On the critical behaviour of the two dimensional spin XY model

1982 ◽  
Vol 60 (3) ◽  
pp. 368-372 ◽  
Author(s):  
Jos Rogiers
Keyword(s):  
Critical Point ◽  
Triangular Lattice ◽  
Second Order ◽  
Critical Behaviour ◽  
Transverse Magnetization ◽  
Xy Model ◽  
Two Dimensional ◽  
Exponential Behaviour ◽  
Transformation Methods

Transformation methods are used to analyse the series for the second order fluctuation of the transverse magnetization for the triangular and square lattices. For the triangular lattice some evidence is found for an exponential behaviour of this quantity near the critical point with a tentative estimate for the exponent [Formula: see text].

scholarly journals Universality class of the motility-induced critical point in large scale off-lattice simulations of active particles.

Soft Matter ◽  
2021 ◽  
Author(s):  
Claudio Maggi ◽  
Matteo Paoluzzi ◽  
Andrea Crisanti ◽  
Emanuela Zaccarelli ◽  
Nicoletta Gnan
Keyword(s):  
Phase Separation ◽  
Critical Point ◽  
Computer Simulations ◽  
Large Scale ◽  
Universality Class ◽  
Critical Behaviour ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Active Particles ◽  
Two Dimensional Model

We perform large-scale computer simulations of an off-lattice two-dimensional model of active particles undergoing a motility-induced phase separation (MIPS) to investigate the systems critical behaviour close to the critical point...

scholarly journals MAGNETIC ORDERING OF ANTIFERROMAGNETS ON A SPATIALLY ANISOTROPIC TRIANGULAR LATTICE

2010 ◽  
Vol 24 (25n26) ◽  
pp. 5011-5026 ◽  
Author(s):  
R. F. BISHOP ◽  
P. H. Y. LI ◽  
D. J. J. FARNELL ◽  
C. E. CAMPBELL
Keyword(s):  
Critical Point ◽  
Triangular Lattice ◽  
Second Order ◽  
Square Lattice ◽  
Cluster Method ◽  
First Order ◽  
Lattice Geometry ◽  
Nearest Neighbours ◽  
Helical State ◽  
Ordered State

We study the spin-1/2 and spin-1 [Formula: see text] Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1 > 0) bonds between nearest neighbours and competing [Formula: see text] bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with [Formula: see text] bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme (κ = 0) and a set of decoupled chains at the other (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For the spin-1/2 [Formula: see text] model, we find a weakly first-order (or possibly second-order) quantum phase transition from a Néel-ordered state to a helical state at a first critical point at κc1 = 0.80 ± 0.01, and a second critical point at κc2 = 1.8 ± 0.4 where a first-order transition occurs between the helical state and a collinear stripe-ordered state. For the corresponding spin-1 model we find an analogous transition of the second-order type at κc1 = 0.62 ± 0.01 between states with Néel and helical ordering, but we find no evidence of a further transition in this case to a stripe-ordered phase.

CRITICAL COARSENING DYNAMICS OF SPIN S =1/2, 3/2 ANTIFERROMAGNETIC ISING MODEL ON TRIANGULAR LATTICE

2006 ◽  
Vol 17 (04) ◽  
pp. 591-600
Author(s):  
KWANGHOON CHUNG ◽  
MOOKYUNG CHEON ◽  
IKSOO CHANG
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Triangular Lattice ◽  
Ground State ◽  
Finite Temperature ◽  
Xy Model ◽  
Two Dimensional ◽  
Type Phase ◽  
Coarsening Dynamics ◽  
Antiferromagnetic Ising Model

The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Connecting Networks for Two-Dimensional Butler Matrices Generating a Triangular Lattice of Beams

2021 ◽  
Vol 1 (2) ◽  
pp. 646-658
Author(s):  
NELSON J. G. FONSECA ◽  
SOPHIE-ABIGAEL GOMANNE ◽  
PILAR CASTILLO-TAPIA ◽  
OSCAR QUEVEDO-TERUEL ◽  
TAKASHI TOMURA ◽  
...  
Keyword(s):  
Triangular Lattice ◽  
Two Dimensional
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