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International Journal of Modern Physics B

SOME PROPERTIES OF THE POSITIVE-U AND NEGATIVE-U HUBBARD MODEL

Francesco Paolo Mancini, M. Marinaro, H. Matsumoto

Published Jun 30, 1996• 13 citations
Publisher

Abstract

In this article we review the properties of the 2D Hubbard model by considering at the same time the cases of repulsive and attractive interaction. The paramagnetic solution is studied by means of the composite operator method in the static approximation for the case of half-filling. Some properties of the two models, as the double occupancy and the spin magnetic susceptibility, are calculated for various values of interaction and temperature and compared. In particular, the different role played by thermal fluctuations is analyzed. Analytical and numerical calculations show that there is a critical value of the interaction, Uc, where the system exhibits a metal-insulator transition. At zero temperature it is found that Uc=−W for the negative-U model and Uc≈1.68W for the positive-U model, where W is the band width. At zero temperature, when the strength of the attractive interaction equals the band width, the system exhibits a phase transition to a pair state, where all the electrons are locally paired. The temperature Tp which controls the crossover to the pair state is calculated as a function of U. For strong attractive interaction χ0 is strongly depressed; increases by increasing T and tends to zero as T→TP.

DOI
10.1142/s0217979296000799

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In this article we review the properties of the 2D Hubbard model by considering at the same time the cases of repulsive and attractive interaction. The paramagnetic solution is studied by means of the composite operator method in the static approximation for the case of half-filling. Some properties of the two models, as the double occupancy and the spin magnetic susceptibility, are calculated for various values of interaction and temperature and compared. In particular, the different role played by thermal fluctuations is analyzed. Analytical and numerical calculations show that there is a critical value of the interaction, Uc, where the system exhibits a metal-insulator transition. At zero temperature it is found that Uc=−W for the negative-U model and Uc≈1.68W for the positive-U model, where W is the band width. At zero temperature, when the strength of the attractive interaction equals the band width, the system exhibits a phase transition to a pair state, where all the electrons are locally paired. The temperature Tp which controls the crossover to the pair state is calculated as a function of U. For strong attractive interaction χ0 is strongly depressed; increases by increasing T and tends to zero as T→TP.
this article we shall consider the case of the Hubbard model with on-site attraction. In Section 2 we derive the explicit expression for the Green's function, in the static approximation. In Section 3 we present some results for the double occupancy and the spin magnetic susceptibility for the case of half filling. By analytic method we can show that at zero temperature the system exhibits a phase transition to a pair state, where all the electrons are locally paired in a singlet state; the critical value U c for the interaction potential at which the phase transition occurs is equal to U c =-8t, 8t being the band width.
We contrast the transport properties (dc resistivity, Seebeck coefficient), optical conductivity, spectral functions, dynamical magnetic susceptibility, and the nuclear magnetic resonance 1/T11/{T}_{1}1/T1​ spin-lattice relaxation rate of the repulsive and attractive infinite-dimensional Hubbard models in the paramagnetic phase for a generic band filling. The calculations are performed in a wide temperature interval using the dynamical mean-field theory with the numerical renormalization group as the impurity solver. The attractive case exhibits significantly more complex temperature dependencies which can be explained by the behavior of the half-filled Hubbard model in external magnetic field with constant magnetization, to which the attractive Hubbard model maps through the partial particle-hole transformation. The resistivity is nonmonotonous for the strongly attractive case: it peaks significantly above the Mott-Ioffe-Regel value at a temperature Tmax{T}_{\mathrm{max}}Tmax​ where the quasiparticle band disappears. For both signs of UUU we find particle-hole asymmetry in the self-energy at low energies, but with the opposite kind of excitations having longer lifetime. This leads to a strong suppression of the slope of the Seebeck coefficient in the attractive case rather than an enhancement as in the repulsive case. The spin-lattice relaxation rate in the strongly attractive case has a nonmonotonic temperature dependence, thereby revealing the pairing fluctuations.
The half-filled Hubbard model is studied in the pair approximation of the cluster variation method (CVM). The use of the SO(4) symmetry of the model makes it possible to give a complete analytical characterization of the ground state, by means of explicit expressions for the double occupancy and the nearest-neighbour correlation functions. The finite-temperature analysis is reduced to the numerical solution of only two coupled transcendental equations. The behaviour of local magnetic moment, specific heat and correlation functions is given for some typical cases in one and two dimensions. The authors obtain good qualitative agreement with exact and numerical results in one dimension. The results for finite temperatures show a rapid evolution, with increasing temperature, from a strongly antiferromagnetic behavior to a disordered one; in the high-temperature region a maximum (which has been related to a 'gradual' metal-insulator transition) is found in the specific heat for very large values of the Coulomb repulsion.
The Mott-Hubbard transition is studied in the context of the two-dimensional Hubbard model. Analytical calculations show the existence of a critical value Uc of the potential strength which separates a paramagnetic metallic phase from a paramagnetic insulating phase. Calculations of the density of states and double occupancy show that the ground state in the insulating phase contains always a small fraction of empty and doubly occupied sites. The structure of the ground state is studied by considering the probability amplitude of intersite hopping. The results indicate that the ground state of the Mott insulator is characterized by a local antiferromagnetic order; the electrons keep some mobility, but this mobility must be compatible with the local ordering. The vanishing of some intersite probability amplitudes at U=Uc puts a constrain on the electron mobility. It is suggested that such quantities might be taken as the quantities which control the order in the insulating phase.
The exact phase diagram and the ground-state properties of the one-dimensional Hubbard model with arbitrary on-site interaction of electrons are calculated over a wide range of magnetic field and electron concentrations by means of the Bethe-ansatz formalism. The ground-state properties, including the total energy, the average spin (magnetization) and spin (magnetic) susceptibility are investigated for both signs of the interaction strength U/t. The critical behavior near the onset of magnetization and magnetic saturation are also analyzed. At the onset of magnetization and near the magnetic saturation the spin susceptibility χ diverges at all U/t for half-filling case n=1, whereas for n≠1 it is always finite. The reverse susceptibility χ-1(U) exhibits anomalous hump, which increases with h or n, and shows discontinuity as U/t→±0 at infinitesimal h→0. The analytical results for the ground-state properties in strong and weak interaction limits are in full agreement with our numerical calculations.