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Journal of Physics Condensed Matter

The one-dimensional Hubbard model with first-, second- and third-nearest-neighbour hopping in the strong-coupling limit

M W Long, C. W. M. Castleton, C A Hayward

Published Jan 10, 1994• 10 citations
Publisher

Abstract

Using an extension of the Jordan-Wigner transformation, we solve the one-dimensional Hubbard model at U = infinity in the limit of dominant nearest-neighbour hopping combined with infinitesimal hopping over slightly longer ranges. We find several possible phases at zero temperature, including ferromagnetism, paramagnetism (with and without a spin gap), and even long-range antiferromagnetism with a particular limiting procedure. Our solutions are always spin-charge separated, and we give evidence that the charge degrees of freedom are best described by bosonic statistics.

DOI
10.1088/0953-8984/6/2/019

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Using an extension of the Jordan-Wigner transformation, we solve the one-dimensional Hubbard model at U = infinity in the limit of dominant nearest-neighbour hopping combined with infinitesimal hopping over slightly longer ranges. We find several possible phases at zero temperature, including ferromagnetism, paramagnetism (with and without a spin gap), and even long-range antiferromagnetism with a particular limiting procedure. Our solutions are always spin-charge separated, and we give evidence that the charge degrees of freedom are best described by bosonic statistics.
In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S -k .S k ) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
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 calculate the magnetic phase diagram of the Hubbard model for a Bethe lattice with nearest-neighbour (NN) hopping t 1 and next-nearest-neighbour (NNN) hopping t 2 in the limit of infinite coordination. We use the amplitude t 2 / t 1 of the NNN hopping to tune the density of states (DOS) of the non-interacting system from a situation with particle–hole symmetry to an asymmetric one with van-Hove singularities at the lower ( t 2 / t 1 >0) respectively upper ( t 2 / t 1 <0) band edge for large enough | t 2 / t 1 |. For this strongly asymmetric situation, we find rather extended parameter regions with ferromagnetic states and regions with antiferromagnetic states.
The infinite-U one-dimensional Hubbard model is 'solvable' in the limit that the nearest-neighbour hopping is dominant and infinitesimal longer-range hopping lifts the one-dimensional spin degeneracy. In this limit the Hubbard model maps onto an effective spin interaction: cyclic or ring exchange with a variety of lengths and strengths. For bipartite geometries we find a region of stable ferromagnetism, which originates from the mechanism intrinsic to Nagaoka's ferromagnetism. We look at connectivities that limit exactly to the square-lattice connectivity as the range of the infinitesimal hopping diverges. When we extend this hopping range, we find that the stable region of ferromagnetism shrinks, eventually vanishing as the infinitesimal hopping range diverges and we limit to the square-lattice geometry. Since the calculation involves a non-trivial density of particles in the thermodynamic limit, it suggests how the proposed region of ferromagnetism may become lost for the case of the two-dimensional square lattice, as numerical simulations predict.