Three-dimensional Ising Systems Research Articles

Phase diagrams of semi-infinite systems by renormalization group theory and Monte Carlo simulation

In this work, we used the Migdal–Kadanoff renormalization group method to investigate two three-dimensional semi-infinite Ising systems. In the first one, the surface and bulk sites are occupied by spin-3/2 and spin-1/2 respectively, whereas those of the second one are occupied by spin-1/2 on the surface and spin-3/2 in the bulk. Based on the ratio of bulk and surface exchange interactions, we explored different topologies of phase diagrams showing various second order phase transitions, namely ordinary, extraordinary, surface and special transitions. We also found that both systems exhibited first order phase transitions, multicritical points and critical end-points. The existence of a first order phase transition at low temperatures was confirmed by plotting the derivative of the free energy. The Monte Carlo simulation was also used to verify and compare the results obtained by the renormalization group for the two semi-infinite systems.

Experimental critical dynamics of 3-methyl pyridine/D2O mixtures without and with antagonistic salt

We observed criticality in the structure and dynamics of a three-dimensional (3D) and two-dimensional (2D) Ising system consisting of 3-methyl pyridine $\text{(3MP)}/{\mathrm{D}}_{2}\mathrm{O}$ without and with antagonistic salt. We could describe both dynamic criticalities by the Kawasaki crossover function. The dynamic critical exponent was $z=0.063\ifmmode\pm\else\textpm\fi{}0.020$ and $0.005\ifmmode\pm\else\textpm\fi{}0.019$ for three and two dimensions, which confirms earlier observations in the 3D case and confirms expectations in the 2D case. The amplitudes of the critical dynamics are governed by the bare viscosities experimentally, and by the coefficient $R$ theoretically [the latter is proportional to ${(4\ensuremath{-}d)}^{\ensuremath{-}1}$ with the dimensionality $d$]. This finding is in accordance with the lubrication effect [N. Gov, A. G. Zilman, and S. Safran, Phys. Rev. E 70, 011104 (2004)], which is also connected to lamellar systems of the Brazovskii criticality [M. Gvaramia et al., Colloid Polym. Sci. 297, 1507 (2019)]. This lubrication effect is tightly connected to a laminar flow enforced by the domain structure, and it also holds for our 2D Ising system. The experimental techniques employed were small-angle neutron and x-rays scattering for the static criticality, and dynamic light scattering and neutron spin echo spectroscopy for the critical dynamics. Furthermore, the criticality of the viscosities was measured.

RKKY-type in-plane ferromagnetism in layered Mn1/4NbS2 single crystals

Placing the layers of the $3d$ magnetic elements in the van der Waals gap pulls the Nb/Ta-based transition metal dichalcogenides family from a nonmagnetic to a magnetic zone. However, the magnetic spectrum is strongly sensitive to the specific combination of host lattice and type of intercalated magnetic element and thus each combination needs a proper explicit study. In this context, we systematically investigate the magnetization of the Mn-intercalated layered ferromagnet ${\mathrm{Mn}}_{1/4}{\mathrm{NbS}}_{2}$ single crystal. Magnetization measurement revealed paramagnetic to ferromagnetic phase transition at around 105 K and an easy in-plane ($H\ensuremath{\perp}c$) significantly high magnetic anisotropy. The in-plane magnetic anisotropic energy (MAE) evaluated via the noncollinear first-principle approach is $\ensuremath{\sim}\ensuremath{-}600 \ensuremath{\mu}\mathrm{eV}$ per unit cell. The position of the Fermi level in between the spin-up and spin-down Mn states suppresses the positive contribution in the overall MAE, which lifts the overall MAE negative (in plane) with a larger size in ${\mathrm{Mn}}_{1/4}{\mathrm{NbS}}_{2}$. The comprehensive analysis of magnetization isotherms measured in the vicinity of ${T}_{c}$ with magnetic field $H$ applied parallel to the $ab$ plane ($H\ensuremath{\perp}c$) yields the asymptotic critical exponents $\ensuremath{\beta}=0.3251(2)$ and $\ensuremath{\gamma}=1.2(1)$, and $\ensuremath{\delta}=4.691(1)$. These critical exponents fulfill the scaling relation and scaling equation of the magnetic state predicted by the scaling theory. The determined critical exponents agree well with those obtained from the results of the renormalization group theory approach for a three-dimensional Ising system coupled with a long-range interaction between spins decaying as $J(r)\ensuremath{\approx}{r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$ with $\ensuremath{\sigma}$ = 1.84. The Ising-type spin state in ${\mathrm{Mn}}_{1/4}{\mathrm{NbS}}_{2}$ is attributed to the higher size of MAE. Our theoretical analysis shows that indirect exchange interaction between Mn-Mn along the $c$ axis, mediated via spins' polarized conduction electrons of Nb atoms, a Ruderman-Kittel-Kasuya-Yoshida (RKKY) type ferromagnetic interaction, dominates over the direct in-plane Mn-Mn exchange interaction (less favorable due to large ${d}_{\text{Mn-Mn}}\ensuremath{\sim}6.67\phantom{\rule{4pt}{0ex}}\stackrel{\ifmmode \mathring{}\else \r{}\fi{}}{\mathrm{A}}$) and is thus responsible for the macroscopic magnetization in ${\mathrm{Mn}}_{1/4}{\mathrm{NbS}}_{2}$. The RKKY-type ferromagnetic interaction in ${\mathrm{Mn}}_{1/4}{\mathrm{NbS}}_{2}$ arises due to the large spin polarization of the conduction electrons associated with the Nb atoms at the Fermi energy and existence of unoccupied Mn $3d$ states well above the Fermi level.

