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arXiv:cond-mat/9605047 · Journal of the Physical Society of Japan

Critical Exponents of the Metal-Insulator Transition in the Two-Dimensional Hubbard Model

Nobuo Furukawa, Fakher F. Assaad, Masatoshi Imada

Published Jul 15, 1996• 3 citations
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Abstract

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 $κ\propto |μ- μ_c|^{-0.58\pm0.08}$ where $μ_c$ is the critical chemical potential. In the insulating phase, the localization length follows $ξ_l \propto |μ- μ_c|^{-ν_l}$ with $ν_l = 0.26 \pm 0.05$. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent $ν_κ= 0.21 \pm 0.04$. Our results show that the exponents $ν_κ$ and $ν_l$ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent $ν= 1/4$ and dynamical exponent $z = 4$. 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 $ν= 1/2$ and $z = 2$.

DOI
10.1143/jpsj.65.2339

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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.
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Thermodynamic and dynamical properties of filling-control metal-insulator transition (MIT) in the Hubbard model are studied by the operator projection method, especially in two dimensions. This is a non-perturbative analytic approach to many-body systems. The present theory incorporates the Mott-Hubbard, Brinkman-Rice and Slater pictures of the MIT into a unified framework, together with reproducing low-energy narrow band arising from spin-charge fluctuations. At half filling, single-particle spectra A(ω,k)A(ω, k)A(ω,k) show formation of two Hubbard bands and their antiferromagnetic shadows separated by a Mott gap in the plane of energy ωωω and momentum kkk with lowering temperatures. These four bands produce splitting to two low-energy narrow bands and two SDW-like bands in the dispersion. Near half filling, the low-energy narrow band persists at low temperatures. This narrow band has a particularly weak dispersion and large weights around (π,0)(π, 0)(π,0) and (0,π)(0, π)(0,π) momenta. The velocity of these low-energy excitations is shown to vanish towards the MIT, indicating the mass divergence as in the Brinkman-Rice picture, but most prominently around (π,0)(π,0)(π,0) and (0,π)(0,π)(0,π) with strong momentum dependence. This reflects the suppression of the coherence near the MIT. Main structures in A(ω,k)A(ω, k)A(ω,k) show remarkable similarities to quantum Monte-Carlo results in two dimensions as well as in the one-dimensional Hubbard model. The charge compressibility appears to diverge with decreasing doping concentration in both one and two dimensions in agreement with the exact and quantum Monte-Carlo results. We also discuss implications of the flat dispersion formed near the Fermi level to the observations in high-$\Tc$ cuprate superconductors.
Transitions between the Mott insulator and metals in clean systems are analyzed with the scaling theory in terms of quantum critical phenomena. In the case of generic transitions controlled by the filling, the scaling theory is established from analyses of the Drude weight and the compressibility based on hyperscaling. In the transition to the Mott insulator, a new universality class with the correlation length exponent ν<1/2 and the dynamical exponent z >2 is derived, which is in contrast to the transition to the band insulator characterized by ν=1/2 and z =2. The unusual exponents lead to various anomalous characters of metals near the Mott insulator such as strong suppression of the degeneracy temperature, and nonlinear dependence of the Drude weight on the doping concentration. Remarkable properties are also found in the specific heat, the compressibility and spin correlations. A mechanism of high temperature superconductivity is discussed in terms of the release from the suppressed coherence.
The Hubbard model on a square lattice in the ground state is investigated. Various aspects of Mott transition at half-filling are clarified from the quantum Monte Carlo study. Critical exponents of the transition for charge and spin correlations are estimated. For the doping concentration δ, the charge susceptibility is proportional to δ -1 indicating the divergence of charge mass for δ→0, while the spin susceptibility is always finite and not singular near δ=0. Several theoretical consequences of the above results are discussed. Incommensurate peak amplitude of spin correlation scales as (δ-δ c ) -1 with δ c <0.01 in disagreement with RPA results ln (δ-δ c ). The antiferromagnetic order at the half-filling also shows strong correlation character and is not sensitive to the shape of the fermi surface, while the incommensurate peak observed away from the half-filling is sensitively suppressed by the loss of partial nesting consistently with the weak coupling picture. Comparisons with experimental indications of high- T c oxides are made.
For doped two-dimensional Mott insulators in their normal state, the challenge is to understand the evolution from a conventional metal at high doping to a strongly correlated metal near the Mott insulator at zero doping. To this end, we solve the cellular dynamical mean-field equations for the two-dimensional Hubbard model using a plaquette as the reference quantum impurity model and continuous-time quantum Monte Carlo method as impurity solver. The normal-state phase diagram as a function of interaction strength UUU, temperature TTT, and filling nnn shows that, upon increasing nnn toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T === 0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.