Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL _q (2) and the SL _h (2) quantum groups. The differential geometry of SL _q (2) is well known. In this Letter, we develop the differential geometry of SL _h (2), and show that the space of left invariant vector fields is three-dimensional.     

Left-covariant differential calculi on SLq(2) and SLq(3)

We study (N²−1)-dimensional left-covariant differential calculi on the quantum group SL_q(N) for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on SL_q(2) (which corresponds to Woronowicz' 3D-calculus on SU_q(2)) and two distinguished calculi on SL_q(3) such that the higher-order calculi give the ordinary differential calculus on SL(2) and SL(3), respectively, in the limit q → 1. Two new differential calculi on SL_q(3) are introduced and developed in detail.     

A q-deformed Lorentz algebra

We derive a q-deformed version of the Lorentz algebra by deforming the algebraSL(2,C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified withSL _q(2,C) generateSU _q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limitq→1 the generators are those of the classical Lorentz algebra plus an additionalU(1). Thus we have a deformation ofSL(2,C)×U(1).     

Phase Transition and Level-Set Percolation for the Gaussian Free Field

We consider level-set percolation for the Gaussian free field on ${\mathbb{Z}^{d}}$ , d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h _*(d) satisfies h _*(d) ≥ 0 for all d ≥ 3 and that h _*(3) is finite, see Bricmont et al. (J Stat Phys 48(5/6):1249–1268, 1987). We prove here that h _*(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h _** ≥ h _*, show that h _**(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h _**. Finally, we prove that h _* is strictly positive in high dimension. It remains open whether h _* and h _** actually coincide and whether h _* > 0 for all d ≥ 3.     

Symmetry and lattice-dynamic aspects of structural phase transitions in (CH3NH3)2MnCL4 and related compounds

A group theoretical analysis of the second-order structural phase transition in (CH₃NH₃)₂MnCL₄ at 394°K and of similar transitions in ethyl and propyl compounds (D¹⁷_4h→D¹⁸_2h) was performed. The soft mode transforms according to the τ^x₅-irreducible representation at the X-point of the Brillouin-zone boundary and its eigenvector is discussed. The transition is of the order-disorder type and is caused by a slowing down of the hindered rotation of NH₃-groups. Knowing the symmetry of the order parameter, a thermodynamic potential expansion was constructed and expected anomalies in material constants around the transition temperature are briefly discussed. The high temperature phase transitions in analogous copper compounds are explained as a sequence D¹⁷_4h→D¹⁸_2h→D¹⁵_2h. The second of these phase transitions is driven by a soft mode transforming as the τ^Y₇-representation at the D¹⁸_2h Brillouin zone boundary.     

Dimensional dependences of the interfacial tension at a solid—liquid interface and the melting temperature of metal nanoparticles

New coordinating relationships are obtained for the dimensional dependences of melting temperature T(r) and interfacial tension ?? _SL (r) of a solid spherical nanoparticle at the boundary with its own melt. The Thomson formula for T(r) and the Tolman formula for ?? _SL (r) follow from the expressions obtained at large radii of curvature. Numerical calculations are performed for metals. Results from calculations on the dimensional dependences of the melting temperature for Pt, Au, and Al that conform fairly well with experimental data are given as examples.     

The random-field Ising model with asymmetric bimodal probability distribution

The Ising model, in the presence of a random field, is investigated within the mean-field approximation based on Landau expansion. The random field is drawn from the bimodal probability distribution P(h)=pδ(h−h₀)+(1−p)δ(h+h₀), where the probability p assumes any value within the interval [0,1], asymmetric distribution. The prevailing transitions are of second-order but, for some values of p and h₀, first-order phase transitions take place for smaller temperatures and higher h₀, thus confirming the existence of a tricritical point. Also, the possible reentrant phenomena in the phase diagram (T−h₀ plane) occur for appropriate values of p and h₀. Using the variational principle, we determine the equilibrium equation for magnetization and solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

Phase front motion in uniaxial ferromagnet

It is well known that in the uniaxial ferromagnet in the presence of an external magnetic field perpendicular to the easy axis (h_x) a continuous phase transition occurs for a critical value of this field. There are metastable and stable states if one includes a small field parallel to the easy axis (h_z). The motion of the relaxation front of the metastable state is investigated. It is found that an “interphase wall of Neel-type” exists, its velocity is proportional to h_z and increases when the critical point is approached.     

