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arXiv:0902.4477

Critical scaling of the renormalized single-particle wave function near the Mott-Hubbard transition

Józef Spałek, J. Kurzyk, Robert Podsiadły, W. Wójcik

Published Feb 25, 2009• 0 citations
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Abstract

We present a quantum critical behavior of the renormalized single-particle Wannier function, calculated in the Gutzwiller correlated state near the insulator-metal transition (IMT) for cubic lattices. The wave function size and its maximum, as well as the system energy scale with increasing lattice parameter $R$ as $R^{n}$. Such scaling is interpreted as the evidence of a dominant role of the Coulomb repulsion. Relation of the insulator-metal transition lattice-parameter value $R=R_{C}$ to the original {\em Mott criterion\} is obtained. The method is tested by comparing our results with the exact approach for the Hubbard chain.

DOI
10.48550/arxiv.0902.4477

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We present a quantum critical behavior of the renormalized single-particle Wannier function, calculated in the Gutzwiller correlated state near the insulator-metal transition (IMT) for cubic lattices. The wave function size and its maximum, as well as the system energy scale with increasing lattice parameter RRR as RnR^{n}Rn. Such scaling is interpreted as the evidence of a dominant role of the Coulomb repulsion. Relation of the insulator-metal transition lattice-parameter value R=RCR=R_{C}R=RC​ to the original {\em Mott criterion} is obtained. The method is tested by comparing our results with the exact approach for the Hubbard chain.
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 have applied our novel numerical scheme combining Lanczos diagonalization in the Fock space with an ab initio renormalization of the single-particle (Wannier) functions, to study the ground state properties of the Extended Hubbard Model. Through the finite-size scaling we determine the discontinuity of the momentum distribution Fermi surface. Our results imply Fermi-liquid behavior for lattice parameter a ≲\lesssim≲ 3a0_{0}0​ (a0_{0}0​ is the Bohr radius) and zero-temperature transition to the localized spin system for larger . Possible further applications of the method are listed at the end.
We use the Bethe ansatz equations to calculate the charge stiffness Dc{\mathit{D}}_{\mathit{c}}Dc​=(L/2)d2{\mathit{d}}^{2}d2E0{\mathit{E}}_{0}E0​/d${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}^{2}$${\mathrm{\ensuremath{\Vert}}}_{\mathrm{\ensuremath{\Phi}}\mathit{c}}$=0 of the one-dimensional repulsive-interaction Hubbard model for electron densities close to the Mott insulating value of one electron per site (n=1), where E0{\mathit{E}}_{0}E0​ is the ground-state energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (\ensuremath{\Elzxh}c/e)${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}$ is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of Dc{\mathit{D}}_{\mathit{c}}Dc​(L) as L\ensuremath{\rightarrow}\ensuremath{\infty} at n=1, which defines and yields an analytic expression for the correlation length \ensuremath{\xi} in the Mott insulating phase of the model as a function of the on-site repulsion U. In the vicinity of the zero-temperature critical point U=0, n=1, we show that the charge stiffness has the hyperscaling form Dc{\mathit{D}}_{\mathit{c}}Dc​(n,L,U)=Y+{\mathit{Y}}_{+}Y+​(\ensuremath{\xi}\ensuremath{\delta},\ensuremath{\xi}/L), where \ensuremath{\delta}=\ensuremath{\Vert}1-n\ensuremath{\Vert} and Y+{\mathit{Y}}_{+}Y+​ is a universal scaling function which we calculate. The physical significance of \ensuremath{\xi} in the metallic phase of the model is that it defines the characteristic size of the charge-carrying solitons, or holons. We construct an explicit mapping for arbitrary U and \ensuremath{\xi}\ensuremath{\delta}\ensuremath{\ll}1 of the holons onto weakly interacting spinless fermions, and use this mapping to obtain an asymptotically exact expression for the low-temperature thermopower near the metal-insulator transition, which is a generalization to arbitrary U of a result previously obtained using a weak-coupling approximation, and implies holelike transport for 01-n\ensuremath{\ll}${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$.
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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.