Ising Square Research Articles

The Witt classes of

We study the Witt classes of the modular categories associated with quantum groups of type Dr at -th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent modulo the subgroup generated by the pointed modular categories. In particular, we produce an example of a simple, completely anisotropic modular category that is not pointed whose Witt class has order 2, answering a question of Davydov, Müger, Nikshych and Ostrik. Our results show that the trivial Witt class has infinitely many square roots modulo the pointed classes, in analogy with the recent construction of infinitely many square roots of the Ising Witt classes modulo the pointed classes constructed in a similar way from certain type Br modular categories. We compare the subgroups generated by the Ising square roots and square roots and provide evidence that they also generate linearly independent subgroups.

Thermodynamic behaviour of magnetocaloric quantities in spin-1/2 Ising square trilayer

A spin-1/2, Ising trilayered ferrimagnetic system on square Bravais lattice is studied, employing Monte-Carlo simulation with the single spin-flip Metropolis algorithm. The bulk of such a system is formed by three layers, each of which is composed entirely either by A or B type of atoms, resulting in two distinct compositions: ABA and AAB and two different types of magnetic interactions: ferromagnetic between like atoms and antiferromagnetic between unlike atoms. For such systems, Inverse Absolute of Reduced Residual Magnetisation is the absolute value of the ratio of the extremum of the magnetisation in between compensation and critical points and the saturation magnetisation. Variation of relative interaction strengths in the Hamiltonian, for a range of values, leads to the shift of compensation point and critical point and changes in the magnitude of Inverse absolute of Reduced Residual magnetisation. Probable mathematical forms of dependences of the Inverse absolute of Reduced Residual magnetisation and temperature interval between the compensation and critical points on controlling parameters were proposed in the absence of applied magnetic field and have obtained phase diagrams for both types of configurations from these relations. This alternative description of the simulated systems may help technologists design magnetocaloric materials according to desired characteristics.

Anisotropic Effect on Local Magnetic Properties of 3D Extended Ising Model

The magnetic properties of anisotropic 3D Ising model on a cubic lattice are studied by Monte Carlo simulation. In particular, we have considered an extended 3D Ising model with spatially uniaxial anisotropic bond randomness on the simple cubic lattice parameterized by exchange interaction parameter [Formula: see text], anisotropy parameter [Formula: see text] and external longitudinal magnetic field [Formula: see text]. The obtained numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model at critical temperature [Formula: see text] that is strongly correlated to [Formula: see text] and [Formula: see text]. Especially, in the limit, [Formula: see text], the spin ½ cubic lattice becomes a collection of noncorrelated Ising chains, whereas in the other limit, [Formula: see text], the system becomes a stack of noncorrelated Ising square lattice.

Decorated Ising Square Lattice in a Magnetic Field

The Ising model on a decorated square lattice with various parameters of exchange interactions in an external magnetic field is studied. A comparison is made with the numerical results obtained for the Ising model on a simple square lattice. It is shown that, if the magnetization increases with increasing field as in an antiferromagnet, then the initial phase transition point gradually decreases, reaching zero when the first frustration field is reached, and, with a further increase in the field, the phase transition does not occur. If the magnetization increases as in a ferromagnet, the phase transition disappears immediately after switching on the field; i.e., an arbitrarily small magnetic field completely suppresses the phase transition.

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

ABSTRACTThe fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.

Quantum Ising model on the frustrated square lattice.

We investigate the role of a transverse field on the Ising square antiferromagnet with first (J_{1}) and second (J_{2}) neighbor interactions. Using a cluster mean-field approach, we provide a telltale characterization of the frustration effects on the phase boundaries and entropy accumulation process emerging from the interplay between quantum and thermal fluctuations. We found that the paramagnetic (PM) and antiferromagnetic phases are separated by continuous phase transitions. On the other hand, continuous and discontinuous phase transitions, as well as tricriticality, are observed in the phase boundaries between PM and superantiferromagnetic phases. A rich scenario arises when a discontinuous phase transition occurs in the classical limit while quantum fluctuations recover criticality. We also find that the entropy accumulation process predicted to occur at temperatures close to the quantum critical point can be enhanced by frustration. Our results provide a description for the phase boundaries and entropy behavior that can help to identify the ratio J_{2}/J_{1} in possible experimental realizations of the quantum J_{1}-J_{2} Ising antiferromagnet.

Phase transitions in a frustrated Ising antiferromagnet on a square lattice.

