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Journal of Physics Condensed Matter

Antiferromagnetism and the metal-insulator transition in the infinite-dimensional Hubbard model

S. Wermbter, G. Czycholl

Published Sep 11, 1995• 7 citations
Publisher

Abstract

We study the Hubbard model in infinite dimension d= infinity within an approximation reproducing simultaneously the atomic limit and the weak-coupling limit up to second order in the interaction strength U. Depending on U we find a paramagnetic metal, a paramagnetic insulator, or an antiferromagnetic insulator. The stability of the antiferromagnetic phase is investigated by calculating the grand canonical potential. We present the phase diagram in the T-U plane and results for the magnetization m as a function of temperature T. For small values of U the antiferromagnetic phase transition is continuous, but for larger values of U the approximation yields a first-order magnetic phase transition. Results for the density of states N( epsilon ) are presented; it either exhibits one single band or an upper and a lower Hubbard band separated by the Mott gap. The upper and lower Hubbard band may exhibit additional magnetic structure, depending on the value of T and U.

DOI
10.1088/0953-8984/7/37/007

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We study the Hubbard model in infinite dimension d= infinity within an approximation reproducing simultaneously the atomic limit and the weak-coupling limit up to second order in the interaction strength U. Depending on U we find a paramagnetic metal, a paramagnetic insulator, or an antiferromagnetic insulator. The stability of the antiferromagnetic phase is investigated by calculating the grand canonical potential. We present the phase diagram in the T-U plane and results for the magnetization m as a function of temperature T. For small values of U the antiferromagnetic phase transition is continuous, but for larger values of U the approximation yields a first-order magnetic phase transition. Results for the density of states N( epsilon ) are presented; it either exhibits one single band or an upper and a lower Hubbard band separated by the Mott gap. The upper and lower Hubbard band may exhibit additional magnetic structure, depending on the value of T and U.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
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.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We study the infinite-dimensional Hubbard model within the self-consistent second-order U-perturbation treatment (SOPT), which is a conserving approximation and the simplest systematic improvement of the Hartree approximation (HA). For half filling we find antiferromagnetic solutions, but even for small Coulomb repulsion U the critical temperature is substantially reduced compared to that obtained in the HA. We construct the phase diagram as a function of doping and find regions of re-entrant behaviour in the temperature dependence of the staggered magnetization away from half filling. The stability of the ordered solutions is investigated by calculation of the grand canonical potential and the filling fraction dependence of the chemical potential; for small values of U the antiferromagnetic phase transition is continuous (as usual), but for larger U the SOPT yields a first-order phase transition. For low temperatures and away from half filling there may be a region of phase coexistence between the antiferromagnetic and homogeneous phases.
Antiferromagnetic and metal to insulator transitions in the ground state of a two dimensional half-filled Hubbard model with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Hartree-Fock approximation. The results indicate that the phase diagram depends on the band structure or t ′ / t ratio; a discontinuous transition between the paramagnetic metallic (PM) and the antiferromagnetic insulator (AI) phases takes place for t ′ / t =0.2, while continuous transitions from PM to AM (antiferromagnetic metal) and then to AI phases take place for t ′ / t =0.5. The results are compared with those of the Gutzwiller approximation and discussed in the light of the other existing theories. The general importance of considering antiferromagnetism in this problem is emphasized.
We study a two-band Hubbard model using the dynamical mean-field theory combined with the exact diagonalization method. At the electron density n=2n=2n=2, a transition from a band-insulator to a correlated semimetal occurs when the on-site Coulomb interaction UUU is varied for a fixed value of the charge-transfer energy ΔΔΔ. At low temperature, the correlated semimetal shows ferromagnetism or superconductivity. With increasing doping ∣n−2∣|n-2|∣n−2∣, the ferromagnetic transition temperature rapidly decreases and finally becomes zero at a critical value of nnn. The second-order phase transition occurs at high temperature, while a phase separation of ferromagnetic and paramagnetic states takes place at low temperature. The superconducting transition temperature gradually decreases and finally becomes zero near n=1n=1n=1 (n=3n=3n=3) where the system is Mott insulator which shows antiferromagnetism at low temperature.