Dynamical susceptibility of the ising model in a transverse magnetic field

An approximate expression is obtained for the dynamical susceptibility, χ_zz(q, ω), of the spin one half, simple cubic Ising model in a transverse magnetic field, which is appropriate to the disordered state (〈S_z〉 = 0). The susceptibility along the direction of the field is shown to contain a thermal part with a relative weight proportional to the difference between the isothermal and adiabatic susceptibilities and a width determined by the thermal diffusion constant.     

Susceptibility divergence of the spin 1 Ising model on the Cayley tree

The temperature T₂(1), below which the zero-field isothermal susceptibility diverges for the spin 1 Ising model on the M-generation Cayley tree, in the limit as M → ∞, is located. The general approach follows closely that of Falk for the spin $\text{1}\text{2}$ problem. Pruning and retraction identities are used to find upper and lower bounds on the bond-length dependence of the general two-spin correlation function and thence on the susceptibility. A conjecture is advanced regarding the divergence of the higher-order susceptibilities.     

Initial perpendicular isothermal susceptibility formulas for the transverse general-spin Ising and Blume-Capel models

Using a Kubo formula and the Suzuki identities, expressions are derived for the initial perpendicular susceptibilities χ_⊥ of the transverse spin-S Ising and spin-S Blume-Capel models on regular and irregular lattices. χ_⊥ is given in terms of the thermal average of a function of the peripheral sumO _i= ε_j J _i,j S _j, where coupling to distant neighbors may be included, as well as arbitrary local parallel magnetic fieldsh _j. For the Ising model on a Bravais lattice, e.g., the susceptibility is given by $$\chi _ \bot = Nm^2 S^{ - 2} \langle B_s (\beta [O_i + h_i ])/[O_i + h_i ]\rangle $$ whereB _s is the Brillouin function. ForS=1/2, the formula of Fisher and the results of Horiguchi and Morita are regained. A connection is made with the general-spin work of Essam and Garelick.     

On the thermodynamics of the random one-dimensional Ising chain in a transverse field

For three simple one-dimensional disordered models: (a) the Ising chain with random magnetic moments in a transverse field, (b) the Ising chain with random coupling constants in a transverse field, and (c) the X-Y model with a special type of disorder, the asymptotic equivalence in the thermodynamic limit is proved and some of its consequences are discussed. The spectral density of the finite chain for the model (a) is calculated by Dean's method for several representative cases and the presence of the local modes is indicated. The expressions for the initial susceptibilities for the models (a) and (b) are reviewed and (in two cases) the derivations are simplified.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

Approximation to the susceptibilities of three-dimensional antiferromagnets

Approximate susceptibilities of simple cubic S=∞ Heisenberg and $\text{S=}\text{1}\text{2}$ Ising antiferromagnets are calculated by using the new expansion method reported previously for the case of a square-plane S=∞ Heisenberg antiferromagnet. The present method gives a good approximation to the susceptibilities of both of the Heisenberg and Ising antiferromagnets above the Néel temperature.     

Approximation to the susceptibilities of two-dimensional antiferromagnets

Approximate susceptibilities of square-plane S = ∞ XY and $\text{S =}\text{1}\text{2}$ Ising antiferromagnets are calculated by using the new expansion method reported previously for the case of a square-plane S = ∞ Heisenberg antiferromagnet. The convergence of the new expansion is examined and it is shown that the present method gives a good approximation to the susceptibilities of all of the Heisenberg, XY and Ising antiferromagnets even below the temperature range where the result exhibits a pronounced broad maximum.     

Note on the difference between the a-cyclic and c-cyclic version of the XY model

The transverse susceptibility of the c-cyclic version of the one-dimensional XY model with respect to an infinitesimal magnetic field in the x-direction is investigated in more detail. Special attention is paid to the c-cyclic version of the one-dimensional Ising model. The c-cyclic susceptibility χ_xx is evaluated explicitly. The autocorrelation function of the magnetization M_x turns out to be time dependent in the c-cyclic Ising model.     

The isothermal and frequency-dependent perpendicular susceptibilities of the one-dimensional alternating ising model

Exact expressions for the isothermal and frequency-dependent perpendicular susceptibilities are obtained for the alternating Ising model in an external longitudinal field. It is shown that even for vanishing field the isolated perpendicular susceptibility is identical to the isothermal one. It is also obtained the longitudinal wave-vector dependent susceptibility, and the results are compared with the known ones.     