The phase transitions occurring in the frustrated Ising square antiferromagnet with first- (J1<0) and second-nearest-neighbor (J2<0) interactions are studied within the framework of the effective-field theory with correlations based on the different cluster sizes and for a wide range of R=J2/|J1|. Despite the simplicity of the model, it has proved difficult to precisely determine the order of the phase transitions. In contrast to the previous effective-field study, we have found a first-order transition line in the region close to R=-0.5 not only between the superantiferromagnetic and paramagnetic (R<-0.5) but also between antiferromagnetic and paramagnetic (R>-0.5) phases.

Structural investigation of a hydrogen bond order–disorder transition in a polar one-dimensional confined ice

The hydrogen-bond arrangement within crystalline 2,3,6,7,10,11-hexahydroxytriphenylene tetrahydrate (HHTP·4H2O) undergoes an order-disorder transition at 240 K, as evidenced by the emergence and disappearance of systematic absence violations in variable-temperature single-crystal X-ray diffraction measurements. The low-temperature ordered phase is polar with ferroelectric coupling between neighbouring one-dimensional ice-like columns of hydrogen-bonded H2O molecules. At temperatures above 240 K the material adopts a paraelectric state characterised by the absence of long-range ordering between column polarisations. We discuss the mapping of this phase transition onto the problem of frustration on the canonical Ising square lattice, and suggest that HHTP·4H2O is an obvious candidate for exploring switchable ferroelectric behaviour in confined ices.

Magnetization of Nano-Size Subsystem in a Two-Dimensional Ising Square Lattice

A two-dimensional Ising square lattice is modeled as a nano-size block array to study by Monte Carlo simulation the magnetic thermal stability of nano-structure magnetic media for data storage, thereon in the blocks J1 > 0 is assigned for the interaction of a pair of nearest-neighbor spins, while 0 J0 J1 for that in regions between the blocks and (J0 + J1)/2 for the nearest-neighbor pairs with one in the block and the other one out of but near-most the block. We show that the magnetic thermal stability of the block accrues with the increase of J1 and with the decrease of J1 - J0 for a given J1, but contrarily, the anchoring ability for the initial magnetic orientation in nano-size block trails off as J1 - J0 diminish. This phenomena and size dependence of such anchoring ability are discussed in detail.

On the spontaneous ordering of the mixed-spin Ising square lattice with singly and triply decorated bonds

The mixed-spin Ising model on a square lattice with singly and triply decorated bonds is exactly solved within the framework of the decoration–iteration transformation, which establishes a precise mapping relationship between the investigated model system and the equivalent spin-1/2 Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy, which acts on the decorating spin-1 atoms, is examined in particular. It is shown that the investigated model displays a very peculiar critical behaviour as a result of the single-ion anisotropy strengthening because it remains spontaneously long range ordered despite its quasi-1D character in a certain range of single-ion anisotropies before it enters the disordered phase. The single-ion anisotropy parameter is also responsible for diverse temperature dependences of sublattice and total magnetization.

Antiphase phenomena in 2d ising square lattice

Monte Carlo simulations of the two dimensional Ising model were carried out to determine the impact of Antiphase Boundaries (APBs) in some thermodynamic functions of the system. We considered several lattice sizes and sometimes both Ferromagnetic (FM) and Antiferromagnetic (AFM) interactions for the same lattice. The analysis of the curves obtained shows that APBs appears only in AFM interaction and their effect is the creation of an energy gapthat prevents the system from reaching the ground state. This happens only in lattices with odd linear size L (L = 9, 15, 31 …). However, these boundaries do not change the critical temperature of the Antiferromagnetic/Paramagnetic (AFM/PARA) phase transition which remains approximately at 2.26 (in unit of J/kB). Nevertheless, they impose on the curve of the specific heat a jittery behaviour as a function of the temperature and with no clear peak as opposed to the case of even L (L = 16 ,20, 48,…) where there is a clear peak and a smooth curve. There is also a fluctuating magnetization around the value zero at low temperature (AFM phase). We show that this energy gap is proportional to 1/L where L is the linear size of the lattice, thus this gap should vanish if L becomes high. A similarity has been established with the role played by APB in some crystal structures and more especially in the binary alloys that are moreover described by the Ising model.

Monte Carlo studies of the Ising square lattice with competing interactions

We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J2 = J1/2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the speciflc heat and the energy distribution for various ratios of J2/1. We find a first order transition for small J2 > J1/2 and the transition temperature suppressed to TC = 0 at the critical point.

