COMPUTER SIMULATION STUDY OF A TWO-DIMENSIONAL NEMATOGENIC LATTICE MODEL WITH LONG-RANGE INTERACTIONS ISOTROPIC IN SPIN SPACE

1995 ◽  
Vol 09 (07) ◽  
pp. 859-873 ◽  
Author(s):  
N. ANGELESCU ◽  
S. ROMANO ◽  
V.A. ZAGREBNOV
Keyword(s):  
Computer Simulation ◽  
Long Range ◽  
Lattice Model ◽  
Molecular Field ◽  
Spin Space ◽  
Rigorous Results ◽  
Pair Potentials ◽  
Long Range Interactions ◽  
Site Cluster ◽  
Ordering Transition

We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3) {uk, k∈Zd}, associated with a d-dimensional lattice Zd, d=1, 2, and interacting via pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T*=kBT/∈), and xk denotes dimensionless lattice-site coordinates. Extending previous rigorous results, one can prove the existence of an ordering transition at finite temperature when 0<σ< d, and its absence when σ≥d. We have studied the case defined by n=3, d=2, σ=1, by means of computer simulation, Molecular Field and Two-Site Cluster theory. The Two-Site Cluster approach was found to bring about a recognizable improvement over Molecular Field; on the other hand, comparison with the Lebwohl-Lasher lattice model shows that the long-range character of the interaction tends to increase the transition temperature towards its Molecular Field limit.

COMPUTER SIMULATION STUDY OF A DISORDERED ONE-DIMENSIONAL NEMATOGENIC LATTICE MODEL WITH LONG-RANGE INTERACTIONS ISOTROPIC IN SPIN SPACE

1995 ◽  
Vol 09 (25) ◽  
pp. 3345-3354 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Computer Simulation ◽  
Long Range ◽  
Temperature Phase ◽  
Spin Space ◽  
Rigorous Results ◽  
One Dimensional ◽  
Pair Potentials ◽  
Long Range Interactions ◽  
Invariant Pair ◽  
Ordering Transition

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] where ∊ is a positive constant setting energy and temperature scales (i.e. T* = kBT/∊). Extending previous rigorous results, one can prove the existence of an ordering transition at finite temperature when 0 < σ < 1, and its absence when σ ≥ 1. We have studied the border case σ = 1, by means of computer simulation. Similarly to the magnetic counterparts of the present model, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ = 0.475 ± 0.005.

Computer Simulation Study of a Disordered Two-Dimensional Nematogenic Lattice Model with Long-Range Interactions Isotropic in Spin Space

1997 ◽  
Vol 11 (07) ◽  
pp. 919-928
Author(s):  
S. Romano
Keyword(s):  
Computer Simulation ◽  
Long Range ◽  
Temperature Phase ◽  
Two Dimensional ◽  
Spin Space ◽  
Pair Potentials ◽  
Long Range Interactions ◽  
Invariant Pair ◽  
Ordering Transition ◽  
Low Temperature Phase

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k∈ Z2}, and interacting via translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T*=k B T /∊), P2 denotes the second Legendre polynomial, and xj are dimensionless coordinates of the lattice sites. Available theorems entail the existence of an ordering transition at finite temperature when 0 < σ < 2, and its absence when σ ≥ 2. We have studied the border case σ=2, by means of computer simulation. Similarly to the nearest-neighbour counterpart of the present model, and to other long-range models, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinski[Formula: see text]–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ=1.112 ± 0.005.

Computer Simulation Study of Classical Spin Models in Two Dimensions with Long-Range Ferromagnetic Interactions Anisotropic in Spin Space

1998 ◽  
Vol 12 (18) ◽  
pp. 1871-1885 ◽  
Author(s):  
S. Romano ◽  
Valentin A. Zagrebnov
Keyword(s):  
Long Range ◽  
Pair Potential ◽  
Two Dimensions ◽  
Spin Space ◽  
Potential Models ◽  
Classical Spin ◽  
Site Cluster ◽  
Invariant Pair ◽  
Ferromagnetic Interactions ◽  
Ordering Transition

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally invariant pair potential, of the long-range ferromagnetic form, anisotropic in spin space [Formula: see text] here a ≥ 0, b ≥ 0, σ > 2, ∊ is a positive constant setting energy and temperature scales (i.e. T*=kBT/∊), xj denotes dimensionless coordinates of lattice sites, and uj,α cartesian spin components; our discussion has been specialized to the extreme, O(2)-symmetric, case 0=a < b. When 2 < σ < 4, the potential model can be proven to support an ordering transition taking place at finite temperature; on the other hand, when σ ≥ 4 a Berezinskiǐ–Kosterlitz–Thouless-like transition takes place. Two potential models defined by σ=3 and σ=4, respectively, have been characterized quantitatively by Monte Carlo simulation. For σ=3, comparison is also reported with other theoretical treatments, such as Molecular Field and Two Site Cluster approximations.

