Antiferromagnetic Ising System Research Articles

Antiferromagnetic Ising model in a triangular vortex lattice of quantum fluids of light.

Vortices are topologically distinctive objects appearing as phase twists in coherent fields of optical beams and Bose-Einstein condensates. Structured networks and artificial lattices of coupled vortices could offer a powerful platform to study and simulate interaction mechanisms between constituents of condensed matter systems, such as antiferromagnetic interactions, by replacement of spin angular momentum with orbital angular momentum. Here, we realize such a platform using a macroscopic quantum fluid of light based on exciton-polariton condensates. We imprint all-optical hexagonal lattice that results into a triangular vortex lattice, with each cell having a vortex of charge l = ±1. We reveal that pairs of coupled condensates spontaneously arrange their orbital angular momentum antiparallel, implying a form of artificial orbital "antiferromagnetism." We discover that correlation exists between the emergent vortex patterns in triangular condensate lattices and the low-energy solutions of the corresponding antiferromagnetic Ising system. Our study offers a path toward spontaneously ordered vortex arrays with nearly arbitrary configurations and controlled couplings.

Short-range character of the antiferromagnetic Ising model with1/rpinteraction

By applying Ewald summation, we analytically compute the Fourier transform of a $d$-dimensional lattice interaction decaying like ${r}^{\ensuremath{-}p}$ for a long-range antiferromagnetic Ising system on a simple cubic lattice with $1lpld$. The resulting analytical form of the dispersion allows to deduce in a simple way that the model's universality class is identical to that of the short-range Ising model.

Short-range order in two-dimensional binary alloys

Using variable-temperature scanning tunneling microscopy, the degree of short-range order of purely two- dimensional (2D) binary alloys on triangular lattices $[{\mathrm{Sn}}_{(1\ensuremath{-}x)}\ensuremath{-}{\mathrm{Si}}_{x},$ ${\mathrm{Pb}}_{(1\ensuremath{-}x)}\ensuremath{-}{\mathrm{Si}}_{x},$ and ${\mathrm{Pb}}_{(1\ensuremath{-}x)}\ensuremath{-}{\mathrm{Ge}}_{x},$ all showing the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}R30\ifmmode^\circ\else\textdegree\fi{}$ reconstruction onto Si(111) or Ge(111)] has been determined quantitatively via a statistical analysis of the atomic positions. The experimental data, also in comparison with Monte Carlo simulations, generally indicate that an effective nearest-neighbor repulsion between substitutional Si (Ge) adatoms explains with good accuracy the short-range-order features observed. This finding implies the occurrence of an ordered phase of the alloys for $x=0.33$ and, in the case of a 1:1 ratio of Sn(Pb) and Si(Ge) adatoms on the surface $(x=0.5),$ demonstrates that the investigated alloys are very good practical realizations of a frustrated antiferromagnetic 2D Ising system on a triangular lattice. The implications of the observed short-range order for the electronic properties of these alloys (as a function of $x)$ are discussed.

Persistence in an antiferromagnetic Ising system with conserved magnetisation

We obtain the persistence exponents for an antiferromagnetic Ising system in which the magnetisation is kept constant. This system belongs to model C (system with non-conserved order parameter with a conserved density) and is expected to have persistence exponents different from that of model A (system with no conservation) but independent of the conserved density. Our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimensions. This non-universal feature is attributed to the presence of the conserved field and is special to the persistence phenomena.

Short-time scaling in the critical dynamics of an antiferromagnetic Ising system with conserved magnetization

We study by Monte Carlo simulation the short-time exponent θ in an antiferromagnetic Ising system for which the magnetization is conserved but the sublattice magnetization (which is the order parameter in this case) is not. This system belongs to the dynamic class of model C. We use nearest-neighbour Kawasaki dynamics so that the magnetization is conserved locally. We find that in three dimensions θ is independent of the conserved magnetization. This is in agreement with the available theoretical studies, but in disagreement with previous simulation studies with a global conservation algorithm. However, we agree with both these studies regarding the result θC≠θA. We also find that in two dimensions, θC = θA.

