Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Some estimates for magnetic fluctuations in a turbulent medium

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B ₀(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b?curlb〉=?B ₀?curlB ₀. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B ₀? curlB ₀/B ₀ ² . The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b ²〉?β/η B ₀ ² for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a ²〉 of the vector potential, rather than 〈b ²〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B ₀ quantities to each other.     

Twinning and short-range order in ordered titanium monoxide

Electron diffraction was used to study the annealed titanium monoxide TiO_1.087 containing the monoclinic ordered phase Ti₅O₅. The diffraction pattern of titanium monoxide in the (112)* _B1 plane of the reciprocal lattice of the parent B1 cubic structure contains not only structural, superstructural, and additional reflections but also a system of planar diffuse strips. It has been established that part of the additional reflections are twins of the superstructure reflections of the monoclinic ordered phase; the twinning plane is the ( $\overline 1 \overline 1 1$ ) plane of the reciprocal lattice of the parent cubic phase. The diffuse scattering contours cover finite plane areas in the reciprocal space characterized by the wave vectors K ₁₀₀ ～ ± (h + 0.07)k ₁₀₀, K ₀₁₀ ～ ±(k + 0.07)k ₀₁₀, and K ₀₀₁ ～ ±(l + 0.07)k ₀₀₁ in the B1 structure. The diffuse scattering is caused by short-range displacement order. Short-range substitution order and the corresponding diffuse scattering are absent.     

Decay of helicity in homogeneous turbulence

The self-similar relaxation of helicity in homogeneous turbulence has been considered taking into account integral invariants ∫ ₀ ^∞ r ^m 〈u(x)ω(x + r)〉 dr = I _m ^h (where ω = curlu and r = |r|). It has been shown that integral invariants with m = 3 for both helicity and energy are possible in addition to helical analogs of Loitsyanskii (m = 4) and Birkhoff-Saffman (m = 2) invariants associated with the conservation laws of momentum and angular momentum, respectively. Helicity always relaxes more rapidly than the energy. Its decay exponent is in the interval from ?3/2 to ?5/2 versus the interval from ?6/5 to ?10/7 for the energy.     

Magnetic phase diagrams of the CrB- and FeB-type HoSi compounds

The temperature magnetic phase diagrams of the dimorphic HoSi compound were studied by neutron diffraction. The sample comprises 35.5% CrB- (Cmcm) and 64.5% FeB-type (Pnma) of structure. Both phases order antiferromagnetically below T_N=25 K and undergo first-order magnetic transitions at T_ic=16.5 K. Their T-phase diagrams comprise a low temperature (LT) 2.7 K−T_ic and a high temperature (HT) range T_ic−T_N with distinct wave vectors.The LT magnetic ordering of the CrB-type HoSi with the wave vector q₁=(1/2, 0, 1/2) corresponds to a uniaxial magnetic structure, with the Ho moments along the shortest axis c. At 2.7 K the ordered moment value is 8.6(2) μ_B/Ho atom. The HT ordering, described by the wave vector q₂=(q_2x, 0, q_2z) with a T-variable length, corresponds to an amplitude modulated structure.The magnetic ordering of the FeB-type HoSi requires two symmetry independent vectors q₃=(0, q_3y, q_3z) for the LT- and q₄=(q_4x, q_4y, 0) for the HT range. Both vectors correspond to sine wave modulated structures with the Ho magnetic moments confined along the shortest axis b. The q₃ vector has an almost invariable length vs. T close to ≈(0, 9/17, 1/11). At 2.7 K the amplitude of the wave is 10.9(1) μ_B/Ho atom. At T_icq₃ jumps to the wave vector q₄=(q_4x, q_4y, 0) with a T-variable length. At 17 K q₄=(0.092(1), 0.538(3), 0). Around T_ic there is a narrow coexistence range of the q₃ and q₄ competing phases. Various models are discussed and compared with the isomorphic RSi (R=rare earth) compounds counterparts of HoSi, a comparison that has led us to briefly review the magnetic structures available in the literature for this interesting class of compounds.     

