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Journal of Physics Condensed Matter

Localization lengths over metal to band insulator transitions

Nicholas D. M. Hine, W. M. C. Foulkes

Published Nov 20, 2007• 12 citations
Publisher

Abstract

Density functional and quantum Monte Carlo methods are used to examine the behaviour of the many-electron localization length near band insulator-to-metal transitions in various one- and two-dimensional model systems. The many-electron localization length is infinite in metals and finite in insulators, and is normally assumed to diverge as an insulator-to-metal transition is approached from the insulating side. Our results show that this is not the case: the band insulator-to-metal transition is normally first order and not associated with a diverging length scale. We also identify examples where the localization length diverges but the system is insulating on both sides of the divergence. The usefulness of the localization length as an indicator of the approach of an insulator-to-metal transition is therefore limited. A comparison of our quantum Monte Carlo and density functional results allows us to draw some general conclusions about the effect of correlation on localization.

DOI
10.1088/0953-8984/19/50/506212

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Density functional and quantum Monte Carlo methods are used to examine the behaviour of the many-electron localization length near band insulator-to-metal transitions in various one- and two-dimensional model systems. The many-electron localization length is infinite in metals and finite in insulators, and is normally assumed to diverge as an insulator-to-metal transition is approached from the insulating side. Our results show that this is not the case: the band insulator-to-metal transition is normally first order and not associated with a diverging length scale. We also identify examples where the localization length diverges but the system is insulating on both sides of the divergence. The usefulness of the localization length as an indicator of the approach of an insulator-to-metal transition is therefore limited. A comparison of our quantum Monte Carlo and density functional results allows us to draw some general conclusions about the effect of correlation on localization.
As a measure to ascertain whether a system is metallic or insulating, localization length λNλ_NλN​, which represents the spread of electron distribution, can be a useful quantity, especially for approaching a metal-insulator transition from the insulator side. We try to calculate λNλ_NλN​ using a variational Monte Carlo method for normal (paramagnetic), superconducting and antiferromagnetic states in the square-lattice Hubbard model. It is found that the behavior of λNλ_NλN​ is consistent with what is expected from other quantities, and gives information complementary to another measure, the Drude weight.
Exact-diagonalization and quantum Monte Carlo methods are used to study the occurrence of a metal-insulator transition and the distribution of charges and magnetic moments in the ground state of a one- and two-dimensional half-filled system of correlated electrons, described by a one-band simplified Hubbard model extended with bond-charge--site-charge interactions. Numerical results for the electrical conductivity, the density of states, and the spin- and charge-density correlation functions strongly suggest that the bond-charge--site-charge interaction plays a fundamental role in the occurrence of a metal-insulator transition and of ferromagnetic order.
We study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as κ∝∣μ−μc∣−0.58±0.08κ\propto |μ- μ_c|^{-0.58\pm0.08}κ∝∣μ−μc​∣−0.58±0.08 where μcμ_cμc​ is the critical chemical potential. In the insulating phase, the localization length follows ξl∝∣μ−μc∣−νlξ_l \propto |μ- μ_c|^{-ν_l}ξl​∝∣μ−μc​∣−νl​ with νl=0.26±0.05ν_l = 0.26 \pm 0.05νl​=0.26±0.05. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent νκ=0.21±0.04ν_κ= 0.21 \pm 0.04νκ​=0.21±0.04. Our results show that the exponents νκν_κνκ​ and νlν_lνl​ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent ν=1/4ν= 1/4ν=1/4 and dynamical exponent z=4z = 4z=4. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents ν=1/2ν= 1/2ν=1/2 and z=2z = 2z=2.
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.
The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase's resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.