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Physical review. B, Condensed matter

Variational wave functions and the Mott transition

Andrew J. Millis, S. N. Coppersmith

Published Jun 1, 1991• 45 citations
Publisher

Abstract

We prove that a large class of generalized Gutzwiller wave functions, previously believed to be useful in the study of the Mott metal-insulator transition, are always metallic in the sense that the response to an electric field includes a \ensuremath{\delta} function at zero frequency (corresponding to free acceleration of carriers in a dc field). We also show that modifying these wave functions to enforce binding of empty and doubly occupied sites is insufficient to produce insulating behavior. We derive a criterion which an insulating wave function must satisfy.

DOI
10.1103/physrevb.43.13770

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We prove that a large class of generalized Gutzwiller wave functions, previously believed to be useful in the study of the Mott metal-insulator transition, are always metallic in the sense that the response to an electric field includes a \ensuremath{\delta} function at zero frequency (corresponding to free acceleration of carriers in a dc field). We also show that modifying these wave functions to enforce binding of empty and doubly occupied sites is insufficient to produce insulating behavior. We derive a criterion which an insulating wave function must satisfy.
We introduce an extension of the Gutzwiller variational wave function able to deal with insulators that escape any mean-field-like description, as, for instance, nonmagnetic insulators. As an application, we study the Mott transition from a paramagnetic metal into a nonmagnetic Peierls, or valence-bond, Mott insulator. We analyze this model by means of our Gutzwiller wave function analytically in the limit of large coordination lattices, where we find that (1) the Mott transition is of first order; (2) the Peierls gap is large in the Mott insulator, although it is mainly contributed by the electron repulsion; and (3) singlet superconductivity arises around the transition.
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.
To distinguish a metal from an insulator, the Drude weight (D, zero-frequency component of conductivity) is a useful measure, as Kohn pointed out long ago: D > 0(=0) for a metal (insulator). Later, Millis and Coppersmith showed that a variational wave function ΨQ in which the key ingredient for a Mott transition (a doublon-holon binding effect) is introduced exhibits D > 0 (metallic) even for sufficiently large correlation strength, namely, a Mott transition is absent from. ΨQ In contrast, variational Monte Carlo studies using ΨQ confirmed, by studying relevant quantities such as doublon density d and a superconducting correlation function, Pd∞ that ΨQ undoubtedly raises a Mott transition. This contradiction has been a long-standing perplexing problem. We definitely settle this problem by adding ΨQ to a configuration-dependent phase factor Pθ, which has been hitherto overlooked. This factor appropriately picks out a negative counterpart in D for insulators, so that D vanishes. Because Pθ does not affect the quantities such as d and Pd∞, the previous results of ΨQ on the Mott transition remain intact for PθΨQ.
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.
The Gutzwiller wave function for a strongly correlated model can, if supplemented with a long-range Jastrow factor, provide a proper variational description of Mott insulators, so far unavailable. We demonstrate this concept in the prototypical one-dimensional t-t(') Hubbard model, where at half-filling we reproduce all known phases, namely, the ordinary Mott undimerized insulator with power-law spin correlations at small t(')/t, the spin-gapped metal above a critical t(')/t and small U, and the dimerized Mott insulator at large repulsion.