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arXiv:cond-mat/0107291 · Journal of the Physical Society of Japan

Filling-Control Metal-Insulator Transition in the Hubbard Model Studied by the Operator Projection Method

Shigeki Onoda, Masatoshi Imada

Published Nov 1, 2001• 33 citations
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Abstract

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 $A(ω, k)$ show formation of two Hubbard bands and their antiferromagnetic shadows separated by a Mott gap in the plane of energy $ω$ and momentum $k$ 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 $(π, 0)$ and $(0, π)$ 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 $(π,0)$ and $(0,π)$ with strong momentum dependence. This reflects the suppression of the coherence near the MIT. Main structures in $A(ω, k)$ 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.

DOI
10.1143/jpsj.70.3398

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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\Tc\Tc cuprate superconductors.
A new non-perturbative framework for many-body correlated systems is formulated by extending the operator projection method (OPM). This method offers a systematic expansion which enables us to project into the low-energy structure after extracting the higher-energy hierarchy. This method also opens a way to systematically take into account the effects of collective excitations. The Mott-Hubbard metal-insulator transition in the Hubbard model is studied by means of this projection beyond the second order by taking into account magnetic and charge fluctuations in the presence of the high-energy Mott-Hubbard structure. At half filling, the Mott-Hubbard gap is correctly eproduced between the separated two bands. Near half filling, a strongly renormalized low-energy single-particle excitations coexisting with the Mott-Hubbard bands are shown to appear. Signifcance of momentum-dependent self-energy in the results is stressed.
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.
We study the transition from paramagnetic metal to paramagnetic insulator by finite temperature Quantum Monte-Carlo simulations for the 2D Hubbard model at half-filling. Working at the moderately high temperature T=0.33*t where the spin correlation length has dropped to 1.5 lattice spacings, and scanning the interaction strength U, we observe an unambiguous metal-insulator transition with onset at U_c=4t. In the metallic phase there are no indications of any `correlation narrowing' of the band width, nor for any decrease of spectral weight at the Fermi energy. The Mott-Hubbard gap opens gradually with increasing U and is accompanied by the formation of side bands near the Fermi energy. In the first stages of gap formation the gap opens at (pi,0), reminiscent of the `pseudogap' phenomenology in cuprate superconductors.
We present results of a systematic quantum Monte Carlo study for the single-band Hubbard model. Thereby we evaluated single-particle spectra (PES and IPES), two-particle spectra (spin and density correlation functions), and the dynamical correlation function of suitably defined diagnostic operators, all as a function of temperature and hole doping. The results allow us to identify different physical regimes. Near half-filling we find an anomalous ``Hubbard-I phase,'' where the band structure is, up to some minor modifications, consistent with the Hubbard-I predictions. At lower temperatures, where the spin response becomes sharp, additional dispersionless ``bands'' emerge due to the dressing of electrons/holes with spin excitations. We present a simple phenomenological fit that reproduces the band structure of the insulator quantitatively. The Fermi surface volume in the low-doping phase, as derived from the single-particle spectral function, is not consistent with the Luttinger theorem, but qualitatively in agreement with the predictions of the Hubbard-I approximation. The anomalous phase extends up to a hole concentration of $\ensuremath{\approx}15%,$ i.e., the underdoped region in the phase diagram of high-Tc{T}_{c}Tc​ superconductors. We also investigate the nature of the magnetic ordering transition in the single-particle spectra. We show that the transition to a spin-density wave-like band structure is not accomplished by the formation of any resolvable ``precursor bands,'' but rather by a (spectroscopically invisible) band of spin-3/2 quasiparticles. We discuss implications for the ``remnant Fermi surface'' in insulating cuprate compounds and the shadow bands in the doped materials.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
Filling-control metal-insulator transition on the two-dimensional Hubbard model is investigated by using the correlator projection method, which takes into account momentum dependence of the free energy beyond the dynamical mean-field theory. The phase diagram of metals and Mott insulators is analyzed. Lifshitz transitions occur simultaneously with metal-insulator transitions at large Coulomb repulsion. On the other hand, they are separated each other for lower Coulomb repulsion, where the phase sandwiched by the Lifshitz and metal-insulator transitions appears to show violation of the Luttinger sum rule. Through the metal-insulator transition, quasiparticles retain nonzero renormalization factor and finite quasi-particle weight in the both sides of the transition. This supports that the metal-insulator transition is caused not by the vanishing renormalization factor but by the relative shift of the Fermi level into the Mott gap away from the quasiparticle band, in sharp contrast with the original dynamical mean-field theory. Charge compressibility diverges at the critical end point of the first-order Lifshitz transition at finite temperatures. The origin of the divergence is ascribed to singular momentum dependence of the quasiparticle dispersion.