Novel critical behavior of magnetization in URhSi: Similarities to the uranium ferromagnetic superconductors UGe2 and URhGe

We study the critical behavior of dc magnetization in the uranium ferromagnet URhSi around the paramagnetic to ferromagnetic phase transition at ${T}_{\mathrm{C}}\ensuremath{\sim}10$ K with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis, and the scaling analysis. URhSi is isostructural to uranium ferromagnetic superconductors URhGe and UCoGe. The critical exponent $\ensuremath{\beta}$ for the temperature dependence of the spontaneous magnetization below ${T}_{\mathrm{C}}$, $\ensuremath{\gamma}$ for the magnetic susceptibility, and $\ensuremath{\delta}$ for the magnetic isotherm at ${T}_{\mathrm{C}}$ in URhSi have been determined as $\ensuremath{\beta}=0.300\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\delta}=4.38\ifmmode\pm\else\textpm\fi{}0.04$ by the scaling analysis and the critical isotherm analysis. These critical exponents fulfill the Widom scaling law $\ensuremath{\delta}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1+\phantom{\rule{0.16em}{0ex}}\ensuremath{\gamma}/\ensuremath{\beta}$. Magnetization has strong uniaxial magnetic anisotropy in the ferromagnetic state of URhSi. However, the universality class of the ferromagnetic transition does not belong to the three-dimensional Ising system with short-range exchange interactions between magnetic moments ($\ensuremath{\beta}=0.325$, $\ensuremath{\gamma}=1.241$, and $\ensuremath{\delta}=4.82$). The obtained exponents in URhSi are similar to those in the uranium ferromagnetic superconductors ${\mathrm{UGe}}_{2}$ and URhGe, and uranium ferromagnets UIr and $\mathrm{U}({\mathrm{Co}}_{0.98}{\mathrm{Os}}_{0.02}$)Al. We have previously reported the unconventional critical behavior of magnetization in the uranium ferromagnetic superconductors [N. Tateiwa et al., Phys. Rev. B 89, 064420 (2014)]. The universality class of the ferromagnetic transition in URhSi may belong to the same one in the uranium ferromagnetic superconductors and the uranium ferromagnets. The unconventional critical behavior of the magnetization in the uranium compounds cannot be understood with previous theoretical interpretations of critical phenomena. The absence of the superconductivity in URhSi is discussed from several viewpoints. The improvement of the sample quality in URhSi could provide a good opportunity to gain a deeper understanding of the ferromagnetic superconductivity in the uranium ferromagnets.

Critical behavior of magnetization in URhAl: Quasi-two-dimensional Ising system with long-range interactions

The critical behavior of dc magnetization in the uranium ferromagnet URhAl with the hexagonal ZrNiAl-type crystal structure has been studied around the ferromagnetic transition temperature T_C. The critical exponent beta for the temperature dependence of the spontaneous magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isotherm at T_C have been obtained with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. We have determined the critical exponents as beta = 0.287 +- 0.005, gamma = 1.47 +- 0.02, and delta = 6.08 +- 0.04 by the scaling analysis and the critical isotherm analysis. These critical exponents satisfy the Widom scaling law delta=1+gamma/beta. URhAl has strong uniaxial magnetic anisotropy, similar to its isostructural UCoAl that has been regarded as a three-dimensional (3D) Ising system in previous studies. However, the universality class of the critical phenomenon in URhAl does not belong to the 3D Ising model (beta = 0.325, gamma = 1.241, and delta = 4.82) with short-range exchange interactions between magnetic moments. The determined exponents can be explained with the results of the renormalization group approach for a two-dimensional (2D) Ising system coupled with long-range interactions decaying as J(r)~r^-(d+sigma) with sigma = 1.44. We suggest that the strong hybridization between the uranium 5f and rhodium 4d electrons in the U-Rh_I layer in the hexagonal crystal structure is a source of the low dimensional magnetic property. The present result is contrary to current understandings of the physical properties in a series of isostructural UTX uranium ferromagnets (T: transition metals, X: p-block elements) based on the 3D Ising model.

