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Physical Review Letters

Finite-Temperature Magnetism in the Hubbard Model

Yolande H. Szczech, Michael A. Tusch, David E. Logan

Published Apr 3, 1995• 28 citations
Publisher

Abstract

A novel and physically transparent approach to the finite-temperature Hubbard model at half filling is proposed. It recovers not only the correct dimensionality dependence of antiferromagnetism but, in $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$, interpolates properly between weak and strong coupling interaction strengths for the N\'eel temperature, and enables extraction of related thermodynamic properties.

DOI
10.1103/physrevlett.74.2804

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A novel and physically transparent approach to the finite-temperature Hubbard model at half filling is proposed. It recovers not only the correct dimensionality dependence of antiferromagnetism but, in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3, interpolates properly between weak and strong coupling interaction strengths for the N'eel temperature, and enables extraction of related thermodynamic properties.
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.
The authors present here a comprehensive and numerically unbiased study for finite-temperature properties of the 3D fermionic Hubbard model. At half-filling, a complete temperature-interaction strength phase diagram of the model is established, including the highly accurate results of the N\'eel transition temperature and a multimessenger characterization of the metal-insulator crossover in the paramagnetic phase. The temperature dependence of various thermodynamic quantities and their underlying physics are thoroughly explored. The persistence of N\'eel AFM ordered phase to finite doping is further revealed.
[[abstract]]The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0 < q < pi) into the state with strong antiferromagnetic correlations (q = pi) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.
The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0<q<π) into the state with strong antiferromagnetic correlations (q=π) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.
The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0<q<π) into the state with strong antiferromagnetic correlations (q=π) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.
We examine the nature of the transition to the antiferromagnetically ordered state in the half-filled three-dimensional Hubbard model using the dual-fermion multiscale approach. Consistent with analytics, in the weak-coupling regime we find that spin-flip excitations across the Fermi surface are important and that the strong-coupling regime is described by Heisenberg physics. In the intermediate-interaction, strong-correlation regime we find aspects of both local and nonlocal correlations. We analyze the critical exponents of the transition in the strong-coupling regime and find them to be consistent with Heisenberg physics down to an interaction of U/t=10U/t=10U/t=10.