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Journal of the Physical Society of Japan

Charge Gap, Charge Susceptibility and Spin Correlationin the Hubbard Model on a Square Lattice

Nobuo Furukawa, Masatoshi Imada

Published Nov 15, 1991• 63 citations
Publisher

Abstract

Quantum Monte Carlo study of the Hubbard model on a square lattice in the ground state is reported. For the on-site interaction U =4 scaled by the transfer, the charge excitation gap at the half-filling is estimated to be Δ c =0.58±0.08. Near half-filling, the charge compressibility κ follows the form κ∝δ -1 for the doping concentration δ, indicating a divergent behavior as the system approaches half-filling, while κ=0 at the half-filling. The incommensurate spin correlation length ξ scales as ξ∝1/\(\sqrt{\delta}\).

DOI
10.1143/jpsj.60.3604

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Quantum Monte Carlo study of the Hubbard model on a square lattice in the ground state is reported. For the on-site interaction U =4 scaled by the transfer, the charge excitation gap at the half-filling is estimated to be Δ c =0.58±0.08. Near half-filling, the charge compressibility κ follows the form κ∝δ -1 for the doping concentration δ, indicating a divergent behavior as the system approaches half-filling, while κ=0 at the half-filling. The incommensurate spin correlation length ξ scales as ξ∝1/(\sqrt{\delta}).
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.
Using the quantum Monte Carlo simulation technique for the lattice fermion systems, we study the charge susceptibility of the two-dimensional Hubbard near the half-filled band. Lattice sizes up to 6×6 and inverse temperatures up to 6 t ( t : hopping parameter) are studied. We find that the behavior of the charge susceptibility is strongly dependent on the electron concentration at low temperature: the charge susceptibility is suppressed at the half-filled band, while it is considerably enhanced in non-half-filled band cases by the electron correlation. The above behavior of the charge susceptibility is quite similar to the recent Bethe-Ansatz result for the one-dimensional Hubbard model.
The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.
We investigate ground state properties of the half-filled staggered-flux Hubbard model on a square lattice. Energy gaps to charge and spin excitations and magnetic as well as dimer orders are calculated as a function of interaction strength U/t by means of a constrained-path quantum Monte Carlo method. It is found that the system is a semimetal at U/t≲5.6 and a Mott insulator, with long-range antiferromagnetic order, at U/t≳6.6. In the range 5.6≲U/t≲6.6, the ground state is a correlated insulator where both magnetic and dimer orders are absent. Furthermore, spin excitation in the intermediate phase appears to be gapless, and the measured spin-spin correlation function exhibits power-law decaying behavior. The data suggest that the nonmagnetic ground state is a possible candidate for the putative algebraic spin liquid.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
The one-dimensional Hubbard model is studied at and close to half-filling using a generalization of Handscomb's quantum Monte Carlo method. Results for spin-correlation functions and susceptibilities are presented for systems of up to 128 sites. The spin-correlation function at low temperature is well described by a recently introduced formula relating the correlation function of a finite periodic system to the corresponding T=0 correlation function of the infinite system. For the T\ensuremath{\rightarrow}0 divergence of the q=2kF{\mathit{k}}_{\mathit{F}}kF​ spin susceptibility, our results are consistent with the form \ensuremath{\chi}(2kF{\mathit{k}}_{\mathit{F}}kF​)\ensuremath{\sim}\ensuremath{\Vert}ln(T)${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\sigma}}}$${\mathit{T}}_{\mathrm{\ensuremath{\rho}}}^{\mathit{K}}$-1, with \ensuremath{\sigma}=1/2.