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Physical review. B, Condensed matter

Microscopic model of the Mott-Hubbard localization

Józef Spałek, W. Wójcik

Published Feb 15, 1992• 9 citations
Publisher

Abstract

We devise a variational approach to the metal-insulator transition (MIT) that combines the Mott and Hubbard aspects of localization. Starting from the Gutzwiller approach, we optimize the ground-state energy not only with respect to the double occupancy \ensuremath{\eta}, but also with respect to the single-particle wave functions, which enter the energy via expressions for the Coulomb interaction U and for the bare bandwidth W. In effect, we obtain a theory of the MIT without resorting to a parametrization in terms of U/W. The effective mass close to the MIT is strongly enhanced even when the transition is discontinuous.

DOI
10.1103/physrevb.45.3799

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We devise a variational approach to the metal-insulator transition (MIT) that combines the Mott and Hubbard aspects of localization. Starting from the Gutzwiller approach, we optimize the ground-state energy not only with respect to the double occupancy \ensuremath{\eta}, but also with respect to the single-particle wave functions, which enter the energy via expressions for the Coulomb interaction U and for the bare bandwidth W. In effect, we obtain a theory of the MIT without resorting to a parametrization in terms of U/W. The effective mass close to the MIT is strongly enhanced even when the transition is discontinuous.
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.
A metal can be driven to an insulating phase through distinct mechanisms. A possible way is via the Coulomb interaction, which then defines the Mott metal-insulator transition (MIT). Another possibility is the MIT driven by disorder, the so-called Anderson MIT. Here we analyze interacting particles in disordered Hubbard chains---thus comprising the Mott-Anderson physics---by investigating the ground-state entanglement with density functional theory. The localization signature on entanglement is found to be a decreasing on the entanglement degree in comparison to the clean (without interaction and disorder) case, with local minima at certain critical densities. Individually, the Mott (Anderson) MIT has a single critical density whose minimum entanglement decreases as the interaction (disorder) enhances. While in the Mott MIT entanglement saturates at finite values, characterizing partial localization, in the Anderson MIT the system reaches full localization, with zero entanglement, for sufficiently strong disorder. In the combined Mott-Anderson MIT, we find three critical densities referring to local minima on entanglement. One of them is the same as for the Anderson MIT, but now the presence of interaction requires a stronger disorder potential to induce full localization. A second critical density is related to the Mott MIT, but due to disorder it is displaced by a factor proportional to the concentration of impurities. The third local minimum on entanglement is unique to the concomitant presence of disorder and interaction, found to be related to an effective density phenomenon, thus referred to as a Mott-like MIT. Since entanglement has been intrinsically connected to the magnetic susceptibility---a quantity promptly available in cold-atom experiments---our detailed numerical description might be useful for the experimental investigation of the Mott-Anderson MIT.
We study ground-state properties of the two-dimensional Hubbard model at half filling by improving variational Monte Carlo method and by implementing quantum-number projection and multi-variable optimization. The improved variational wave function enables a highly accurate description of the Mott transition and strong fluctuations in metals. We clarify how anomalous metals appear near the first-order Mott transition. The double occupancy stays nearly constant as a function of the on-site Coulomb interaction in the metallic phase near the Mott transition in agreement with the previous unbiased results. This unconventional metal at half filling is stabilized by a formation of ``electron-like pockets'' coexisting with an arc structure, which leads to a prominent differentiation of electrons in momentum space. An abrupt collapse of the ``pocket'' and ``arc'' drives the first-order Mott transition.
A generalized Hubbard model with correlated hoppings is studied at half filling using exact diagonalization methods. For certain values of the hopping parameters our results for several static properties, the Drude weight and the single-particle spectra function, suggest the occurrence of a metal-insulator transition (MIT) at a finite value of the local Coulomb interaction Uc{\mathit{U}}_{\mathit{c}}Uc​. We identify the regions of the hopping parameters where the MIT is of the Mott type. In these regions, for large UUc{\mathit{U}}_{\mathit{c}}Uc​, we find a ferromagnetic ground state.
An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase (U<Uc)(U<{U}_{c})(U<Uc​) corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the U>UcU>{U}_{c}U>Uc​ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.