X iv : c on d - m at / 9 50 80 05 v 1 1 A ug 1 99 5 The Spin Stiffness and the Transverse Susceptibility of the Half - filled Hubbard Model

The T = 0 spin stiffness ρs 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 . PACS numbers: 71.27.+a, 75.10.Jm, 75.40.Cx Typeset using REVTEX 1 Recent discovery of high-Tc superconductivity in the Cuprate materials has generated tremendous interest in the subject of strongly correlated electrons. In these systems, the phenomena of unusual metallic behavior, antiferromagnetism and superconductivity occur in a narrow parameter range. It is widely believed that these phenomena have a common microscopic origin. The Hubbard model is one of the simplest models to describe correlated electron behavior in a solid. It consists of a single band of electrons, with nearest neighbor hopping parameter t and an on-site Coulomb repulsion between opposite spin electrons of magnitude U . The model is best understood at half-filling, that is, when there is one electron per unit cell, where the system becomes an antiferromagnetic insulator. At large values of U this system is well described by the Heisenberg model. At small U one expects the spindensity-wave (SDW) mean field description to become accurate. The possibility of d-wave superconductivity, away from half-filling, has been widely explored [1]. Direct calculations of the superconducting transition temperatures in the Hubbard model are beyond present numerical capabilities. Thus, phenomenological approaches, where the Hubbard model is used to determine the parameters in a scaling theory of superconductivity, are appropriate. If spin-fluctuations are important to the mechanism of superconductivity in the Cuprates, and if the magnetic excitations in the doped Cuprates are related to those in the stoichiometric insulating phases, the magnetic excitations in the half-filled system are clearly important for understanding superconductivity in these materials. The magnetic ground state of the two-dimensional (2D) square lattice Hubbard model has been investigated by quantum Monte Carlo simulations [2–4] and Lanczos diagonalization [5]. These studies have mainly consisted of a finite size scaling analysis of the ground state properties of the model. They confirm the existence of long-range antiferromagnetic order at T = 0. To our knowledge, they have not been used to calculate the spin-stiffness constant or the spin-wave velocity for this model. Accurate numerical calculations of these quantities exist for the Heisenberg model. The Spin-density wave theory combined with the random phase approximation (RPA) has been used by Schrieffer et al. [6] to calculate the spin wave velocity for the Hubbard model. This calculation should become exact as U → 0. Somewhat 2 surprizingly, it was found that the result of this calculation is also accurate in the Heisenberg limit (U/t ≫ 1). In this Letter we first derive an expression for the spin stiffness constant of the Hubbard model by applying a slow twist in the ordering direction. We then introduce a one-parameter family of Hamiltonians which interpolate between the half-filled Hubbard and Ising models. This allows us to develop series expansions for the spin-stiffness. In addition, we develop series expansions for several thermodynamic parameters of this model, such as the uniform susceptibility and the sublattice magnetization. The spin wave velocity is calculated from the hydrodynamic relation [7] v s = ρs/χ⊥. At large U our results extrapolate smoothly to the Heisenberg values. They also show good agreement with the spin-density wave theory at small U . The variation of the magnetic susceptibility with the Hubbard parameter ratio U/t shows a relatively well defined crossover between a χ⊥ ∼ U behavior at large U and χ⊥ decreasing with increase in U at small U . This crossover between the strong coupling ( Heisenberg model behavior) and weak coupling SDW behavior occurs at U/t ≈ 4. The Hubbard model is defined by the lattice Hamiltonian H0 = −t ∑ ,σ (ciσ cjσ + c + jσ ciσ) + U ∑ i (ni↑ − 1 2 ) (ni↓ − 1 2 ) − μ ∑ i (ni↑ + ni↓) , (1) where ciσ and ciσ are the creation and annihilation operators for electrons with a z−component of spin σ at lattice site i, and ni,σ = ciσ ciσ. U is the on site repulsive interaction, μ the chemical potential, and t the nearest-neighbor hopping amplitude. If we rotate the ordering direction by an angle θ along a given direction such as y axis, then the spin stiffness constant ρs can be defined through the increase of the ground state energy: Eg(θ) = Eg(θ = 0) + 1 2 ρs θ 2 + O(θ). This rotation can be carried out by the following transformation applied to the fermion operators: