The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈
Siz(t)
Siz〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈
Six(t)
Six〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.
In studies of dynamic correlation functions the focus is, in general, on their long-time asymptotic behavior and the singularity structure of the associated spectral densities. It turns out that the analysis of the same dynamic correlation functions from a quite different perspective is equally useful and revealing. The focus of our study is on the properties of spectral densities at high frequencies, specifically their decay law, expressible asΦ0■ω■ ∼exp■ − ω2■λ■, in terms of a characteristic exponent λ. That decay law governs the growth rate of the sequence of recurrents which determine the relaxation function in the continued-fraction representation. The value of λ contains valuable information on the underlying dynamical processes taking place in the system, which, in some sense, is complementary to that inferred from the singularity structure of spectral densities. A detailed study of the dynamics of various quantum and classical spin models has prompted us to introduce the concept of ‘‘universality class’’ for a classification of dynamical behavior on the basis of the characteristic exponent λ. For the equivalent-neighbor XYZ model1 we are able to demonstrate four different prototype universality classes: λ=0 (compact support), λ=1 (Gaussian decay), λ=2 (exponential decay), λ=3 (stretched exponential decay), which can be interpreted in terms of basic notions of classical dynamics. Finally, we present the exact T=∞ dynamic correlation functions for a two-sublattice Heisenberg antiferromagnet with uniform intersublattice interaction and zero intrasublattice interaction.
The dynamics of the classical XXZ model with uniform interaction is nonlinear for N≥2 spins and nonintegrable for N≥3. However, the nonlinearities disappear in the thermodynamic limit N→∞, and the spin autocorrelation functions can be determined exactly for infinite temperature. The function 〈Szi(t)Szi〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈Sxi(t)Sxi〉 decays algebraically to zero or like a Gaussian, depending on the type (easy axis or easy plane) and amount of uniaxial anisotropy.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the
S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the
S=1 bilinear-biquadratic chain and the
S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the
S=1/2 Heisenberg chain with a non-uniform magnetic field, the
S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
We introduce an integrable spin ladder model and study its exact solution, correlation functions, and entanglement properties. The model supports two particle types (corresponding to the even and odd sublattices), such that the scattering phases are constants: Particles of the same type scatter as free fermions, whereas the interparticle phase shift is a constant tuned by an interaction parameter. Therefore, the spin ladder bears similarities with anyonic models. We present exact results for the spectrum and correlation functions, and we study the sublattice entanglement by numerical means.