Van Der Waerden's Coloring Theorem and Classical Spin Systems

We find a non-invertible matrix representation for Van der Waerden's coloring theorem for two distinct colors in a one-dimensional periodic lattice. Using this, an infinite one-dimensional antiferromagnetic Ising system is mapped to a pseudo-ferromagnetic one, thereby relating the couplings. All this is reminiscent of renormalization group.

Exact ground state spin configurations for 2D and 3D lattices with nearest neighbor bilinear exchange

A systematic research of the bilinear exchange Hamiltonians providing an exactly known ground state spin configuration is performed for two and three dimensional lattices. Apart from the well known ferromagnetic Heisenberg and ferro or antiferromagnetic Ising systems, particular attention is paid to the anisotropic Heisenberg model H ij = −J xS ixS jx − J yS iyS jz − J zS izS jz for a bond along the z axis. It is shown that, under special conditions, this pair interaction leads to a two sublattice antiferromagnetic configuration which is an eigenstate of the system at T = 0 K, i.e. without spin deviation. All the possible structures displaying this remarkable property are derived.

Exact determination of all ground states of random field systems in polynomial time

An algorithm is developed which allows calculating all ground states of ferromagnetic and unfrustrated antiferromagnetic Ising systems with arbitrary site-dependent fields by transforming the system into an equivalent network and calculating the maximal flow. By a trial and error scheme a minimum cut is constructed which corresponds to a spin configuration. In this way each ground state is calculated with a finite probability. The algorithm is applied to site-diluted antiferromagnets in external magnetic fields. It is found that in this case its time complexity is approximately quadratic in the lattice size. As an application we calculate the distribution of overlaps between ground states of the site-diluted antiferromagnet in a strong magnetic field and we analyse the fractal structure of these ground states.

Residual entropies of the Ising antiferromagnets on fractals: II. d-dimensional checkerboard type of fractals

We study the residual entropies of the antiferromagnetic Ising systems, in the maximum critical field, situated on checkerboard (CB) type of fractals embedded in d-dimensional Euclidean space. For a given d, the CB fractals constitute a family, whose each member is labelled by b, where b is an odd integer (3⪕ b<∞). Each family itself furnishes a crossover to the corresponding d-dimensional hypercubic lattice. By calculating explicitly residual entropies σ( b) for finite b, and by using a specific method of extrapolating the obtained results, we were able to establish the crossover behavior of σ( b). It turns out that the established crossover is of the same type as the one found in the case of fractal families of the Sierpinski gasket type.

Residual entropies of the Ising antiferromagnets on fractals: I. d-dimensional sierpinski gasket type of fractals

We apply a new method to study the problem of the residual entropies of the antiferromagnetic Ising systems, in the maximum critical field, situated on Sierpinski gasket (SG) type of fractal lattices. The SG fractal lattices comprise a family, so that each member is labelled by an integer b (2⪕ b<∞) and is embedded in d-dimensional Euclidean space. In the two-dimensional (2D) and three-dimensional (3D) case, we find expressions for the residual entropy σ( b) for arbitrary b, and show that in the limit b→∞ the calculated values should converge to the values pertinent to the corresponding Euclidean lattices. In particular, these results confirm recent findings concerning the residual entropy of the 2D Sierpinski gasket type of fractals, obtained by numerical fitting. We also study the d-dimensional case and find a general expression for σ( b) for arbitrary b⪖3.

Statics and Dynamics of Antiferromagnetic Ising System on a Triangular Lattice. II

Statics and Dynamics of Antiferromagnetic Ising System on a Triangular Lattice. II

Magnetic excitations in the Ising-like layer compound (Co( γ-CH3-C5H4NO)6)(ClO4)2

The results of ESR experiments on the 2D antiferromagnetic Ising system (Co( gamma -CH3-C5H4NO)6)(ClO4)2 in the temperature range 1.2-4.2 K are reported. The observed absorption signals are found to originate from single-spin reversals and can be associated with nearest-neighbour configurations of different forms. The magnetic field dependence of the excitation energies has been interpreted on the basis of a rectangular (2D) Ising lattice with nearest-neighbour exchange interactions. The signal intensities show a behaviour that can qualitatively be explained in terms of the predictions for a 2D Ising model.