Magnetostriction and thermal expansion of HoAl3(BO3)4

The magnetostriction and thermal expansion of rare-earth aluminoborate HoAl₃(BO₃)₄ have been studied theoretically. The calculated field and temperature dependences of the multipole moments of the Ho³⁺ ion in HoAl₃(BO₃)₄ made it possible to describe the known experimental data and to predict possible anomalies of thermal expansion. It has been shown that, for the direction of the field B ‖ c, the nonmonotonic character of magnetostriction along the axis a is determined by the multipole moments, the main of which is β _J 〈O ₄ ⁰ 〉. For B ‖ a and B ‖ b, the maximum moments are β _J 〈O ₄ ² 〉and α _J 〈O ₂ ² 〉; their variation with the field and temperature explain well the form of magnetostriction. It has been established that the greater value of magnetostriction Δa/a for B ‖ b than for B ‖ a and the greater value of magnetostriction for the field in the basal plane than for B ‖ c are caused by greater variations in the field of actual multipole moments.     

Probability Law for the Euclidean Distance Between Two Planar Random Flights

We consider two independent symmetric Markov random flights Z ₁(t) and Z ₂(t) performed by the particles that simultaneously start from the origin of the Euclidean plane $\mathbb{R}^{2}$ in random directions distributed uniformly on the unit circumference S ₁ and move with constant finite velocities c ₁>0, c ₂>0, respectively. The new random directions are taking uniformly on S ₁ at random time instants that form independent homogeneous Poisson flows of rates λ ₁>0, λ ₂>0. The probability distribution function $\varPhi(r,t)= \operatorname{Pr} \{ \rho(t)<r \}$ of the Euclidean distance $$\rho(t)=\big\Vert \mathbf{Z}_1(t) - \mathbf{Z}_2(t) \big\Vert , \quad t>0, $$ between Z ₁(t) and Z ₂(t) at arbitrary time instant t>0, is derived. Asymptotics of Φ(r,t), as r→0, and a numerical example are also given.     

Exact solution of the Boltzmann equation in the relaxation approximation,existence of negative differential conductivity for certain types of energy bands and its implication for the fluctuation spectrum and the noise temperature

The Boltzmann equation for the distributionf _k of a system of charged particles obeying classical statistics in a uniform fieldF, $$\frac{{\partial f_k }}{{\partial t}} + F\frac{{\partial f_k }}{{\partial k}} = \smallint d^3 k'(W_{kk'} f_{k'} - W_{k'k} f_k ),$$ will be solved analytically for a special class of transition ratesW _kk′=const·h _k ·ν _k ·ν _k′ for any initial distribution.h _k is the Maxwell distribution andν _k >0 can be interpreted as ak-dependent relaxation frequency. The constant relaxation approximation (ν _k =ν) will be used to discuss the drift velocitiesu for all the fields and temperaturesT for certain types of band structuresE(k). Bands with lineark-dependence for largek give rise to drift velocities saturating for large fields. For bands with the periodicity of the reciprocal lattice, the zero drift-theorem has been proved. It states that $$\mathop {\lim }\limits_{F \to \infty } u (F,T) = \mathop {\lim }\limits_{T \to \infty } u (F,T) = 0$$ for all the periodic band structures. This theorem is even correct for a generalW _kk′ if certain restrictions are made. Finally, making use of the Markov character of the conditional probability (Green's function) solution of the Boltzmann equation, the velocity fluctuation spectrumS is calculated forE(k)=A(1?cosa k). It will be shown thatS(F, T, 0) remains positive for the critical field and all temperatures, and therefore the noise temperature diverges on approaching the critical field.     

Mössbauer absorption by soft ferromagnets in radio-frequency magnetic field

The simultaneous influence of periodical magnetic field reversals at the nucleus between the values ±h ₀ and of magnetostrictive vibrations on the shape of the Mössbauer absorption spectrum is analyzed. The effect of a constant external magnetic field is taken into account by assuming unequal durations of the states ?h ₀ and +h ₀. It is shown that such asymmetric reversals of the magnetic field lead to splitting of the absorption lines into Zeeman patterns corresponding to the time-averaged magnetic field h ₀ R, where R is the asymmetry parameter of the reversals. The calculations agree well with experiment.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

Quantum Bundle Description of Quantum Projective Spaces

We realise Heckenberger and Kolb??s canonical calculus on quantum projective (N ? 1)-space C _q [C p ^N?1] as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group C _q [SU _N ]. We introduce a calculus on the quantum sphere C _q [S ^2N?1] in the same way. With respect to these choices of calculi, we present C _q [C p ^N?1] as the base space of two different quantum principal bundles, one with total space C _q [SU _N ], and the other with total space C _q [S ^2N?1]. We go on to give C _q [C p ^N?1] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb??s calculus as an associated vector bundle to the principal bundle with total space C _q [SU _N ]. Finally, we construct strong connections for both bundles.     

