We present an exact solution for the spin-spin correlation function of the dynamical spherical model (the spherical-model equilibrium state together with the classical-Heisenberg-model equations of motion) in one dimension at $T=0$. The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}=1$ in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.
We present an exact solution for the spin-spin correlation function of the dynamical spherical model (the spherical-model equilibrium state together with the classical-Heisenberg-model equations of motion) in one dimension at T=0. The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint S\ensuremath→i\ifmmode⋅\else\textperiodcentered\fiS\ensuremath→i=1 in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
The usual spin-wave theory, when applied to the calculation of rotationally invariant averages in the Heisenberg model, permits exact calculation of static and dynamical correlation functions in one dimension. The results are also believed to be exact in two dimensions, although this has not been show rigorously here. The one-dimensional ferromagnet in the hydrodynamic regime and the two-dimensional ferromagnet at all wave vectors do not satisfy the dynamical scaling hypothesis. There is no characteristic exponent in these cases. The present results contradict the prediction of the mode-coupling theory.
The properties of the time-dependent spin correlation functions are investigated for the spherical model of the classical magnetic system. Equations for the correlation functions are derived by a diagrammatic analysis. The lowest-order approximation is for T>Tc identical to the equation derived by Resibois and De Leener for the Weiss limit of the Heisenberg model. Attention has been given to the hydrodynamic regime along the line of spontaneous magnetization from T=0 up to T=Tc, where spin-wave frequencies and damping rates as well as the "diffusive" behavior have been calculated to lowest order in the wave number. The dynamical scaling problem is examined within this diagrammatic analysis. The scaling properties and the equations determining the homogeneous correlation function are established explicitly.
In this work we present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighbourhood of a critical point keeping, at the same time, fuli information on the non universal properties of the model. A simple approximation to our formally exact equations is studied for the spin-S Heisenberg model where explicit results for critical exponents, critical temperature and coexistence curve are obtained. In three dimensions the results are in good agreement with available quantum Monte Carlo simulations and series expansions.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.