Spin Van Der Waals Model Research Articles

Exact time evolution of the total spin components and its application to the classical spin van der Waals model in transverse magnetic fields

The exact time evolution expressions for the total spin components are obtained by solving the equations of motion for the classical spin van der Waals model in transverse magnetic fields. They are given in terms of Jacobi elliptic functions and are examined in detail. They are also used to obtain the time-dependent total spin correlation functions by adopting equilibrium configurations acquired via a heat bath algorithm. The time-dependent total spin correlation functions are interpreted by utilizing the motion of the tip of the total spin vector on the surface of a Bloch sphere.

Dynamic finite-size effect in an infinite-range classical antiferromagnetic spin model

As a further test of the dynamic scaling method proposed by us previously, dynamic finite-size effect in an infinite-range classical antiferromagnetic spin model, which is an anisotropic version of two-sublattice Lieb-Mattis model, is studied. Application of our dynamic scaling method to the model yields time scaling exponents similar to those of the infinite-range ferromagnetic classical spin van der Waals model. With the hereby obtained time scaling variables, we can successfully collapse the time-dependent correlation functions of the model into universal curves.

Dynamic finite-size effect in the classical spin van der Waals model.

A phenomenological dynamic scaling method is proposed and utilized to interpret finite-size effects in the time-dependent correlation function for the classical spin van der Waals model. We claim that the equations of motion for the classical spin van der Waals model are well defined only within a certain characteristic time. The order of magnitude for the size-dependent maximum time interval of the time-dependent correlation function, analytically obtained in the large N limit via the method of Laplace by Dekeyser and Lee, is found to be related to our characteristic time. As a by-product of this work, time scaling variables are found with which we can completely collapse the time-dependent correlation functions of the classical spin van der Waals model into one curve. \textcopyright{} 1996 The American Physical Society.

Wolff algorithm and anisotropic continuous-spin models: An application to the spin-van der Waals model.

The problem of applying Wolff's cluster algorithm to anisotropic classical spin models is resolved by modifying a part of the Wolff algorithm. To test the effectiveness of our modified algorithm, the spin--van der Waals model is investigated in detail. Our estimate of the dynamical exponent of the model is z=0.19\ifmmode\pm\else\textpm\fi{}0.04.

Nonequilibrium statistical mechanics of the spin-1/2 van der Waals model. I. Time evolution of a single spin.

The spin van der Waals model has received some attention in recent years as a soluble model, useful for demonstrating collective behavior analytically. By solving the Heisenberg equation of motion, we obtain the time evolution of a single spin in this model when N\ensuremath{\rightarrow}\ensuremath{\infty}, where N is the number of spins in the model. Our solution is characterized by rotations in spin space having the appearance of a Larmor precession. In the accompanying paper, we use this time-evolution solution to obtain the spin autocorrelation function and determine its long-time behavior.

The relaxation function of the spin van der Waals model in a weak magnetic field

We consider the spin van der Waals model in a weak magnetic field. For this model we obtain the relaxation function by applying the perturbative method that was developed for the evaluation of the relaxation function represented as an infinite continued fraction. To show the applicability of the perturbative method, we make comparisons between an exact result and various degrees of approximation.

Memory functions and relaxation functions of some spin systems

The method of recurrence relations has proved to be a powerful tool in the understanding of time evolutions of many-body systems such as an electron gas, classical harmonic chains, and spin systems on a lattice. We review the method and its applications to the dynamics of spins in the one-dimensional XY and transverse Ising models, and the spin van der Waals model.

Method of Recurrence Relations and Applications to Many-Body Systems

The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation.The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.

Critical properties of pseudospin Hamiltonians

The thermodynamic critical properties of a simple class of pseudospin model Hamiltonians are discussed. This class of models includes the spin van der Waals model and the Meshkov–Glick–Lipkin model as particular cases. Second-order thermodynamic phase transitions occur when the spin–spin interaction contributes negatively in a particular direction and the linear interaction term is orthogonal to the direction(s) of greatest energy gain through the spin–spin interaction.

Time-Dependent Behavior of the Spin-½ Anisotropic Heisenberg Model in Infinite Lattice Dimensions

The spin van der Waals model may be regarded as an infinite- lattice dimensional limit of the spin-(1/2) anisotropic Heisenberg model. By solution of the generalized Langevin equation, time-dependent behavior is obtained. The geometry of realized Hilbert spaces lends a simple interpretation for critical dynamics.

Critical behavior of the spin van der Waals model

Thermodynamic behavior in the critical region of the spin van der Waals model is considered when the number of spins in the system is large but finite (1≪N<∞). The specific heat curve is shown to possess a steep, yet smooth, maximum which turns into a singularity as the number of spins becomes infinite. In the case of the finite system, this maximum occurs at ε=ε*=O(N−1/3), where ε=(T−Tc)/Tc, Tc being the critical temperature of the infinite system.

Long-range order in the spin van der Waals model

Long-range order in the spin van der Waals model is considered when the number of spins is finite and also when infinite. We show explicitly that a finite system cannot support long-range order. An infinite system at high temperatures is found to be dominated by the entropy of degenerate states of the system and, as a result, the system behaves essentially like an ideal system. In an infinite system at low temperatures, long-range order exists, fully reflecting the spin symmetry of the Hamiltonian. For the XY-like regime (J≳Jz ), mx is finite but mz vanishes, where mx and mz are reduced order parameters for the transverse and longitudinal directions, respectively. For the Ising-like regime (J<Jz), mx vanishes but mz is finite. The isotropic interaction (Jz = J) behaves as a singularity and it must be considered separately. A physical interpretation of the behavior of long-range order is offered using the geometry of spin space.

Fluctuation and susceptibility for the spin van der Waals model and the bounds of Falk and Bruch

Fluctuation and susceptibility for the spin van der Waals model and the bounds of Falk and Bruch

The bounds of Falk and Bruch in the low temperature Van Der Waals model

For many-body systems, the susceptibility and the fluctuation in the long-range order can be different. There are bounds on the ratio of the susceptibility to the fluctuation due to Falk and Bruch, according to which the two correlation functions cannot be entirely independent. For the spin van der Waals model, we obtain exact expressions for the susceptibility and fluctuation in the low temperature region. We find that the susceptibility and fluctuation are the same in the XY-like regime of the model but they are different in the Ising-like regime. In both cases the bounds do not merge. Thus, the bounds alone are not sufficient to explain the relative behavior of the susceptibility and fluctuation.

Statics and dynamics of higher-order constant-coupling spin Hamiltonian

We discuss the spin Van der Waals model and its generalization of 4-spin interaction. The susceptibility is described with respect to the fluctuation and our main conclusions on the static dynamic properties of the 4-spin model are presented. We also discuss the usefulness of this model to the renormalization group methods.