The role of adsorption of sodium bis(2-ethylhexyl) sulfosuccinate in wetting of glass and poly(methyl methacrylate) surface

Advancing contact angles, θ, for aqueous solutions of the anionic surfactant, sodium bis(2-ethylhexyl) sulfosuccinate (AOT) were measured on glass and poly(methyl methacrylate) (PMMA) surface. Using the obtained results we determined the properties of aqueous AOT solutions in wetting of these surfaces. It occurs that the wettability of glass and PMMA by these solutions depends on the concentration of AOT in solution. There is almost linear dependence between the contact angle (θ) and concentration of AOT (log C) in the range from 5 × 10⁻⁴ to 2.5 × 10⁻³ M/dm³ (value of the critical micelle concentration of AOT—CMC) both for glass and PMMA surface. For calculations of AOT adsorption at solid (glass, PMMA)-solution drop-air system interfaces the relationship between the adhesion tension (γ_LV cos θ) and surface tension (γ_LV) and the Gibbs and Young equations were taken into account. From the measurement and calculation results the slope of the γ_LV cos θ − γ_LV curve was found to be constant and equal 0.7 for glass and −0.1 for PMMA over the whole range of AOT concentration in solution. From this fact it can be concluded that if Γ_SV is equal zero then Γ_SL > 0 for the PMMA-solution and Γ_SL < 0 for glass-solution systems. It means that surfactant concentration excess at PMMA-solution interface is considerably lower than at solution-air interface, but this excess of AOT concentration at glass-solution interface is lower than in the bulk phase. By extrapolating the linear dependence between the adhesion and surface tension the value of the critical surface tension (γ_c) of wetting for glass and PMMA was also determined, that equaled 25.9 and 25.6 mN/m for glass and PMMA, respectively. Using the value of the glass and PMMA surface tension as well as the measured surface tension of aqueous AOT solutions in Young equation, the solid-liquid interface tension (γ_SL) was found. There was a linear dependence between the γ_SL and γ_LV both for glass and PMMA, but there were different slope values of the curves for glass and PMMA, i.e. −0.7 and 0.1, respectively. The dependence between the work of adhesion (W_A) and surface tension (γ_LV) was also linear of different slopes for glass and for PMMA surface.     

Elastic properties of D<Subscript>2</Subscript>O ices in solid-state amorphization and transformations between amorphous phases

Nonequilibrium phase transformations in D₂O ices, including the solid-state amorphization of ice 1h (1h-hda) and the heating-induced transition cascade hda-lda-1c-1h from high-density amorphous (hda) ice to low-density amorphous (lda) ice followed by crystallization in cubic ice 1c and phase transition to ordinary hexagonal ice 1h, were studied using an ultrasonic technique. It has been shown that, as in H₂O ice, the softening of a crystal lattice or an amorphous network precedes nonequilibrium transformations. However, noticeable isotopic differences in the behavior of the elastic properties of H₂O and D₂O, in particular, their 1h and hda modifications, call for a more detailed study of the structural features of these H₂O and D₂O phases.     

Ternary generalization of Pauli’s principle and the <Emphasis Type="Italic">Z</Emphasis><Subscript>6</Subscript>-graded algebras

We show how the discrete symmetries Z ₂ and Z ₃ combined with the superposition principle result in the SL(2,C) symmetry of quantum states. The role of Pauli’s exclusion principle in the derivation of the SL(2,C) symmetry is put forward as the source of the macroscopically observed Lorentz symmetry; then it is generalized for the case of the Z ₃ grading replacing the usual Z ₂ grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally in the case of two generators only, as the symmetry group preserving these forms. The wave equation generalizing the Dirac operator to the Z ₃-graded case is introduced, whose diagonalization leads to a sixthorder equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.     

Propagation of super-Lorentzian beams through a paraxial ABCD optical system

Based on the Collins integral, analytical propagation formulae of super-Lorentzian (SL) SL₀₁, SL₁₀, and SL₁₁ beams passing through a paraxial ABCD optical system are derived by means of the convolution theorem of the Fourier transform. The propagation properties of the SL₀₁ and SL₁₁ beams in free space are graphically illustrated with numerical examples. The power in the bucket of the SL₀₁ and SL₁₁ beams has been also examined in the far-field plane. This research is useful to the applications of super-Lorentzian beams.     

14N NMR in NH4I and NH4Br

¹⁴N magnetic resonance data show that the α → β transition in NH₄I is connected with a discontinuous change in the ¹⁴N chemical shift, Δδ = 20.5 ppm, whereas the order-disorder type β(O_h) → γ(D_4h) transition is accompanied by a small quadropole splitting of the ¹⁴N spectrum. A quadrupole splitting of the ¹⁴N line was as well found in the tetragonal “antiferromagnetically” ordered γ-phase of NH₄Br, but not in the “ferromagnetically” ordered δ(T_d) phase of NH₄Cl.     

Glass formation in the As2Se3-As2Te3-Sb2Te3 system

Chalcogenide glasses from the As₂Se₃-As₂Te₃-Sb₂Te₃ system were synthesized for the first time. The glass-forming region was determined by X-ray diffraction and electron microscopic analyses.The basic physicochemical parameters such as density (d), microhardness (HV) and temperatures of phase transformations (glass transition T_g, crystallization T_cr and melting T_m) were measured. Compactness and some thermomechanical characteristics such as volume (V_h) and formation energy (E_h) of micro-voids in the glassy network as well as the elasticity module (E) were calculated. The glass-forming ability was evaluated according to Hruby's criteria (K_G). The correlation between composition and properties of the (As₂Se₃)_x(As₂Te₃)_y(Sb₂Te₃)_z glasses was established and comprehensively discussed.     

