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arXiv:cond-mat/9710139 · Physical review. B, Condensed matter

π excitation of the t−J model

Eugene Demler, Hiroshi Kohno, Shoucheng Zhang

Published Sep 1, 1998• 53 citations
arXivHTMLPDF

Abstract

In this paper, we present analytical and numerical calculations of the \ensuremath{\pi} resonance in the $t\ensuremath{-}J$ model. We show in detail how the $\ensuremath{\pi}$ resonance in the particle-particle channel couples to the dynamical spin correlation function in a superconducting state. The contribution of the \ensuremath{\pi} resonance to the spin excitation spectrum can be estimated from general model-independent sum rules, and it agrees with our detailed calculations. The results are in overall agreement with the exact diagonalization studies of the $t\ensuremath{-}J$ model. Earlier calculations predicted the correct doping dependence of the neutron resonance peak in the ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ superconductor, and in this paper detailed energy and momentum dependence of the spin correlation function is presented. The microscopic equations of motion obtained within current formalism agree with that of the SO(5) nonlinear \ensuremath{\sigma} model, where the \ensuremath{\pi} resonance is interpreted as a pseudo-Goldstone mode of the spontaneous SO(5) symmetry breaking.

DOI
10.1103/physrevb.58.5719

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In this paper, we present analytical and numerical calculations of the \ensuremath{\pi} resonance in the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. We show in detail how the \ensuremathπ\ensuremath{\pi}\ensuremathπ resonance in the particle-particle channel couples to the dynamical spin correlation function in a superconducting state. The contribution of the \ensuremath{\pi} resonance to the spin excitation spectrum can be estimated from general model-independent sum rules, and it agrees with our detailed calculations. The results are in overall agreement with the exact diagonalization studies of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. Earlier calculations predicted the correct doping dependence of the neutron resonance peak in the YBa2Cu3O6+x{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}YBa2​Cu3​O6+x​ superconductor, and in this paper detailed energy and momentum dependence of the spin correlation function is presented. The microscopic equations of motion obtained within current formalism agree with that of the SO(5) nonlinear \ensuremath{\sigma} model, where the \ensuremath{\pi} resonance is interpreted as a pseudo-Goldstone mode of the spontaneous SO(5) symmetry breaking.
In this work, we present numerical results which support SO(5) symmetry as a concept unifying superconductivity and antiferromagnetism in the high-temperature superconductors. Using exact cluster diagonalization, we verify the recently proposed SO(5) multiplet structures for a widely used microscopic model, the t-J model. Our results show that the d-wave superconducting ground states away from half-filling are obtained from the higher spin states at half-filling through SO(5) rotations, that the dominant low energy resonance in the dynamical spin correlation function corresponds to a `pi-resonance' in the entire low doping region, and that the SO(5) symmetry is robust against inclusion of longer ranged Coulomb repulsion and next-nearest neighbor hopping.
A relaxation-function theory for the dynamic spin susceptibility in the $t\text{\ensuremath{-}}J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperatures. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on ${\text{La}}_{2\ensuremath{-}\ensuremath{\delta}}{\text{Sr}}_{\ensuremath{\delta}}{\text{CuO}}_{4}$. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its $\ensuremath{\omega}/T$ scaling behavior are calculated in a reasonable agreement with experimental and ED data.
We present numerical evidence for the approximate SO(5) symmetry of the Hubbard model on a 10-site cluster. Various dynamic correlation functions involving the $\ensuremath{\pi}$ operators, the generators of the SO(5) algebra, are studied using exact diagonalization, and are found to possess sharp collective peaks. Our numerical results also lend support to the interpretation of the recent resonant neutron scattering peaks in the YBCO superconductors in terms of the Goldstone modes of the spontaneously broken SO(5) symmetry. Perturbations such as longer-ranged hoppings and interactions do not suppress the $\ensuremath{\pi}$ resonance in interesting parameter regimes.
We present a systematic study of spin dynamics in a superconducting ground state, which itself is a doped Mott insulator and can correctly reduce to an antiferromagnetic (AF) state at half-filling with an AF long-range order (AFLRO). Such a doped Mott insulator is described by a mean-field theory based on the phase string formulation of the $t\text{\ensuremath{-}}J$ model. We show that the well-known spin wave excitation in the AFLRO state at half-filling evolves into a resonancelike peak at a finite energy in the superconducting state, which is located around the AF wave vectors. The width of such a resonancelike peak in momentum space decides a spin correlation length scale which is inversely proportional to the square root of doping concentration, while the energy of the resonancelike peak scales linearly with the doping concentration at low doping. These properties are consistent with experimental observations in the high-Tc{T}_{c}Tc​ cuprates. An important prediction of the theory is that, while the total spin sum rule is satisfied at different doping concentrations, the weight of the resonancelike peak does not vanish, but is continuously saturated to the weight of the AFLRO at zero-doping limit. Besides the low-energy resonancelike peak, we also show that the high-energy excitations still track the spin wave dispersion in momentum space, contributing to a significant portion of the total spin sum rule. The fluctuational effect beyond the mean-field theory is also examined, which is related to the broadening of the resonancelike peak in energy space. In particular, we discuss the incommensurability of the spin dynamics by pointing out that its visibility is strongly tied to the low-energy fluctuations below the resonancelike peak. We finally investigate the interlayer coupling effect on the spin dynamics as a function of doping, by considering a bilayer system.
In this series of papers we present a detailed study of the particle-particle collective excitations of the Hubbard model, and their contribution to the density and spin excitation spectrum. In the first paper, we shall investigate the singlet particle-particle pair with momentum (π, π), the η particle, of the negative-U Hubbard model. We review three previously obtained theorems about the η particle and develop a self-consistent linear response theory which takes into account its contribution to the density excitation spectrum in the superconducting state. We show that this self-consistent theory agrees with the exact theorems as well as the results of numerical Monte Carlo simulations.
We present a new method for studying the low energy excitations in finite cluster systems and apply it to clarify the nature of the low energy spin excitations in the moderately doped t-J model. We find that the character of the low energy spin excitations changes within the Brillouin zone, with the dominant excitation for momentum transfer (pi,pi) being a spin wave-like collective mode. Spin excitations with momentum transfer different from (pi,pi) have particle-hole character. We give an explicit construction of a low energy collective spin mode in a system with `almost broken symmetry' and demonstrate its validity by explicit numerical check.