Self-Induced Glassiness and Pattern Formation in Spin Systems Subject to Long-Range Interactions.

We study the glass formation in two- and three-dimensional Ising and Heisenberg spin systems subject to competing interactions and uniaxial anisotropy with a mean-field approach. In three dimensions, for sufficiently strong anisotropy the systems always modulate in a striped phase. Below a critical strength of the anisotropy, a glassy phase exists in a finite range of temperature, and it becomes more stable as the system becomes more isotropic. In two dimensions the criticality is always avoided and the glassy phase always exists.

Surface phase transitions of the transverse Ising model with modified surface–bulk coupling

The phase diagram of the three-dimensional transverse Ising system with antiferromagnetic intrasurface J S and surface–bulk J ⊥ interactions modified with respect to the ferromagnetic bulk exchange interaction J B is studied by the use of an effective field method in the framework of a single-site cluster theory. In this investigation, the state equations are derived by means of differential operator technique. Using these coupled equations within the four-layer approximation, it has been shown that the new parameter J ⊥ strongly affects the phase diagram obtained recently in the case J ⊥ = J B. Indeed, our calculations reveal some qualitatively interesting features both in the absence and in the presence of the surface and bulk transverse fields.

Dynamical transitions of a driven Ising interface

We study the structure of an interface in a three-dimensional Ising system created by an external nonuniform field H(r,t) . H changes sign over a two-dimensional plane of arbitrary orientation. When the field is pulled with velocity v(e) , [i.e., H(r,t)=H(r-v(e)t) ], the interface undergoes several dynamical transitions. For low velocities it is pinned by the field profile and moves along with it, the distribution of local slopes undergoing a series of commensurate-incommensurate transitions. For large v(e) the interface depins and grows with Kardar-Parisi-Zhang exponents.

First excitations in two- and three-dimensional random-field Ising systems

We present results on the first excited states for the random-field Ising model. These arebased on an exact algorithm, with which we study the excitation energies and theexcitation sizes for two- and three-dimensional random-field Ising systems with a Gaussiandistribution of the random fields. Our algorithm is based on an approach of Frontera andVives which, in some cases, does not yield the true first excited states. Using the correctedalgorithm, we find that the order–disorder phase transition for three dimensions isvisible via crossings of the excitation energy curves for different system sizes,while in two dimensions these crossings converge to zero disorder. Furthermore,we obtain in three dimensions a fractal dimension of the excitation cluster ofds = 2.42(2). We also provide analytical droplet arguments to understand the behavior of theexcitation energies for small and large disorder as well as close to the criticalpoint.

Critical sound attenuation of three-dimensional Ising systems

The field-theoretic description of dynamic critical effect s of the disorder on acoustic anomalies near the temperature of second-order phase transition is considered for three-dimensional Ising-like systems. The calculations of sound attenuation in high temperature phase for pure and diluted Ising-like systems near the critical point are presented. The dynamic scaling function for critical attenuation coefficient is calculated. The effect of the quenched disorder on asymptotic behaviour of the critical ultrasonic anomalies is discussed.

Microcanonical analysis of small systems

The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.

Critical Casimir Effect in Three-Dimensional Ising Systems: Measurements on Binary Wetting Films

The critical Casimir force (CF) is observed in thin wetting films of a binary liquid mixture close to the liquid/vapor coexistence. X-ray reflectivity shows thickness (L) enhancement near the bulk consolute point. The extracted Casimir amplitude Delta(+-)=3+/-1 agrees with the theoretical universal value for the antisymmetric 3D Ising films. The onset of CF in the one-phase region occurs at L/xi approximately 5 regardless of whether the bulk correlation length xi is varied with temperature or composition. The shape of the Casimir scaling function depends monotonically on the dimensionality.

