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Physical review. B, Condensed matter

Spirals and spin bags: A link between the weak- and the strong-coupling limits of the Hubbard model

David M. Frenkel, Werner R. Hanke

Published Oct 1, 1990• 41 citations
Publisher

Abstract

We reconsider some of the previous studies of the interactions between the holes and the spin background in the Hubbard model in the small-doping limit. We show that the hole-hole interaction due to the transverse magnetization fluctuations (i.e., spin waves) is finite at arbitrarily small hole density. It may be comparable to the coupling due to the longitudinal magnetization fluctuations (which are the essence of the ``spin-bag'' mechanism of Schrieffer, Wen, and Zhang). This suggests that the two couplings should be considered simultaneously when studying the phase diagram of the lightly doped quantum antiferromagnets at the values of U/t relevant to the high-${\mathit{T}}_{\mathit{c}}$ superconducting materials. We also show that in the large-(U/t) limit the Hubbard-model expressions for the coupling of holes reduce to those given by Shraiman and Siggia for the t-J model. This provides a link between the weak- and the strong-coupling theories of the doped quantum antiferromagnets, in the small-doping limit.

DOI
10.1103/physrevb.42.6711

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Physical review. B, Condensed matter•Feb 1, 1991•65 citations
We reconsider some of the previous studies of the interactions between the holes and the spin background in the Hubbard model in the small-doping limit. We show that the hole-hole interaction due to the transverse magnetization fluctuations (i.e., spin waves) is finite at arbitrarily small hole density. It may be comparable to the coupling due to the longitudinal magnetization fluctuations (which are the essence of the ``spin-bag'' mechanism of Schrieffer, Wen, and Zhang). This suggests that the two couplings should be considered simultaneously when studying the phase diagram of the lightly doped quantum antiferromagnets at the values of U/t relevant to the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconducting materials. We also show that in the large-(U/t) limit the Hubbard-model expressions for the coupling of holes reduce to those given by Shraiman and Siggia for the t-J model. This provides a link between the weak- and the strong-coupling theories of the doped quantum antiferromagnets, in the small-doping limit.
We study spin fluctuations in the spiral phase of the two-dimensional Hubbard model at low doping on the basis of the spin-particle-hole coherent-state path integral. In the strong correlation limit, we obtain an analytical expression of the spin-wave excitations over the entire Brillouin zone except in the vicinity of q=0{\bf q}=0q=0. We discuss the validity of the Hartree-Fock and random-phase approximations in the strong-coupling limit, and compare our results with previous numerical and analytical calculations. Although the spiral phase is unstable, as shown by a negative mean-field compressibility and the presence of imaginary spin-fluctuation modes, we expect the short-wavelength fluctuation modes (with real energies) to survive in the actual ground-state of the system.
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.
The $t\ensuremath{-}{J}_{z}$ model is the strongly anisotropic limit of the $t\ensuremath{-}J$ model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of t/J.t/J.t/J. Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic $t\ensuremath{-}J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.
The weak-coupling form of the effective interaction between two holes in the antiferromagnetic state of the two-dimensional Hubbard model near half-filling is derived. It is shown to predict a Cooper instability in the singlet ddd-wave channel, with binding temperature Tc>100{T}_{c}>100Tc​>100 K. Our results differ from those of the "spin-bag" theory of Schrieffer, Wen, and Zhang.
We calculate the spin correlation function and the magnetic longitudinal and transverse susceptibilities of a two-dimensional antiferromagnet doped with a small concentration of holes, in the $t\text{\ensuremath{-}}J$ model. We find that the motion of holes generates spin fluctuations that add to the quantum fluctuations, the spin correlations decaying with the inverse of the spin distance, while increasing with doping as the critical hole concentration, where the long-range order disappears, is approached. Moreover, the longitudinal susceptibility becomes finite in the presence of doping due to the strong damping effects induced by the hole motion, while the transverse susceptibility is renormalized by softening effects. Both the longitudinal and transverse susceptibilities increase with doping, the former more significantly than the latter. Our results imply that doping destroys the long-range order while local antiferromagnetic spin correlations persist. This is consistent with experiments on doped copper oxide superconductors.
A path-integral approach to the Hubbard model is developed for the whole range of the coupling strength U. At half filling, the strong-coupling results are readily reproduced within the simple Gaussian fluctuations. The low-lying spin wave is shown to be described by the nonlinear \ensuremath{\sigma} model. The effective coupling of the doped hole with the background fluctuations also agrees with that obtained from the t-J model in the small-doping limit. At finite doping, such a formalism may provide a starting point for investigating the short-range spin-liquid state.