Accurate determination of the interchain exchange in CsCoCl3 · 2H2O from detailed low-temperature susceptibility data

Detailed measurements of the b and c axis susceptibilities of CsCoCl₃ · 2H₂O have been made between 1.5 and 30 K. The results support the canted antiferromagnetic linear chain model previously used to describe this determine the value of the intrachain exchange. The results give J/k = ?52 ± 2 K and g_b = 4.18 ± 0.08. This value for J/k is significantly larger than what other investigators have reported for CsCoCl₃ · 2H₂O and if the Ising model is applicable it will be more accurate because of the simplicity of the analysis.     

Magnetic properties of 2D nano-islands II: Ising spin model with out-of-plane magnetic field

An Ising effective field theory model is presented to calculate the magnetic properties of 2D nano-islands on a nonmagnetic substrate, subject to an externally out-of-plane applied magnetic field. The system Hamiltonian contains nearest neighbor exchange interactions, single-atom magnetic anisotropies, and the Zeeman term. The calculations yield, in particular, the single site spin correlations, the magnetizations, and the isothermal susceptibilities, for the core and periphery domains of the nano-island. The choice of a spin S=1 for the atoms of the system permits the analysis of local spin fluctuations via the single site spin correlations. We investigate in this respect the effects due to the different magnetocrystalline anisotropies and reduced dimensionalities, for the core and periphery domains, and in particular the critical influence of the applied magnetic field. Detailed theoretical results are presented for the square and hexagonal lattice symmetries, with numerical applications for the 2D monolayer Co nano-islands on a Pt substrate. It is shown that the remarkable differences between the magnetic properties of the core and periphery domains in zero field are washed out when an out-of-plane field is applied. The applied field also provokes critical discontinuities for the spin correlations and magnetization reversals, for the core and periphery domains, which are especially evident for the hexagonal lattice nano-island in the range of fields of interest. The discontinuities and magnetization reversals occur over elementary temperature widths, and shift to lower temperatures with increasing field. The field-dependant isothermal susceptibilities show new features very different from those for the susceptibilities in zero field. The present Ising model does not show any blocking temperature transition to superparamagnetism.     

Magnetic properties of 2D nano-islands I: Ising spin model

An Ising spin effective field theory (EFT) is developed as a framework for a detailed analysis of the magnetic properties of two-dimensional (2D) nano-islands on a nonmagnetic substrate with an out of plane magnetization. The Hamiltonian with nearest neighbor exchange interactions and single-atom magnetic anisotropy defines the ground state. The calculation yields the single site spin correlations, the magnetizations, and the isothermal susceptibilities for the core and periphery domains, and the island core phase diagrams. The choice of a spin S=1 for the atoms permits the analysis of the effects of spin fluctuations via the single site spin correlations. In particular we investigate the effects due to the different anisotropies and reduced dimensionalities for the core and periphery domains. The present model calculations are developed for different 2D nano-islands lattices. Detailed theoretical results are presented for the square and hexagonal lattices, with numerical applications for the 2D Co nano-islands on Pt. The derived transition temperature for the hexagonal lattice nano-islands is in good agreement with the experimental data for Co nano-islands on Pt. Though both the core and the periphery domains have the same order-disorder transition temperature, the magnetization of each domain attains this transition differently. The temperature behavior of the spin correlations is also fundamentally different for the periphery and core sites, which entails distinctly different isothermal susceptibilities, and yields statistically averaged nano-islands susceptibilities that do not correspond to a second order phase transition. The experimental susceptibility results for 2D Co nano-islands on Pt can be interpreted within our EFT Ising model without reference to a transition from a blocking state of the particle to a superparamagnetic behavior. The results for the different lattices are formally comparable, and demonstrate the robustness and general character of the model.     

Measurements of magnetic susceptibilities,their anisotropies and crystal field investigations of monoclinic single crystals of Ho2GeO5

Monoclinic single crystals of Ho₂GeO₅ were grown by a flux technique. The molar principal susceptibilities (χ₁ and χ₃) have been measured in the temperature range 300–13.5 K. The principal magnetic anisotropies (χ₁–χ_2,χ₂–χ₃ and χ₁–χ₃) are obtained from the measured anisotropies in different modes of suspensions (b-axis vertical, c-axis vertical and (1 0 0) plane horizontal) of the crystal. The molar principal susceptibility (χ₂) is obtained from the observed values of the principal anisotropies. The observed principal susceptibilities together with their anisotropies are explained by invoking the crystal field interaction of the Ho³⁺ ion with its diamagnetic neighbors in non-axial symmetry. The molar susceptibilities are correlated to the ionic values by finding out the proper orientation of the principal molar susceptibilities, i.e., crystal susceptibilities with respect to that of the ionic susceptibilities (K_x, K_y and K_z). The electronic heat capacities are calculated from the crystal field analysis showing Schottky type anomalies.     

