulty when we consider the thermodynamic limit. In order to solve this difficulty, we investigate, at first, the change of the magnetization for systems with various boundary conditions, and we find that even for the even numbers of holes, the ground states are saturated ferromagnetic under the antiperiodic boundary condition. Furthermore, we explicitly investigate the spin-spin correlation functions < S z i S z j > to see the spin ordering. These calculations shows that all ground states in this filling region are the spiral state under a suitable boundary condition, and that the singlet states in these ground states, therefore, have a ferromagnetic order within, approximately, a half of the system size, although the total spin is still zero. From these results, we conclude that the properties in this region are by no means ill, and all ground states are essentially ferromagnetic in the sense that the spins in the singlet states are also ordered ferromagnetically if we take in the half of the system size.
The solution of a three-body Schr\"odinger equation (for two holes and a spin flip) shows that in the U=\ensuremath{\infty} Hubbard model the saturated ferromagnetic state with two holes is unstable (in a finite volume and for periodic boundary conditions). The ground state carries a finite momentum, and becomes degenerate with the Nagaoka state in the thermodynamic limit. This is in agreement with exact diagonalization results for small lattices where, in addition, we find that the true two-hole ground state is a singlet. As a result, our notions of ferromagnetism in the Hubbard model might need revision.