The longitudinal and transverse space and time dependent spin correlation functions for the one-dimensional Heisenberg-Ising model are derived asymptotically in the weak-coupling-low-frequency-long-wavelength limit within the framework of ordinary many-body theory (field theory). By means of the Jordan-Wigner transformation and using methods developed previously for the Tomonaga model (1950), the zero-temperature correlation functions is expressed as functional integrals. The associated one-body problem is explicitly soluble in the asymptotic regions and the correlation functions assume the form of Gaussian integrals which are evaluated analytically in the weak-coupling-low-frequency-long-wavelength limit. Apart from a linear correction to the transverse exchange constant agreement with Luther and Peschel (1975) is obtained.
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.
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 study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.
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.