We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.
We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional
S=21 isotropic
XY model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$,
m=0,1,2,3,…, where ${\ensuremath{\omega}}_{0}=J$ for the
XY model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the
XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the
XY-like substances
Cs2Co
Cl4 and Pr
Cl3 and the
S=21 Ising-like substance CsCo
Cl3${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.
Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-\textonehalf{} Ising model with a transverse field $H=\ensuremath{-}\ensuremath{\Gamma}{\ensuremath{\Sigma}}_{i}{S}_{i}^{x}\ensuremath{-}J{\ensuremath{\Sigma}}_{i}{S}_{i}^{z}{S}_{i+1}^{z}$. Explicit results are obtained for $〈{{S}^{x}(q, \ensuremath{\omega}){S}^{x}(\ensuremath{-}q)}〉$, $〈{E(q, \ensuremath{\omega}), E(\ensuremath{-}q)}〉$, and their Fourier transforms in the limits $T=\ensuremath{\infty}$ and
T=0 and for the autocorrelation functions
〈Six,Six(t)〉 and
〈Ei,Ei(t)〉 in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.
The effects of Gaussian disorder on the dynamics of the one-dimensional spin-1/2 random transverse Ising model are studied in the high-temperature limit by the recursion method. Both spin autocorrelation functions and the corresponding spectral densities are calculated for three types of disordered cases. It is found that when the standard deviation ${\ensuremath{\sigma}}_{J}$ of the exchange coupling (or the standard deviation ${\ensuremath{\sigma}}_{B}$ of the random transverse field) is small, the dynamics of the system undergoes two crossovers in sequence as the mean value of the exchange coupling (or the random transverse field) varies: from a central-peak behavior to a collective-mode one, and then to the precession of free spins in an external transverse magnetic field. However, when ${\ensuremath{\sigma}}_{J}$ or ${\ensuremath{\sigma}}_{B}$ are large enough, there is no crossover, i.e., the dynamics of the system shows a central-peak behavior and a disordered behavior, respectively.
New exact results are presented on the long-time asymptotic expansions of the
T=0 autocorrelation functions, and on the leading singularities of their frequency-dependent Fourier transforms, for the one-dimensional
S=21 isotropic
XY model and the
S=21 transverse Ising model at the critical field. High-precision numerical calculations of the latter functions are also given, and experiments are proposed to observe the functional behaviors that are found.
We investigate the dynamics of the one-dimensional S=(1/2) isotropic XY model and transverse Ising model in the high-temperature limit by using the method of recurrence relations. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin
Sjx in these models, namely, the memory functions and the random forces. We find that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale.