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Journal of the Physical Society of Japan

Partial Ferromagnetism in the Hubbard Chain with Next-Nearest-Neighbor Hopping

Hiroki Nakano, Yoshinori Takahashi

Published May 15, 2003• 10 citations
Publisher

Abstract

We study the one-dimensional Hubbard model with next-nearest-neighbor hopping by means of the numerical diagonalization for finite-size clusters. It is found that the ground state is evidently partially spin-polarized in a wide region of intermediate values of on-site Coulomb repulsion, which provides the first example as a minimal model without orbital degeneracy. Analysis of correlation functions makes it clear that the partially spin-polarized states form a spin polaron. Polaron's size grows with the increasing interaction.

DOI
10.1143/jpsj.72.1191

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Journal of the Physical Society of Japan•Apr 15, 2004•9 citations
We study the one-dimensional Hubbard model with next-nearest-neighbor hopping by means of the numerical diagonalization for finite-size clusters. It is found that the ground state is evidently partially spin-polarized in a wide region of intermediate values of on-site Coulomb repulsion, which provides the first example as a minimal model without orbital degeneracy. Analysis of correlation functions makes it clear that the partially spin-polarized states form a spin polaron. Polaron's size grows with the increasing interaction.
Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.
Using exact diagonalization techniques, the Hubbard model with hopping between nearest (t) and next-nearest neighbors ${(t}^{\ensuremath{'}})$ is studied in a two-dimensional cluster with $4\ifmmode\times\else\texttimes\fi{}4$ sites. Within a range of values of ${t}^{\ensuremath{'}}<0$ and low densities, states with different total spin are degenerate in the noninteracting system. At a finite value of the Coulomb repulsion U, states with high spin are energetically favored and the ground state can be fully polarized. The behavior of the density of states differs from that obtained within the Hartree-Fock approximation. The low energy region of the density of states (DOS) for the majority spins remains weakly modified by U. Our results bring further support to the idea that itinerant ferromagnetism occurs in strongly correlated systems with high noninteracting DOS at the bottom of the band.
In a t-J model on a chain, an additional next-nearest-neighbor hopping term destroys the complete spin-charge separation at J/t\ensuremath{\ll}1 and induces the formation of stable spin-polarized S>0 states. The properties of a single ferromagnetic polaron are investigated by the exact diagonalization of small systems and by the approximative treatment of an effective model, describing a spinless fermion coupled to the spin background. Both methods give consistent results on the critical Jc{\mathit{J}}_{\mathit{c}}Jc​ for the polaron formation on the increase of the spin S and the polaron size with decreasing J. A very large enhancement of the polaron mass with increasing S is observed. Systems with a finite concentration of holes are also studied within the same framework. Analytical considerations indicate on the bound many-hole states and consequently on phase-separated regions in the phase diagram.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
Metal–insulator transition in the ground state of the Hubbard chain with the next-nearest-neighbor hopping is studied by exact diagonalization method. This model shows the partially ferromagnetic metal near the Mott insulator for large Coulomb repulsion while the model reveals the non-magnetic metal for small but finite Coulomb repulsion. We find the transition between the partially ferromagnetic metal and the Mott insulator without phase separation. A new critical exponent of the diverging charge compressibility with respect to the hole density is observed when the system approaches the metal–Mott-insulator transition point from the side of the partially ferromagnetic metal. We also find the occurrence of the phase separation between partially ferromagnetic metal and the Mott insulator for even larger Coulomb repulsion. The model proposed here is interesting in view of the magnetism as well as the metal–insulator transition.