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Journal of the Physical Society of Japan

Properties of New Saturated Ferromagnetism in the 1D Flat-Band Hubbard Model as the Spiral State

Yuuki Watanabe, Seiji Miyashita

Published Dec 15, 1997• 12 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.66.3981

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Magnetism in the two-dimensional Hubbard model (abstract)

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Journal of Applied Physics•Apr 15, 1985•0 citations
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.
We investigate the magnetization of the ground state of the one-dimensional flat-band Hubbard model as a function of the electron filling, using exact diagonalization techniques. Our calculation shows that the systems with odd n holes added to half-filling bands for a sufficiently large U (the on-site Coulomb repulsion energy) have a saturated ferromagnetic ground state, which is nondegenerate except for the trivial spin degeneracy. Dependences of the critical value of U on the size and on λ (a parameter of the system) are also investigated. For all values of λ, it is found that the ground state is a saturated ferromagnetic state if U is sufficiently large. Moreover, we find that the critical U for the saturated ferromagnetic ground state is almost independent on the system size, at least, for the system near quarter-filling. These facts indicate that even if the system size go to infinity, the ferromagnetic ground state can be exist for a finite U .
In order to study conditions for the appearance of ferromagnetism in a wide filling region, we investigate numerically three types of one-dimensional Lieb-ferrimagnetic Hubbard models: a periodic diamond (PD) chain, a periodic alternately-attached leg (PAAL) chain and an open diamond (OD) chain. All of these models have a flat band (or equivalently, degenerate single-electron eigenvalues). The PD and OD chains commonly have a local-loop structure. Nagaoka's theorem holds only in the PD chain. At half-filling, it have been rigorously proven that all of these models are ferrimagnet. Away from half-filling, however, quite different magnetic properties are found. In the fillings 1/3< rho_e <1/2, the ground state of the PD chain for a infinitely-large U is the extended ferromagnetic state, that is, the saturated ferromagnetic state or the spiral state for odd or even number of electrons, respectively. In the PAAL chain, on the other hand, there is no magnetic order. Thus, the flat band is found to be not a sufficient condition of the extended ferromagnetic state. We find, moreover, that the saturated ferromagnetism appears in the OD chain, although the Nagaoka theorem does not hold on this chain. This indicates that the local-loop structure plays an important role on the appearance of the extended ferromagnetic state.
We explore how the spiral spin (SP) state, a spin singlet known to accompany fully polarized ferromagnetic (F) states in the Hubbard model, is related with the F state in the thermodynamic limit using the density-matrix renormalization group and exact diagonalization. We first obtain an indication that when the F state is the ground state the SP state is also eligible as the ground state in that limit. We then follow the general argument by Koma and Tasaki [J. Stat. Phys. 76, 745 (1994)] to find that (i) the SP state possesses a kind of order parameter, and (ii) although the SP state does not break the SU(2) symmetry in finite systems, it does so in the thermodynamic limit by making a linear combination with other states that are degenerate in that limit. We also calculate the one-particle spectral function and dynamical spin and charge susceptibilities for various one-dimensional finite-size lattices. We find that the excitation spectra of the SP state and the F state are almost identical. The present results suggest that the SP and the F states are equivalent in the thermodynamic limit. These properties may be exploited to determine the magnetic phase diagram from finite-size studies.
We consider the question of ferromagnetism in the one-band Hubbard model on the square lattice with infinite on-site repulsion using exact diagonalization techniques. Our results show a nonmonotonic behavior as a function of the number of holes Nh{N}_{h}Nh​ away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.
Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.