Induced moment antiferromagnet in an external magnetic field—I: Molecular field theory

The behavior of a two-level induced moment antiferromagnet in an external magnetic field is investigated in the molecular field approximation. A significant reduction in the critical field and in the sublattice magnetizations is shown. However, the total magnetization rises more rapidly with field and can remain at large value in an external field even at T = 0. The magnetic susceptibility also remains finite at T = 0 in contrast to the case of a permanent moment Ising antiferromagnet. The effects of a ferromagnetic next-nearest neighbor interaction are then examined. It is shown that, in contrast to the usual Ising antiferromagnets, the ferromagnetic coupling has to exceed a certain value depending on the crystal field strength and the antiferromagnetic interaction, to allow for a first order phase transition in a field to occur even at zero temperature.     

The superparamagnetic response of a Co54Ga46 alloy to low d.c. fields

The low d.c. field susceptibilities and thermoremanence of a Co₅₄Ga₄₆ alloy are presented. A broad peak is observed in the zero field cooled susceptibility at a temperature T_B, which decreases with increasing magnetic field, while the thermoremanence is found to persist to temperatures greater than T_B. The behaviour is discussed in terms of the growth and subsequent blocking of superparamagnetic assemblies.     

Frequency dependence of the complex susceptibility for a spin-1 Ising model

The complex susceptibility or the dynamic susceptibility (χ(ω)=χ′(ω)−iχ″(ω)) for a spin-1 Ising system with bilinear and biquadratic interactions is obtained on the basis of Onsager theory of irreversible processes. If the logarithm of the susceptibilities is plotted as a function of the logarithm of frequency, then the real part (χ′) displays a sequence of plateau regions and the imaginary part (χ″) has a sequence of maxima in the ordered or ferromagnetic phase. On the other hand, only one plateau region in χ′ and one maximum in χ″ is observed in the disordered or paramagnetic phase. Argand or Cole-Cole plots (χ″−χ′) for a selection of temperatures are also shown, and a sequence of semicircles is illustrated in the ordered phase and only one semicircle for the disordered phase in these plots.     

Pinning of spiral fluxons by giant screw dislocations in YBa2Cu3O7 − δ single crystals: Josephson analog of the fishtail effect

By using a highly sensitive homemade AC magnetic susceptibility technique, the magnetic flux penetration has been measured in YBa₂Cu₃O_{7 ? δ} single crystals with giant screw dislocations (having the structure of the Archimedean spirals) exhibiting a = 3 spiral turnings, the pitch b = 18.7 μm and the step height c = 1.2 nm (the last parameter is responsible for creation of extended weak-link structure around the giant defects). The magnetic field applied parallel to the surface enters winding around the weak-link regions of the screw in the form of the so-called spiral Josephson fluxons characterized by the temperature dependent pitch b _f (T). For a given temperature, a stabilization of the fluxon structure occurs when b _f (T) matches b (meaning an optimal pinning by the screw dislocations) and manifests itself as a pronounced low-field peak in the dependence of the susceptibility on magnetic field (applied normally to the surface) in the form resembling the high-field (Abrikosov) fishtail effect.     

Quantum effects in the critical properties of the transverse Ising model for 2 < d ⩽ 3

The crossover behaviour of a d-dimensional (2<d?3) Ising model in a transverse field Г is investigated near the multicritical point $\text{[Г, T] = [Г}{}_{\mathrm{c}}\text{(0), 0].}$ A renormalization scheme which removes divergences in the zero-temperature limit is presented. The crossover exponent and scaling function for the longitudinal susceptibility are found.     

Spin and energy coupling in the Ising model with a transverse field: One dimension,T=∞

The dynamics of the one dimensional Ising model with a transverse field is studied in the limit T = ∞. Numerical studies of the classical chain and exact calculations for the spin $\text{1}\text{2}$ chain indicate the presence of an energy density term in the dynamic spin susceptibility along the direction of the field.     

Clusters and Ising droplets in the antiferromagnetic lattice gas

A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if P_b = 1 - exp(-|K|/2). Along this line these clusters called “droplets” diverge with Ising exponents.