Nonequilibrium dynamical phase transition of 3D kinetic Ising/Heisenberg spin system

We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase \linebreak boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. TTCP=1.33×exp(−ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H0, reduced temperature T/TC and the frequency ω as τ = C.ωα.H0−β.(T/TC)−γ.

Phase Transitions in Three Dimensional Ising Systems under Free and Periodic Boundary Conditions

The Ising model (and also the Ising model in a transverse field) has been used extensively to investigate ferroelectric phase transitions. Numerical Monte Carlo simulations of phase transitions in three-dimensional Ising systems with variable size (2 ≤ L ≤ 100) have been performed. The susceptibility maximum and peak temperature Tc 3D(L,Lc) for free and periodic boundary conditions have been determinated and have been observed to be well fitted by simple expressions as a function of L. For L → ∞, as expected [Tc 3D]fbc = [Tc 3D]pbc being fbc = free boundary conditions and pbc = periodic boundary conditions. Scaling plots MLβ/v vs |ϵ|Ll/v for larger L with both boundary conditions were obteined and compared. The scaling function for our data up to L ≈ 100 was examined closely and an excellent fit at T → Tc was obtenied.

Bound states and glueballs in three-dimensional Ising systems

We study the spectrum of massive excitations in three-dimensional models belonging to the Ising universality class. By solving the Bethe–Salpeter equation for 3D φ 4 theory in the broken symmetry phase we show that recently found non-perturbative states can be interpreted as bound states of the fundamental excitation. We show that duality predicts an exact correspondence between the spectra of the Ising model in the broken symmetry phase and of the Z 2 gauge theory in the confining phase. The interpretation of the glueball states of the gauge theory as bound states of the dual spin system allows us to explain the qualitative features of the glueball spectrum, in particular, its peculiar angular momentum dependence.

NONCOMMUTATIVE GEOMETRY, GEOMETRICAL ASPECTS OF SKYRMIONS AND STRING THEORY

It is pointed out here that noncommutative geometry having the space–time manifold X = M4× Z2leads to the description of a massive fermion as a skyrmion when the discrete space–time is incorporated as an internal variable. The inherent anisotropic feature of the internal space gives rise to the Skyrme term as well as the Wess–Zumino term. An array of such skyrmions may be considered as a three-dimensional Ising system. It is shown that the composite system of such skyrmions may be viewed as the (3 + 1)-dimensional relative of the Polyakov string theory when the associated Liouville field corresponds to the attachment of topological fixtures like vortex lines at the end points of an open string. These two systems are found to satisfy certain duality principles through conformal duality which are similar to T duality and S duality in conventional string picture.

Five-loop renormalization-group expansions for the three-dimensionaln-vector cubic model and critical exponents for impure Ising systems

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pade-Borel-Leroy resummation under n = 0. These exponents are found to be: \gamma = 1.325 +/- 0.003, \eta = 0.025 +/- 0.01, \nu = 0.671 +/- 0.005, \alpha = - 0.0125 +/- 0.008, \beta = 0.344 +/- 0.006. For the correction-to-scaling exponent, the less accurate estimate \omega = 0.32 +/- 0.06 is obtained.

Effects of Shape and Boundary Conditions on Finite-Size Scaling Functions for Anisotropic Three-Dimensional Ising Systems

Effects of Shape and Boundary Conditions on Finite-Size Scaling Functions for Anisotropic Three-Dimensional Ising Systems

Criticality of D=2 and D=3 Ising models: cluster structure versus populations.

The energy and the specific heat of two- and three-dimensional Ising systems are analyzed in terms of cluster properties. The energy and the specific heat are decomposed into two components, which are defined by quantities pertaining to cluster populations and cluster structure expressed in terms of average cluster perimeters. It is shown that the structural component of the energy as well as of the specific heat represents the dominant contribution. Indications are presented that the critical exponent of structural and populational components of specific heat matches the exponent of the entire specific heat.

Self-averaging of random and thermally disordered diluted Ising systems.

Self-averaging of singular thermodynamic quantities at criticality for randomly and thermally diluted three-dimensional Ising systems has been studied by the Monte Carlo approach. Substantially improved self-averaging is obtained for critically clustered (critically thermally diluted) vacancy distributions in comparison with the observed self-averaging for purely random diluted distributions. Critically thermal dilution, leading to maximum relative self-averaging, corresponds to the case when the characteristic vacancy ordering temperature (theta) is made equal to the magnetic critical temperature for the pure three-dimensional (3D) Ising systems (T(3D)(c)). For the case of a high ordering temperature (theta>>T(3D)(c)), the self-averaging obtained is comparable to that in a randomly diluted system.