A new type of peak effect in the magnetization of anisotropic superconductors

Experimental and theoretical studies of the magnetization curves M(_H) obtained for plates of an anisotropic hard superconductor are performed. The samples are YBCO single-domain textured crystals with the c axis lying in the sample plane. It is shown that, for some orientations of the magnetic field in the sample plane, the magnetization curves contain additional extrema; at the same time, the virgin magnetization curve M(H) exhibits features characteristic of the peak effect. This proves that the anisotropy of the current-carrying ability can give rise to a new type of peak effect.     

The discovery of 3-(1-aminoethylidene)quinoline-2, 4(1H,3H)-dione derivatives as novel PSII electron transport inhibitors

Based on the structures of the 4-hydroxyphenylpyruvate dioxygenase inhibitor mesotrione and natural product fischerellin A, a series of imine derivatives of ( $E$ )-3-acyl-quinoline-2,4(1H,3H)-dione (6, 12 and 16) were designed, synthesized and systematically evaluated for their herbicidal activity. The bioassay results indicated that most of the synthesized compounds displayed good to excellent herbicidal activity, of which 6e, 6g, 6h, 6q and 6t exhibited more than 50 % inhibition against Brassica napus L., Amaranthus retroflexu or Digitaria adscendens at a dosage of $94\,\hbox {g}\,\hbox {ha}^{-1}$ or lower. The symptom of injured leaves in vivo, the high Hill reaction inhibitory activity of 6h in vitro ( $\hbox {IC}_{50}\,0.1\, \upmu \hbox {g}\,\hbox {mL}^{-1})$ and the computer-based binding model of compound 6h with D1 protein in photosystem II (PSII) reaction centre suggest this novel structure to likely be a new type of PSII electron transport inhibitor. Thus, we have found a novel type of diketone enamine structure targeted at the PSII reaction centre.     

Anisotropic resonant magnetoplasticity of NaCl crystals in the Earth’s magnetic field

Resonant relaxation of the dislocation structure under the action of crossed magnetic fields, i.e., constant magnetic field of the Earth (B _Earth) and alternating radio-frequency field ( $\tilde B$ ), has been experimentally studied in a series of dielectric (NaCl) crystals with various compositions of impurities under variations in the frequency, direction of the pumping field $\tilde B$ , and orientation of the samples in the Earth’s magnetic field. The frequency dependence of the dislocation path length l(ν) exhibits peaks with various heights (l _max) and resonant frequencies (ν_res). The maximum resonant effect has been observed for dislocations with the direction L orthogonal to the plane of crossed magnetic fields in a configuration of mutually perpendicular vectors {L, $\tilde B$ , B _Earth} belonging, together with sample edges {a, b, c}, to the 〈100〉 system. Variation of the concentration C of calcium impurity in crystals of the NaCl_Ca series only influenced the resonant peak height as $l_{\max } \propto 1/\sqrt C $ . Rotation of the magnetic field $\tilde B$ in the (b, c) plane from direction $\tilde B$ ⊥ B _Earth to $\tilde B$ ∥ B _Earth also did not influence the frequency of the resonance but changed its amplitude. Depending on the crystal type, this influence changed from rather insignificant (in crystals of the NaCl_LOMO series) to complete suppression of the effect for $\tilde B$ ∥ B _Earth (in the NaCl_Nik series). The resonant frequency ν_res is sensitive to orientation of the sample with respect to B _Earth. Upon rotation of the crystal by the angle θ = ∠(c, B _Earth) about the a ⊥ B _Earth edge, the initial peak for dislocations L ‖ a at the crystal orientation θ = 0 and the frequency ν _res ⁰ is replaced by a pair of peaks at frequencies ν_{1, 2} ≈ ν _res ⁰ cosθ_{1, 2}, where θ₁ = 90° ? θ and θ₂ = θ. Previously, these peaks were observed separately in NaCl_Nik crystals for $\tilde B$ ∥ c and $\tilde B$ ∥ b. In the present study, these peaks have been observed simultaneously for both orientations of $\tilde B$ in NaCl_LOMO and NaCl_Ca crystals, where the resonance is not completely suppressed for $\tilde B$ ∥ B _Earth.     

