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arXiv:cond-mat/0303410 · Physical review. B, Condensed matter

Scaling study of the metal-insulator transition in one-dimensional Fermion systems

Shi-Jian Gu, Vitor M. Pereira, Nuno M. R. Peres

Published Dec 19, 2002• 10 citations
arXivHTMLPDF

Abstract

We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on a finite-size lattice with N sites. We compute the large-$N$ behavior of the spin stiffness, obtaining the correlation length $\ensuremath{\xi}.$ We use our results to discuss the scaling behavior of the metal-insulator transitions in one-dimensional systems, taking into account the mapping between the XXZ Heisenberg chain and the spinless fermion model, and known results for the Hubbard model. We study the scaling properties of both the Hubbard model and the XXZ Heisenberg chain by solving numerically the Bethe ansatz equations. We find that for some range of values of $\ensuremath{\xi}/N,$ the scaling behavior may be observed for the Hubbard model but not for the XXZ Heisenberg chain. We show how $\ensuremath{\xi}$ can be obtained from the scaling properties of the spin stiffness for small system sizes. This method can be applied to models having not an exact solution, illuminating their transport properties.

DOI
10.1103/physrevb.66.235108

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We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on a finite-size lattice with N sites. We compute the large-NNN behavior of the spin stiffness, obtaining the correlation length \ensuremathξ.\ensuremath{\xi}.\ensuremathξ. We use our results to discuss the scaling behavior of the metal-insulator transitions in one-dimensional systems, taking into account the mapping between the XXZ Heisenberg chain and the spinless fermion model, and known results for the Hubbard model. We study the scaling properties of both the Hubbard model and the XXZ Heisenberg chain by solving numerically the Bethe ansatz equations. We find that for some range of values of \ensuremathξ/N,\ensuremath{\xi}/N,\ensuremathξ/N, the scaling behavior may be observed for the Hubbard model but not for the XXZ Heisenberg chain. We show how \ensuremathξ\ensuremath{\xi}\ensuremathξ can be obtained from the scaling properties of the spin stiffness for small system sizes. This method can be applied to models having not an exact solution, illuminating their transport properties.
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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.
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The dc conductivity for the t-J model is exactly obtained at t=J and for the Hubbard model in one dimension. Our calculation is carried out for arbitrary band filling by use of the finite-size-scaling method based on the Bethe-ansatz solution under twisted boundary conditions. The effective transport mass defined as the inverse of the conductivity serves as a sensitive probe of the metal-insulator transition. The effective mass is extremely enhanced near half-filling both for the t-J model at t=J and the repulsive Hubbard model, which is a direct evidence of the metal-insulator transition due to the electron correlation. We observe that the large antiferromagnetic coupling J in the t-J model increases the effective mass. On the other hand, the conductivity in the attractive Hubbard model exhibits quite different properties from the repulsive models.