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Journal of Physics Condensed Matter

Analytic correlation functions for the one-dimensional Hubbard model with short-range hopping in the strong-coupling limit

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

Published Aug 15, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0953-8984/6/33/007

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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.
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.
Two aspects of strongly correlated electron systems are studied in this thesis. Hubbard-type models are thought to be at the basis of high-critical-temperature superconductivity. One interaction which has not had much study is the nearest-neighbour hopping of on-site singlet pairs. We refine earlier renormalization group arguments and, using the density-matrix renormalization group method, numerically confirm their prediction that a spin gap opens at infinitesimal pair-hopping amplitude V > 0 in the one-dimensional tight-binding model. We also find a phase separation transition at a finite V > 0 as well as a spin-gap transition at a finite V < 0. The exotic magnetic behaviour of heavy-fermion materials involves an interplay between the screening of local moments from each other and the formation of a magnetic state of long range order. While the single-impurity Kondo Hamiltonian is thought to model some aspects of this behaviour, the properties of the Kondo lattice model away from half-filling are largely unknown. We determine the presence of a spin-gap region in the phase diagram of the one-dimensional Heisenberg-Kondo lattice model and make predictions about certain concealed dimerization order parameters.
Exact results related to extended Hubbard model containing couplings up to next-nearest-neighbour terms were deduced for restricted regions of parameter space in arbitrary D > 1 dimensions at half-filling and T = 0 temperature. Different types of commensurate charge-density waves and spin-density waves, a ferromagnetic domain and a phase separation region were found, the last being for the first time signalled at the level of approximation-free results in dimensions higher than one. In connection with other presented phases we extended their stability domains in comparison with previously published data. The deduced phase diagram is complete in the localized limit. The dependences on the interaction range and dimension were studied. The obtained results emphasize the importance of next-nearest neighbour terms in the development of ordered phases within the studied system.
Exact results related to extended Hubbard model containing couplings up to next-nearest-neighbour terms were deduced for restricted regions of parameter space in arbitrary D > 1 dimensions at half-filling and T = 0 temperature. Different types of commensurate charge-density waves and spin-density waves, a ferromagnetic domain and a phase separation region were found, the last being for the first time signalled at the level of approximation-free results in dimensions higher than one. In connection with other presented phases we extended their stability domains in comparison with previously published data. The deduced phase diagram is complete in the localized limit. The dependences on the interaction range and dimension were studied. The obtained results emphasize the importance of next-nearest neighbour terms in the development of ordered phases within the studied system.
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.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.