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arXiv:0704.1389 · Physical Review B

Gutzwiller description of non-magnetic Mott insulators: Dimer lattice model

Michele Fabrizio

Published Oct 9, 2007• 62 citations
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Abstract

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.

DOI
10.1103/physrevb.76.165110

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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.
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.
The main flaw of the well-known Brinkman-Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high-energy virtual processes that generate, for instance, the antiferromagnetic exchange $J\ensuremath{\sim}{t}^{2}/U$. Here, we propose a way to capture those processes by combining the Brinkman-Rice approach with a variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal-to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller approximation becomes exact. We indeed find for the Mott transition a description very close to the real one provided by the dynamical mean-field theory, an encouraging result in view of possible applications to more involved models.
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.
We investigate the phase diagram of the square lattice bilayer Hubbard model at half-filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping and on-site Coulomb repulsion U . With this study we resolve some discrepancies in previous calculations based on the dynamical mean-field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: an antiferromagnetic Mott insulator at small for any value of U and a band insulator at large . At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing in the range of . Finally, we discuss the phase diagrams obtained in relation to findings from previous studies based on different many-body approaches.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2Z_2Z2​ symmetry breaking that can be identified as the Mott insulator-to-metal transition.
The ionic Hubbard model is a paradigmatic setup for studying the competition between band and Mott insulating behavior. Within the variationally exact in infinite dimensions Gutzwiller approximation, we derive a compact analytic expression for the phase boundary between the Mott and band insulators. While the method reproduces the expected band–Mott insulator phenomenology, it does not capture the correlated metallic state at finite staggered potential found, for example, in dynamical mean-field theory. This absence highlights that the metallic phase originates from incoherent Hubbard-band physics rather than Fermi-liquid behavior well captured by the Gutzwiller approximation. Our formulation establishes a concise variational framework for the ionic Hubbard model, with natural extensions to nonequilibrium setups and spin-exchange dynamics. Abstract Published by the Jagiellonian University 2026 authors