We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelation function (ACF) is more subtle. Unlike the classical Heisenberg trimer where the ACF approaches a unique non-zero limit for long times, for the quantum trimer the ACF is periodic in time. We present exact values of the time average over one period of the quantum trimer for s⩽7 and for infinite temperature. These averages differ from the long-time limit, (940)ln3+(730), of the corresponding classical trimer by terms of order 1/s2. However, upon applying the Levin u-sequence acceleration method to our quantum results we can reproduce the classical value to six significant figures.
We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelation function (ACF) is more subtle. Unlike the classical Heisenberg trimer where the ACF approaches a unique non-zero limit for long times, for the quantum trimer the ACF is periodic in time. We present exact values of the time average over one period of the quantum trimer for s⩽7 and for infinite temperature. These averages differ from the long-time limit, (940)ln3+(730), of the corresponding classical trimer by terms of order 1/s2. However, upon applying the Levin u-sequence acceleration method to our quantum results we can reproduce the classical value to six significant figures.
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.
Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where six, siy, and siz are components of the unit vector si, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions Jijx=Jijy=J and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.
Starting from the von Neumann equation for the spin density matrix of a Heisenberg system, we analyze the perturbation expansion of the spin autocorrelation function by the diagrammatic technique previously applied to quantum gases. We demonstrate a number of theorems which allow us to express this perturbation series in terms of renormalized graphs only; we then derive a kinetic equation for the autocorrelation function. The main feature of this equation is that the kernel, which is highly nonlinear in the autocorrelation function itself, tends to zero in the limit of long times. The results, which are exact in the high-temperature region and in the Weiss limit (number of neighbors $Z\ensuremath{\rightarrow}\ensuremath{\infty}$), allow us to consider the behavior of the autocorrelation function for times both short and long. This model is a typical example of a system with a discrete unperturbed spectrum showing an irreversible behavior.
Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function G(t) for a linear chain of N spins (S=21) interacting by nearest-neighbor Heisenberg exchange for N=5,7,9and11 by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for M22 to M30 of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to M10. G(t) does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for N=11 to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at t=0.
A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.
The two-time spin-pair autocorrelation function $\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{0},t)$ and the Fourier-space transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the two-time spin-pair correlation function $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$, are expressed in terms of the "friction function" occurring in the generalized Langevin equation for the spin operator ${s}_{i}^{\ensuremath{\sim}}$ and its Fourier-space transform ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{z}$, respectively. The friction function is determined in the form of a product of a Gaussian distribution function and a power series as a function of time t for the isotropic Heisenberg magnet and the XY magnet of spin \textonehalf{} at infinite temperature. By truncating the power series to the exactly known term, satisfactory results are obtained for $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the isotropic Heisenberg magnet of the linear chain and sc lattice. Satisfactory results for $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$ are achieved by taking an inverse Fourier-space transform of the thus determined $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$.