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arXiv:cond-mat/9208021 · Modern Physics Letters B

MOTT TRANSITION IN THE HUBBARD MODEL

B. SRIRAM SHASTRY

Published Oct 10, 1992• 8 citations
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Abstract

In this article, I discuss W. Kohn’s criterion for a metal insulator transition, within the framework of a one-band Hubbard model. This and related ideas are applied to 1-dimensional Hubbard systems, and some interesting.miscellaneous results discussed. The Jordan-Wigner transformation converting the two species of fermions to two species of hardcore bosons is performed in detail, and the “extra phases” arising from odd-even effects are explicitly derived. Bosons are shown to prefer zero flux (i.e., diamagnetism), and the corresponding “happy fluxes” for the fermions identified. A curious result following from the interplay between orbital diamagnetism and spin polarization is highlighted. A“spin-statistics” like theorem, showing that the anticommutation relations between fermions of opposite spin are crucial to obtain the SU(2) invariance is pointed out.

DOI
10.1142/s0217984992001137

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In this article, I discuss W. Kohn’s criterion for a metal insulator transition, within the framework of a one-band Hubbard model. This and related ideas are applied to 1-dimensional Hubbard systems, and some interesting.miscellaneous results discussed. The Jordan-Wigner transformation converting the two species of fermions to two species of hardcore bosons is performed in detail, and the “extra phases” arising from odd-even effects are explicitly derived. Bosons are shown to prefer zero flux (i.e., diamagnetism), and the corresponding “happy fluxes” for the fermions identified. A curious result following from the interplay between orbital diamagnetism and spin polarization is highlighted. A“spin-statistics” like theorem, showing that the anticommutation relations between fermions of opposite spin are crucial to obtain the SU(2) invariance is pointed out.
In this article, I discuss W.Kohn’s criterion for a metal insulator transition, within the framework of a one band Hubbard model. This and related ideas are applied to 1-dimensional Hubbard systems, and some intersting miscellaneous results discussed. The Jordan Wigner transformation converting the two species of fermions to two species of hardcore bosons is performed in detail, and the “extra phases” arising from odd-even effects are explicitly derived. Bosons are shown to prefer zero flux (i.e. diamagnetism), and the corresponding “happy fluxes ” for the fermions identified. A curious result following from the interplay between orbital diamagnetism and spin polarization is highlighted. A “spin-statistics ” like theorem, showing that the anticommutation relations between fermions of opposite spin are crucial to obtain the SU(2) invariance is pointed out. 1
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
An intuitive interpretation of the relationship between the dispersion relation of the single-particle excitation in a metal and that of the spin excitation in a Mott insulator is presented, based on the results for the one- and two-dimensional Hubbard models obtained by using the Bethe ansatz, dynamical density-matrix renormalization group method, and cluster perturbation theory. The dispersion relation of the spin excitation in the Mott insulator is naturally constructed from that of the single-particle excitation in the zero-doping limit in both one- and two-dimensional Hubbard models, which allows us to interpret the doping-induced states as the states that lose charge character toward the Mott transition. The characteristic feature of the Mott transition is contrasted with the feature of a Fermi liquid and that of the transition between a band insulator and a metal.
An intuitive interpretation of the relationship between the dispersion relation of the single-particle excitation in a metal and that of the spin excitation in a Mott insulator is presented, based on the results for the one- and two-dimensional Hubbard models obtained by using the Bethe ansatz, dynamical density-matrix renormalization group method, and cluster perturbation theory. The dispersion relation of the spin excitation in the Mott insulator is naturally constructed from that of the single-particle excitation in the zero-doping limit in both one- and two-dimensional Hubbard models, which allows us to interpret the doping-induced states as the states that lose charge character toward the Mott transition. The characteristic feature of the Mott transition is contrasted with the feature of a Fermi liquid and that of the transition between a band insulator and a metal.
We are interested in the description of the Mott-Hubbard transition from a perturbation theory arising from a two-dimensional Hubbard model. We calculate the self-energy within the random phase approximation and a double-Lorentzian as a non-interacting density of states. The paramagnetic ground state is unstable for realistic values of U, since the self-energy presents some inconsistencies; one of them is that its imaginary part presents more than one zero, which is a violation of the Luttinger theorem. We use a Bogolyubov transformation and within a spin density wave mean field, the antiferromagnetic correlations of wave vector Q = (π/a,π/a) are included. We recalculate the self-energy in the new ground state and then it is able to describe the Mott-Hubbard transition, and the Luttinger theorem is satisfied.
We study the metal-insulator transition and magnetic ordering in the Hubbard model using the particle-hole mapping. This representation facilitates the exact solution of the half-filled band in the limit of high spin. We find that the two-dimensional Hubbard model has a charge-excitation gap at half filling for any finite U in the high-spin region on both the bipartite square lattice and the nonbipartite triangular lattice. We show, using the Kohn insulator criteria, that this bound-state phase is indeed insulating. Using the information from the single-pair solution, we construct a generalized BCS type of many-body wave function and derive the gap equation for lower-spin states. We find that in some cases, the system goes through a phase transition to become a metal as the total spin Sz{\mathit{S}}_{\mathit{z}}Sz​ is lowered.