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arXiv:cond-mat/0405349 · Physical review. B, Condensed matter

Finite-size scaling for Mott metal-insulator transition on a half filled nonpartite lattice

J. X. Wang, Sabre Kais

Published Aug 14, 2002• 8 citations
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Abstract

We combine the finite-size scaling method with a multistage real-space renormalization-group procedure to examine the Mott metal-insulator transition (MIT) on a nonpartite lattice. Based on the Hubbard model, we have found that there exists a critical point $U/t=12.5$ for the MIT with the correlation length exponent $\ensuremath{\nu}=1.$ At the critical point, the charge gap scales with the system size as ${▵}_{g}\ensuremath{\sim}{1/L}^{0.91}.$

DOI
10.1103/physrevb.66.081101

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We combine the finite-size scaling method with a multistage real-space renormalization-group procedure to examine the Mott metal-insulator transition (MIT) on a nonpartite lattice. Based on the Hubbard model, we have found that there exists a critical point U/t=12.5U/t=12.5U/t=12.5 for the MIT with the correlation length exponent \ensuremathν=1.\ensuremath{\nu}=1.\ensuremathν=1. At the critical point, the charge gap scales with the system size as ▵g\ensuremath∼1/L0.91.{▵}_{g}\ensuremath{\sim}{1/L}^{0.91}.▵g​\ensuremath∼1/L0.91.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions UUU and VVV at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed UUU, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value Vc{V}_{c}Vc​ and the coefficients ${c}_{\ensuremath{\Delta}}$ and cK{c}_{K}cK​ are functions of UUU. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite UUU case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when UUU is varied for a fixed VVV.
In this paper, we have investigated the metal-insulator transition (MIT) in a two-dimensional half-filled extended Hubbard model on an isotropic triangular lattice with a real space block renormalization group technique. It has been found that the MIT can be driven nontrivially by either the on-site interaction U or the nearest-neighbor one V, but with different critical exponents. Depending upon the values of V, the system could have one, two or three MIT critical points. Moreover, for the metallic regime, we have also studied the competition effect from the spin density wave and charge density wave phases by using a mean-field theory based upon Hartree-Fock approximations. Finally, the single-site entanglement is also calculated and its first derivative with respect to U shows a jump along the critical line of the MIT.
Correlated electrons in a binary alloy ${A}_{x}{B}_{1\ensuremath{-}x}$ are investigated within the Hubbard model and dynamical mean-field theory (DMFT). The random energies ${\ensuremath{\epsilon}}_{i}$ have a bimodal probability distribution and an energy separation $\ensuremath{\Delta}.$ We solve the DMFT equations by the numerical renormalization-group method at zero temperature and calculate the spectral density as a function of disorder strength $\ensuremath{\Delta}$ and interaction U at different fillings. For filling factors $\ensuremath{\nu}=x$ or 1+x1+x1+x the lower or upper alloy subband is half filled and the system becomes a Mott insulator at strong interactions, with a correlation gap at the Fermi level. At the metal-insulator transition hysteresis is observed. We also analyze the effective theory in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty}$ limit and find good agreement between analytical and numerical results for the critical interaction Uc{U}_{c}Uc​ at which the metal-insulator transition occurs.
We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder Δc≈U/2Δ_c \approx U/2Δc​≈U/2. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.
We report a systematic study of the transition from a band insulator (BI) to a Mott insulator (MI) in a one-dimensional Hubbard model at half-filling with an on-site Coulomb interaction U and an alternating periodic site potential V. We employ both the zero-temperature density matrix renormalization group (DMRG) method to determine the gap and critical behavior of the system and the finite-temperature transfer matrix renormalization group method to evaluate the thermodynamic properties. We find two critical points at U=Uc{U=U}_{c}U=Uc​ and U=Us{U=U}_{s}U=Us​ that separate the BI and MI phases for a given V. A charge-neutral spin-singlet exciton band develops in the BI phase (U<Uc)(U<{U}_{c})(U<Uc​) and drops below the band gap when U exceeds a special point Ue.{U}_{e}.Ue​. The exciton gap closes at the first critical point Uc{U}_{c}Uc​ while the charge and spin gaps persist and coincide between Uc<U<Us{U}_{c}<U<{U}_{s}Uc​<U<Us​ where the system is dimerized. Both the charge and spin gaps collapse at U=Us{U=U}_{s}U=Us​ when the transition to the MI phase occurs. In the MI phase (U>Us)(U>{U}_{s})(U>Us​) the charge gap increases almost linearly with U while the spin gap remains zero. These findings clarify earlier published results on the same model, and offer insights into several important issues regarding an appropriate scaling analysis of DMRG data and a full physical picture of the delicate nature of the phase transitions driven by electron correlation. The present work provides a comprehensive understanding for the critical behavior and phase diagram for the transition from BI to MI in one-dimensional correlated electron systems with a periodic alternating site potential.