Quark jet as a trigger for gluon jet in deep inelastic muon scattering

We calculate, to first order in QCD, the probability for a fast hadron observed at laboratory angle θ _h from the virtual photon directionq, to originate from the quark of a quark-gluon final state. Given this probability, the average gluon energy and angle with respect toq are then calculated. This approach provides a method for gluon jet direction in forthcoming experiments.     

Bloch electron dynamics

New equations of motion for a Bloch electron [momentum p=h k,energy ε _n(p),zone number n, charge -e]: $$m_j \frac{{dv_j }}{{dt}} = - e(E + v \times B)_j $$ are proposed, where v≡?ε_n(p)/?p is the velocity, and {m_j}are the principal masses m _j ^? 1=?²ε_n/?p _j ² along the normal and the two principal axes of curvatures at each point of the constant-energy surface represented by ε=ε_n(p).Their advantages over the prevalent equations of motion where the left-hand-side is replaced by hk _j are demonstrated by examining de Haas-van Alphen oscillations and orientation-dependent cyclotron resonance peaks.     

Magneto-optical signature of broken mirror symmetry of two-dimensional conductors

An unusual aspect of macroscopic electrodynamics of two-dimensional mirror-odd conducting structures bound up with the band spin–orbit coupling H _so = α(p × c) · σ of current carriers (where c is one of two none-quivalent normals to a given structure) is pointed out. Namely, it is shown that due to the spin–orbit coupling the presence of the in-plane magnetic field H ₀ gives rise to a dependence of the reflection/transmission amplitudes on the structure orientation c, the wave vector of the incident radiation q, and H ₀ of the form q · (c × H ₀). This q- and H ₀-odd dependence can be the foundation of the optical way to determine the value of the spin–orbit coupling α.     

Effect of renormalization of magnetic fluctuations on the kinetic coefficients of high-T c superconductors

Temperature dependences of resistivity, ρ(T), and Hall coefficient, R _H (T), in a 2D doped antiferromagnet are studied for various forms of the dynamic spin susceptibility X(q, θ) (in the mean-field approximation, taking into account attenuation and renormalization of the magnetic excitation spectrum θ_q, and for so-called strongly overdamped magnons). Doped CuO₂ planes in cuprates are considered in the one-band model of the Kondo lattice. Charge carrier scattering anisotropy, which strongly depends on temperature, is taken into account using the density matrix formalism and seven-moment approximation for the nonequilibrium distribution function. It is shown that the behavior of ρ(T) and R _H (T) is completely determined by the renormalization θ_q → $\omega _q \to \tilde \omega _q $ of the spin wave spectrum (the renormalization is essentially controlled by the fulfillment of the sum rule for X(q, θ) and by the strong temperature dependence of the gap δ(T). The resultant ρ(T) and R _H (T) dependences match the experimental data for optimally doped high-T _c superconductors.     

On the temperature behavior of second sound and Poiseuille flow

The linearized Peierls equation for the phonon densityN (k λ,r t) is solved by replacing the collision operator in the subspace orthogonal to the collision invariants byk-dependent relaxation rates. For the normal process relaxation time the behaviorτ _N (k λ)∝|k|^?p for smallk is assumed. Taking into account thisk-dependence ofτ _N explicitly and avoiding an expansion with respect toΩτ _N (kλ) before performing the necessary integration overk yields new, non-analytic, terms in the hydrodynamic equations describing second sound and Poiseuille flow. It is shown that this may lead to a temperature dependence of second sound damping and thermal conductivity in the Poiseuille flow region differing from the usual theoretical predictions and in better agreement with experiments.