IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 19 (2007) 506212 (19pp) doi:10.1088/0953-8984/19/50/506212 Localization lengths over metal to band insulator transitions N D M Hine and W M C Foulkes Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ, UK Received 16 August 2007, in final form 31 October 2007 Published 20 November 2007 Online at stacks.iop.org/JPhysCM/19/506212 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. 1. Introduction The question of whether the many-body wavefunction of a crystalline solid represents a metal or an insulator is easily answered for the determinantal wavefunctions used in one-electron band theory: if the density of one-electron states is finite at the Fermi level, the system is a metal; if not, it is an insulator. An alternative approach notes that one can apply a unitary transformation to the determinant of Bloch functions of an insulator to obtain an equivalent determinant of exponentially localized [1] one-electron Wannier functions. The exponential localization of the Wannier functions implies a corresponding exponential localization of the one-electron density matrix ρ(r, r ′ ) as a function of r − r ′ , and it has been shown [2] that any system for which the second moment of ρ(r, r ′ ) is finite must be insulating. In the one-electron theory of disordered systems, the energy eigenfunctions may be localized and a finite density of states at the Fermi level need not imply conducting behaviour, but the more general idea that insulating behaviour results from wavefunction localization survives. In fact, more than forty years ago, Kohn [3] argued that all metal to insulator transitions, even those in strongly correlated interacting solids, are accompanied by a specific 0953-8984/07/506212+19$30.00 © 2007 IOP Publishing Ltd Printed in the UK 1