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Physical review. B, Condensed matter

Onsager-field study of the magnetism of the Hubbard model

Hiroyuki Kaga, M. Cyrot

Published Nov 1, 1998• 4 citations
Publisher

Abstract

We show that the Onsager-field theory for itinerant systems is also quite useful for arbitrary filling of the two-dimensional (2D) and 3D Hubbard models in the weak to intermediate coupling of $U,$ yielding various magnetic properties in agreement with the results of quantum Monte Carlo simulations and the two-particle self-consistent method of Vilk et al. Any small doping into the antiferromagnetic ground state of the half-filled 3D model introduces an incommensurate phase with a significant reduction due to fluctuations.

DOI
10.1103/physrevb.58.12267

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We show that the Onsager-field theory for itinerant systems is also quite useful for arbitrary filling of the two-dimensional (2D) and 3D Hubbard models in the weak to intermediate coupling of U,U,U, yielding various magnetic properties in agreement with the results of quantum Monte Carlo simulations and the two-particle self-consistent method of Vilk et al. Any small doping into the antiferromagnetic ground state of the half-filled 3D model introduces an incommensurate phase with a significant reduction due to fluctuations.
We study the magnetic properties of the three-dimensional (3D) Hubbard model at half filling in the two particle self-consistent formalism, previously developed for the 2D model. We focus on the N\'eel transition approached from the disordered side and on the paramagnetic phase. We find a very good quantitative agreement with dynamical mean-field (DMF) results for the isotropic 3D model. Calculations on finite size lattices also provide satisfactory comparisons with Monte Carlo results up to the intermediate coupling regime. We point out a qualitative difference between the isotropic 3D case, and the 2D or anisotropic 3D cases for the double occupation factor. Even for this local correlation function, 2D or anisotropic 3D cases are out of reach of DMF: this comes from the inability of DMF to account for antiferromagnetic fluctuations, which are crucial.
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.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
Magnetic properties of the two-dimensional Hubbard-model are inferred from results of Monte Carlo simulations. Lattice sizes up to 8\ifmmode\times\else\texttimes\fi{}8 and temperatures down to T=t20T=\frac{t}{20}T=20t​ (t=hoppingt=\mathrm{hopping}t=hopping) were studied. The half-filled system is found to exhibit antiferromagnetic long-range order for all values of the Coulomb repulsion UUU. The low-temperature magnetic properties are found to be well described by spin-wave theory with renormalized local moment and spin-wave velocity. Numerical evidence presented suggests that when doped the system loses the long-range order immediately away from half filling.
The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations.Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state-of-the-art quantum many-body methods.Upon cooling into its insulating antiferromagnetic ground state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate-temperature metallic regime, and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations.We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo methods) serving as a benchmark.By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes.Based on this analysis, we explain how quasiparticles can coexist with increasingly longrange antiferromagnetic correlations and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime.We also critically
The observed magnetic spin susceptibility of high-temperature superconductors such as ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​ increases when x increases from zero, i.e., as one dopes away from half-filling. Recent Monte Carlo simulations of A. Moreo [Phys. Rev. B 48, 3380 (1993)] suggest that this behavior can be reproduced by the two-dimensional Hubbard model only at large coupling, namely, U/t of order 10. Using longer runs, our Monte Carlo simulations show that the same behavior as for U/t=10 is obtained even in the intermediate coupling regime (U/t=4), as long as the temperature is low enough (T=t/6) that strong antiferromagnetic correlations are building up at half-filling. These results are consistent with the fact that in two dimensions, the generalized random phase approximation should fail in the parameter range where it predicts a magnetic phase transition.