COMPUTER SIMULATION STUDY OF LATTICE SPIN MODELS IN ONE DIMENSION WITH LONG-RANGE INHOMOGENEOUS INTERACTIONS POSSESSING O(2) SYMMETRY

1996 ◽  
Vol 10 (09) ◽  
pp. 1095-1109 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Computer Simulation ◽  
Long Range ◽  
Mean Field ◽  
Upper And Lower Bounds ◽  
One Dimension ◽  
Pair Potentials ◽  
Qualitative Similarity ◽  
Lattice Spin Models ◽  
Two Parameters ◽  
Ordering Transition

We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, in general anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a ≥ 0, b ≥ 0, and the symbols uj, α denote cartesian components of the spins. For some specific values of the two parameters a and b, and on the basis of available theoretical results, one can prove the existence of an ordering transition taking place at finite temperature, and obtain rigorous upper and lower bounds on the transition temperatures. This holds, for example, when n = 2, 3, a > b = 0 (the models studied in our previous paper), as well as for n = 2, a = b > 0 and n = 3, b > a = 0, where a continuous O(2) symmetry of the interaction is involved. We have studied these two latter cases by computer simulation, and made comparison with mean-field treatment; simulation results show a broad qualitative similarity between the four models, and a closer, quantitative one, between pairs of models with the same number of spin components, especially for n = 2.

COMPUTER SIMULATION STUDY OF ONE-DIMENSIONAL LATTICE SPIN MODELS WITH LONG-RANGE INHOMOGENEOUS INTERACTIONS

1995 ◽  
Vol 09 (22) ◽  
pp. 1447-1459 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Computer Simulation ◽  
Long Range ◽  
Mean Field ◽  
Upper And Lower Bounds ◽  
Pair Potentials ◽  
The Mean ◽  
Lattice Spin Models ◽  
Ordering Transition ◽  
Theoretical Results ◽  
Unit Vectors

We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a > 0, b ≥ 0, and the symbols uj,α denote Cartesian components of the spins. On the basis of available theoretical results, one can prove the existence of an ordering transition at a finite temperature for n ≥ 2, b = 0, as well as upper and lower bounds on the transition temperatures. We have studied the two cases n = 2, 3, b = 0, by computer simulation, and made a comparison with the Mean Field treatment.

scholarly journals Kink dynamics in a lattice model with long-range interactions

2001 ◽  
Vol 34 (20) ◽  
pp. 4269-4280 ◽  
Author(s):  
T Ioannidou ◽  
J Pouget ◽  
E Aifantis
Keyword(s):  
Long Range ◽  
Lattice Model ◽  
Long Range Interactions

COMPUTER SIMULATION STUDY OF A CLASSICAL HEISENBERG MODEL IN TWO DIMENSIONS WITH LONG-RANGE FERROMAGNETIC INTERACTIONS ISOTROPIC IN SPIN SPACE

1996 ◽  
Vol 10 (21) ◽  
pp. 2687-2698 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Long Range ◽  
Pair Potential ◽  
Spherical Model ◽  
Two Dimensions ◽  
Continuous Transition ◽  
Site Cluster ◽  
Invariant Pair ◽  
Ferromagnetic Interactions ◽  
Ordering Transition ◽  
Rotationally Invariant

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.

MEAN FIELD, TWO-SITE CLUSTER, AND COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL

2001 ◽  
Vol 15 (03) ◽  
pp. 259-280 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Lattice Gas ◽  
Mean Field ◽  
Three Dimensional ◽  
Lattice Gas Model ◽  
Rigorous Results ◽  
Positive Quantity ◽  
Occupation Numbers ◽  
Site Cluster ◽  
Ordering Transition ◽  
Unit Vectors

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearest-neighbouring sites interact via the nematogenic potential [Formula: see text] here P2(τ) denotes the second Legendre polynomial, νj = 0, 1 are occupation numbers, uj are unit vectors (classical spins), and ∊ is a positive quantity setting energy and temperature scales (i.e. T* = k B T/∊); the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model defines the extensively studied Lebwohl–Lasher model, possessing a transition to an orientationally ordered phase at low temperature; according to available rigorous results, there exists a μ0 < 0, such that, for all μ > μ0, the system supports an ordering transition at a finite, μ-dependent, temperature. Continuing along the lines of our previous communication [S. Romano, Int. J. Mod. Phys.B14, 1195 (2000)], we present here a detailed study of the case μ = 0, using Monte Carlo simulation, Mean Field and Two Site Cluster treatments; the latter significantly improves the agreement with simulation results.

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