Magnetic Phase Diagram of a 2D $XY$ Model with Random Easy Axis OrientationMagnetic Phase Diagram of a 2D $XY$ Model with Random Easy Axis Orientation

Donald Priour

Bulletin of the American Physical SocietyOct 25, 20140 citations

Correlation functions and computer simulations

Giorgio Parisi A

Nuclear Physics BMay 1, 1981242 citations

Stochastic Models for Many-Body Systems. II. Finite Systems and Statistical Nonequilibrium

Robert H. Kraichnan

In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.

Journal of Mathematical PhysicsMay 1, 196220 citations

Correlation functions and how they enter series expansions†

Graeme Walter Milton

Cambridge University Press eBooksMay 6, 20020 citations

Lectures on correlation functions

Alexander L. Fetter

Lecture notes in physicsJan 1, 19832 citations

Statistical mechanics of classical one-dimensional Heisenberg ferromagnets with single-site anisotropy

Kazuo Torii Nakamura, Tomohei Sasada

The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.

Journal of Physics C Solid State PhysicsJan 28, 197861 citations

Magnetic Phase Diagram of a 2D

XY

Model with Random Easy Axis Orientation