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arXiv:cond-mat/9508005 · Europhysics Letters (EPL)

The Spin Stiffness and the Transverse Susceptibility of the Half-Filled Hubbard Model

Zhu-Pei Shi, Rajiv R. P. Singh

Published Aug 1, 1995• 8 citations
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Abstract

The T = 0 spin stiffness ρ s and the transverse susceptibility χ ⊥ of the square lattice half-filled Hubbard model are calculated as a function of the Hubbard parameter ratio U / t by series expansions around the Ising limit. We find that the calculated spin stiffness, transverse susceptibility, and sublattice magnetization for the Hubbard model smoothly approach the Heisenberg values for large U / t . The results are compared for different U / t with RPA and other numerical studies. The uniform susceptibility data indicate a crossover around U / t ≈ 4 between weak-coupling (spin density wave) behavior at small U and strong-coupling (Heisenberg) behavior at large U .

DOI
10.1209/0295-5075/31/4/006

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The T = 0 spin stiffness ρ s and the transverse susceptibility χ ⊥ of the square lattice half-filled Hubbard model are calculated as a function of the Hubbard parameter ratio U / t by series expansions around the Ising limit. We find that the calculated spin stiffness, transverse susceptibility, and sublattice magnetization for the Hubbard model smoothly approach the Heisenberg values for large U / t . The results are compared for different U / t with RPA and other numerical studies. The uniform susceptibility data indicate a crossover around U / t ≈ 4 between weak-coupling (spin density wave) behavior at small U and strong-coupling (Heisenberg) behavior at large U .
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.
The velocity of long-wavelength spin waves for the two-dimensional Hubbard model on a square lattice at half-filling is calculated using the variational Monte Carlo method with a Gutzwiller-type wave function. We show that the result from the Random Phase Approximation (RPA) for the spin wave velocity can also be found as the square root of the ratio of the helicity modulus and the perpendicular susceptibility as calculated in the Hartree-Fock approximation (HFA). Since the Gutzwiller wave function improves upon the HFA, our result for the spin wave velocity improves upon the RPA result. For the on-site repulsion U between 3 and 12 times the hopping integral t the spin wave velocity increases by 34 to 15% compared to the RPA result.
Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t-TN{\mathrm{T}}_{\mathrm{N}}TN​ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.
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.
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.
We analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random-phase approximation in a magnetically ordered state. For a sizable next-to-nearest-neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a Néel antiferromagnet in the electron-doped regime at and above half-filling and a planar circular spiral state in the hole-doped regime below half-filling. Upon electron doping, the Néel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero and then increases again very slowly upon increasing hole doping. At finite temperatures, the Néel-to-spiral transition is shifted into the hole-doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron-doped Hubbard model remains Néel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole-doped regime, giving rise to a quantum disordered state with a spin gap.