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We consider a bilayer version of the extended $t$-$J$ model, with a view to computing the form of certain experimentally observable properties. Using the slave-boson decomposition, we show at the mean-field level that in the bilayer system the existence of in-plane $d$-wave singlet pairing excludes any interplane singlet order for reasonable values of the interplane superexchange parameter. Restricting the analysis to the regime of no interplane singlet pairing, we deduce parameter sets reproducing the Fermi surfaces of YBCO- and BSCCO-like bilayer systems. From these we calculate the form of the dynamic susceptibility $chi( {bf q}, omega )$ in both systems, and of the anomalies in frequency and linewidth of selected phonon modes in YBCO. We compare the results with experiment, and discuss the features which differ from the single-layer case.
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A relaxation-function theory for the dynamic spin susceptibility in the $t$--$J$ 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 temperaturs. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on La_{2-delta}Sr_delta)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 (omega/T) scaling behavior are calculated in a reasonable agreement with experimental and ED data.
Dynamic spin susceptibility is calculated for the t-J model in the paramagnetic phase by applying the memory function method in terms of the Hubbard operators. A self-consistent system of equations for the memory function is obtained within the mode coupling approximation. Both itinerant hole excitations and localized spin fluctuations give contributions to the memory function. Spin dynamics have a diffusive character in the hydrodynamic limit; spin-wave-like excitations are regained in the high-frequency region.
Drude weight of optical conductivity is calculated at zero temperature by exact diagonalization for the two-dimensional t-J model with the two-particle term, $W$. For the ordinary t-J model with $W$=0, the scaling of the Drude weight $D propto delta^2$ for small doping concentration $delta$ is obtained, which indicates anomalous dynamic exponent $z$=4 of the Mott transition. When $W$ is switched on, the dynamic exponent recovers its conventional value $z$=2. This corresponds to an incoherent-to-coherent transition associated with the switching of the two-particle transfer.
Stripe order where electrons self-organize into alternating periodic charge-rich and magnetically-ordered charge-poor parallel lines was proposed as a way of optimizing the kinetic energy of holes in a doped Mott insulator. Static stripes detected as extra peaks in diffraction patterns, appear in a number of oxide perovskites as well as some other systems. The more controversial dynamic stripes, which are not detectable by diffraction, may be universally present in copper oxide superconductors. Thus it is important to learn how to detect dynamic stripes as well as to understand their influence on electronic properties. This review article focuses on lattice vibrations (phonons) that might show signatures of the charge component of dynamic stripes. The first part of the article describes recent progress in learning about how the phonon signatures of different types of electronic charge fluctuations including stripes can be distinguished from purely structural instabilities and from each other. Then I will focus on the evidence for dynamic stripes in the phonon spectra of copper oxide superconductors.
We consider the implications of spin-phonon coupling within the slave-boson, mean-field treatment of the extended t-J model of a high-temperature superconductor. In bilayer cuprates such as YBaCuO, where the $CuO_2$ plane is buckled, this interaction is linear in $O$ displacement along the c-axis, and the coupling constant is found to be large. The formation of a spin singlet causes additional contributions to the phonon self-energy, and we calculate from these the superconductive phonon anomalies. The magnitude and sign of the frequency shift and linewidth broadening for various mode symmetries correspond well with Raman and infra-red light scattering experiments, and with neutron scattering studies. In the t-J model, spin singlet formation and superconductivity do not coincide in the low-doping regime, giving rise to spin-gap features and a variety of temperature scales in the spin response observed by NMR and neutron investigations. Phonon anomalies in underdoped YBaCuO compounds indeed show evidence of spin-gap phenomena with the same characteristic temperature, suggesting that the theory may offer the possibility of a unified understanding of the anomalies in magnetic and lattice properties. While the origin of the superconducting interaction is electronic, this spin-phonon coupling affords the possibility of a small isotope effect, and our estimate is in good agreement with recent site-selective $O$-substitution experiments.
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