Nonstandard Poincaré and Heisenberg Algebras

Nonstandard deformations of the Poincaré group Fun(P(1+1)) and its dual enveloping algebra U (p(1+1)) are obtained as a contraction of the h-deformed (Jordanian) quantum group Fun( SL _h (2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated.     

Interfacial energies of carbon tetrabromide

The equilibrated grain boundary groove shapes for solid carbon tetrabromide (CTB) in equilibrium with its melt were directly observed by using a horizontal temperature gradient stage. From the observed grain boundary groove shapes, Gibbs–Thomson coefficient (Γ) and solid–liquid interfacial energy (σ_SL) and grain boundary energy (σ_gb) of CTB have been determined to be (7.88 ± 0.8) × 10⁻⁸ K m, (6.91 ± 1.04) × 10⁻³ J m⁻² and (13.43 ± 2.28) × 10⁻³ J m⁻², respectively. The ratio of thermal conductivity of equilibrated liquid phase to solid phase for CTB has also been measured to be 0.90 at its melting temperature. The value of σ_SL for CTB obtained in present work was compared with the values of σ_SL determined in the previous works for same material and it was seen that the present result is in good agreement with previous works.     

A DFT study the solvent effects of H2 adsorption on Cu(h k l) surface

Density functional theory (DFT) combined with conductor-like solvent model (COSMO) have been performed to study the solvent effects of H₂ adsorption on Cu(h k l) surface. The result shows H₂ can not be parallel adsorbed on Cu(h k l) surface in gas phase and only vertical adsorbed. At this moment, the binding energies are small and H₂ orientation with respect to Cu(h k l) surfaces is not a determining parameter. In liquid paraffin, when H₂ adsorbs vertically on Cu(h k l) surface, solvent effects not only influences the adsorptive stability, but also improves the ability of H₂ activation; When H₂ vertical adsorption on Cu(h k l) surface at 1/4 and 1/2 coverage, H-H bond is broken by solvent effects. However, no stable structures at 3/4 and 1 ML coverage are found, indicating that it is impossible to get H₂ parallel adsorption on Cu(h k l) surfaces at 3/4 and 1 ML coverages due to the repulsion between adsorbed H₂ molecules.     

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.     

Hexagonal spin structure of A-phase in MnSi: Densely packed skyrmion quasiparticles or two-dimensionally modulated spin superlattice?

We have studied in detail the A-phase region in the field-temperature (H-T) phase diagram of the cubic heli-magnet MnSi using small angle neutron diffraction. The A-phase revealed itself as a two-dimensional hexagonal pattern of Bragg spots with k _{h(1, 2, 3)} ⊥ H. The directions and magnitudes of the wave vectors k _{h(1, 2, 3)} are well preserved over the whole crystal of the size of 100 mm³, but in the small room of the (H-T) phase diagram just below T _c = 29 K. The droplets of the orientationally disordered, presumably hexagonal, spin structure with k _h ⊥ H are observed in the wide range beyond the A-phase boundaries in the field range from B _T1 ≈ 0.1 T to B _T2 ≈ 0.25 T at temperatures down to 15 K. No melting of these droplets into individual randomly located skyrmions is observed for all temperatures and magnetic fields. The wave vector of two-dimensional modulations k _h is equal to the wave vector of the cone phase k _c . We conclude that observable is a two dimensionally modulated hexagonal spin superlattice built on the same competition of interactions (ferromagnetic exchange and Dzyaloshinskii-Moriya interactions) similar to a case of one-dimensionally modulated simple spin spiral.     

Ground-state phase diagrams of Ising metamagnet in a mixed longitudinal and transverse magnetic field

The ground-state magnetic properties of a two-sublattice Ising metamagnet in a mixed longitudinal and transverse magnetic field are studied within the effective-field theory. A parameter j₂=J₂/J₁ is introduced, which reflects the strength ratio of spin coupling between adjacent planes and in each plane. In addition to the second-order transition lines, the first-order transition lines are also presented, since the ground-state energy can be calculated numerically. The ground-state phase diagrams in h_x–h_z are presented. The results show that when j₂<0 the phase transition of the system is always first-order for h_x<2.751, and when −1000?j₂<0 it is always second-order for h_x>4.36. For the given h_x (0<h_x<14.71), the longitudinal critical magnetic field increases as j₂ decreases. The reentrant phenomenon occurs in the range of j₂<−11.89, h_x>14.71. There is no fourth-order critical point in the phase diagrams given by using EFT as found by using mean field theory (MFT).