Locally converging algorithms for determining the critical temperature in Ising systems

We introduce a class of algorithms that converge to criticality automatically, in a way similar to the invaded cluster algorithm. Unlike the invaded cluster algorithm which uses global percolation as a test for criticality, these local algorithms use an average over local observables, specifically the number of satisfied bonds, in a feedback loop which drives the system toward criticality. Two specific algorithms are introduced, the average algorithm and the locally converging Wolff algorithm. We apply these algorithms to study the Ising square lattice and the Ising Bethe lattice. We find reasonable convergence to the critical temperature for both systems under the locally converging Wolff algorithm. We also re-examine the phase diagram of the dilute two-dimensional (2D) Ising model and find results supporting our previously reported conclusions regarding the existence of a local regime of magnetization below the percolations threshold. In addition, the presented algorithms are computationally more efficient than the invaded cluster algorithm, requiring less CPU time and memory.

Phase diagram of the Ising square lattice with competing interactions

We restudy the phase diagram of the 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds via Monte-Carlo simulations. We present the finite temperature phase diagram and introduce computational methods which allow us to calculate transition temperatures close to the criticalpoint at J2 = J1/2. Further on we investigate the character of the different phase boundariesand find that the transition is weakly first order formoderate J2 > J1/2.

Properties of the Ising magnet confined in a corner geometry

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ± h are applied, i.e., at the two boundary rows ending at the lower left corner a field + h acts, while at the two boundary rows ending at the upper right corner a field − h acts. For temperatures T less than the critical temperature T c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T f ( h ) runs from the upper left corner to the lower right corner, while for T < T f ( h ) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T = T f ( h ) the magnetization profile m ( z ) in the z-direction normal to the interface simply is linear and the interfacial width scales as w ∝ L , while for T > T f ( h ) it scales as w ∝ L . The distribution P ( ℓ ) of the interface position ℓ (measured along the z-direction from the corners) decays exponentially for T < T f ( h ) from either corner, is essentially flat for T = T f ( h ) , and is a Gaussian centered at the middle of the diagonal for T > T f ( h ) . Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.

Temperature effects in the magnetic properties of two-dimensional Ising square lattices: A Monte Carlo investigation

The magnetic behavior of a two-dimensional nearest-neighbor Ising model with the presence of linear temperature variation in a thermal steady state was studied using the Wolff Monte Carlo simulation. The technique consists of fixing the temperatures of boundary spins, while the temperature field in the interior linearly varies with distance. It is found that with increasing the temperature difference between the two boundaries, the magnetization greatly reduces in magnitude while the susceptibility peaks tend to spread out over a temperature range. The detailed descriptions of these magnetization and susceptibility behaviors are elucidated from their spatial variation. The extraction of the ``critical temperatures'' is taken via the fourth-order cumulant of the magnetization. The critical temperatures are found to reduce slightly with increasing the temperature difference. This implies the vulnerability of the magnetization and susceptibility properties to the temperature variation in ferromagnetic materials, and to use such materials in temperature variation environments must be done with caution.

Magnetic spin interactions observed by heat capacity measurements for layered compounds: Na LnTiO 4 ( Ln = Sm, Eu, Gd, Tb, Dy, Ho and Er)

Magnetic properties of a series of orthorhombic Na LnTiO 4 ( Ln = Sm, Eu, Gd, Tb, Dy, Ho and Er) characterized by heat capacity measurements are reported. Magnetic transitions are observed for all Na LnTiO 4 investigated except for Ln = Eu. The change in magnetic entropy closely agrees with the expected theoretical values except for Ln = Sm and Gd. Peak profiles in temperature dependence of heat capacity indicate that Ln = Sm, Tb, Dy, Ho and Er phases possess Ising square lattices. Furthermore, Ln = Gd also exhibits the similar property regardless of its S state; there seems some degree of anisotropy in its magnetic lattice. Magnetization and heat capacity data indicate that superexchange through oxygen is the dominant spin interaction in this system.

Magnetic Study of an Anisotropic Mixed Spin-1/2 and Spin-1 Ising Model on a Square Lattice with the Uniaxial Crystal-field Anisotropy

Magnetic properties of an anisotropic mixed spin-1/2 and spin-1 Ising square lattice are studied within the framework of the linear chain approximation (LCA) method. LCA is formulated on the basis of Gibbs-Bogolyubov variational principle in order to obtain all thermodynamic quantities in a self-consistent manner.

The Fuchsian differential equation of the square lattice Ising model (3)susceptibility

Using an expansion method in the variables $ x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the 3-particle contribution $\chi^{(3)}$, using a $ N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $\chi^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.

Multi-Information in the Thermodynamic Limit

A multivariate generalization of mutual information, multi-information, is defined in the thermodynamic limit. The definition takes phase coexistence into account by taking the infimum over the translation-invariant Gibbs measures of an interaction potential. It is shown that this infimum is attained in a pure state. An explicit formula can be found for the Ising square lattice, where the quantity is proved to be maximized at the phase-transition point. By this, phase coexistence is linked to high model complexity in a rigorous way.