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Journal of the Physical Society of Japan

Charge Susceptibility of the Two-Dimensional Hubbard Model near the Half-Filled Band -A Monte Carlo Study-

Hiromi Otsuka

Published Aug 15, 1990• 10 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.2916

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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.
Using quantum Monte Carlo techniques we study the magnetic susceptibility of the Hubbard model on a two-dimensional square lattice for values of the coupling U/t ranging from 4 to 20, and as a function of temperature and doping. For U/t=4 the magnetic susceptibility has a maximum at half filling and decreases with doping, in qualitative agreement with recent random-phase-approximation calculations. However, for U/t=10 the susceptibility increases with hole doping near half filling resembling the experimental results obtained for ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. At low electronic density the temperature dependence of the susceptibility is compatible with that of a Fermi liquid.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
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 present results of Monte Carlo simulations of the three-dimensional half-filled Hubbard model on simple-cubic 43{4}^{3}43 lattices and some 63{6}^{3}63 lattices with periodic boundary conditions, as well as of small clusters embedded in an effective medium. Magnetic properties were studied for values of the Hubbard repulsion U ranging from 0 to the bandwidth W=12t (t is the hopping matrix element). Our results show that antiferromagnetism is enhanced up to U\ensuremath{\sim}(5/6)W and then suppressed. The q=0 magnetic susceptibility is enhanced by U, although much less than predicted by the random-phase approximation. The transition temperature to an antiferromagnetic state is found to have a broad maximum around U\ensuremath{\sim}(5/6)W, with a maximum value of approximately W/18. The implications of our results for superconductivity in the attractive Hubbard model are also discussed.
The magnetic susceptibility of the two-dimensional repulsive Hubbard model with nearest-neighbor hopping is investigated using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. At half-filling, the calculated zero-frequency susceptibility and the square of the site spin reproduce adequately the results of Monte Carlo simulations. Also, in agreement with numerical simulations, no evidence of ferromagnetic correlations was found in the considered range of electron concentrations $0.8\ensuremath{\lesssim}\overline{n}\ensuremath{\lesssim}1.2$ for the repulsion parameters $8\ensuremath{\mid}t\ensuremath{\mid}\ensuremath{\leqslant}U\ensuremath{\leqslant}16\ensuremath{\mid}t\ensuremath{\mid}$. However, for larger $U∕\ensuremath{\mid}t\ensuremath{\mid}$ and $\ensuremath{\mid}1\ensuremath{-}\overline{n}\ensuremath{\mid}\ensuremath{\approx}0.2$, the nearest-neighbor correlations become ferromagnetic. For $\overline{n}\ensuremath{\lesssim}0.94$ and $\overline{n}\ensuremath{\gtrsim}1.06$, the imaginary part of the real-frequency susceptibility becomes incommensurate for small frequencies. The incommensurability parameter grows with departure from half-filling and decreases with increasing frequency. This behavior of the susceptibility can explain the observed low-frequency incommensurate response observed in normal-state lanthanum cuprates.
We use a theory developed in a previous paper [T. Kopp and F. Mila, Phys. Rev. B 43, 12 980 (1991)] to interpret recent Monte Carlo data relative to the doping dependence of the zero-temperature susceptibility of the Hubbard model on a square lattice. According to this theory, the susceptibility does not decrease, like the density of states, but increases upon doping, the essential reason being that the hopping processes that contribute to the susceptibility are suppressed close to half-filling because the wave functions on different sublattices have a vanishing overlap in that limit. We show that, for U/t=4, this theory is in very good agreement with recent Monte Carlo data.