Statics and Dynamics of Antiferromagnetic Ising System on a Triangular Lattice.I

The Ising model on a triangular lattice with antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions is investigated by treating the master equation by molecular field approximation(MFA) and the cluster decoupling approximation(CDA). For the fully frustrated case, CDA gives no phase transition in contrast with the MFA result which gives the finite Néel temperature. The change of states (i.e., phase transition) is explained by words of bifurcation theory and the relaxation process is investigated by studying the flow in the space of sublattice magnetizations in each approximation.

Surface Phase Diagram of a Layered Antiferromagnetic Ising System with a Site‐Diluted Free Surface

Surface Phase Diagram of a Layered Antiferromagnetic Ising System with a Site‐Diluted Free Surface

Evidence for a two-dimensional Ising system from magnetic dilution studies on bis(isothiocyanato)bis(1,2,4-triazole)iron(II)

The magnetic dilution problem has been studied in a series of substances of general formula Fe pZn 1- ptrz 2(NCS) 2 ( p = 1.00-0.60; trz = 1,2,4-triazole) with the aid of low-temperature magnetic-susceptibility measurements. The data can be successfully interpreted on the basis of a two-dimensional antiferromagnetic Ising system. The critical concentration p C of Fe 2+ ions at which T C( p) vanishes is estimated to be about 0.60, in agreement with theory.

Antiferromagnetic Ising Model on a Triangular Lattice

An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.

Critical temperature of mixed ferromagnetic and antiferromagnetic Ising systems with arbitrary spin

Using the high-temperature series expansion method, a system of randomly mixed ferromagnetic and antiferromagnetic Ising bonds is considered in quenched and annealed systems for an arbitrary value of spin and a BCC lattice. The dependence of the critical temperature on the concentration p of ferromagnetic bonds is determined. It is found that the resulting phase diagrams depend on the system (whether quenched or annealed), the spin value and the strength of the coupling constants. Also the dependence of the critical exponent gamma on p and the crossover exponent phi are estimated.

Critical behaviour of ising systems on extremely anisotropic lattices: IV. Antiferromagnetic cases

The critical behaviour of the reduced magnetic susceptibility X'≡( kT μ 2 )X of spin- 1 2 Ising systems on d-dimensional cartesian lattices in which the coupling constants along d-1 lattice axes, J 2,… J d, are much smaller than the coupling constant along the remaining lattice axis, J 1, and in which some of the J i are negative, is investigated on the basis of the series expansion for X' in the variables t i = tanh βJ i , i = 1,… d. For 1 + t 1 《 1 ; |t 2 |,… |t d | 《 1 , X' is found to behave as X'(t 1 ,…t d )≈X' 0 (t 1 ,…t d )≡( 1+t 1 ) ∑ n 2=o ∞ … ∑ n d=o ∞ c n 2…n d 0 Φ i=2 d t 1 1+t 1 , where the c n2… n d0 are constants. In the limit ( t 1,… t d)→(-1, 0,…0), taken in such a way that the power series in this expression remains convergent, X' 0 behaves through the factor 1+ t 1 as the susceptibility of the antiferromagnetic linear Ising chain. The d-dimensional nature of the system is contained in the power series. It is rigorously shown that X' 0 = 1 2 ( 1+<σ 1 σ 2 ) 0 , where < σ 1 σ 2 is the spin-spin correlation function for nearest neighbours along the first lattice axis, and ( ) 0 is the leading-order term for ( t 1, t 2,… t d)→(−1, 0,…0) of the quantity within the brackets. Using this relation it is shown that for the quadratic lattice with ( t 1, t 2) ≈ (−1, 0) X' 0 is given by X' 0 = ( 1+t 1 ) 1 π ƒ 0 π 2 1 − 4 ( t 2 1 + t 2 ) 2sin 2Ψ 1 2 . The fact that the integral, and hence X' 0, displays the same critical behaviour as the internal energy is discussed in connection with the critical behaviour of X' for antiferromagnetic Ising systems in general. A numerical analysis indicates that for the quadratic lattice with t 1 ↑ 1, t 2 ↑ 0 the critical behaviour of the leading-order term of X' is also similar to that of the internal energy. Finally, the first-order correction X' 1 on X' 0 is calculated explicitly for the quadratic lattice with ( t 1, t 2) ≈ (-1, 0), and the region of validity of the approximation X'≈ X' 0 is discussed.