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.

     

Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction

The uniqueness and the global Markov property for the regular Gibbs measure corresponding to the interaction $$U_\Lambda (\varphi ): = \lambda \int\limits_\Lambda {d_2 x\int {d\varrho (\alpha ):e^{\alpha \varphi } :_0 (x)} } $$ [forλ>0,d?(α) a probability measure with support in \(( - 2\sqrt {\pi ,} 2\sqrt \pi )\) ] is proved.     

An inequality for Hilbert-Schmidt norm

For the absolute value |C|=(C*C)^1/2 and the Hilbert-Schmidt norm ∥C∥_HS=(trC*C)^1/2 of an operatorC, the following inequality is proved for any bounded linear operatorsA andB on a Hilbert space $$|| |A|---|B| ||_{HS} \leqq 2^{1/2} ||A - B||_{HS} $$ . The corresponding inequality for two normal state ? and ψ of a von Neumann algebraM is also proved in the following form: $$d(\varphi ,\psi ) \leqq ||\xi (\varphi ) - \xi (\psi )|| \leqq 2^{1/2} d(\varphi ,\psi )$$ . Here ξ(χ) denotes the unique vector representative of a state χ in a natural positive coneP ^? forM, andd(?, ψ) denotes the Bures distance defined as the infimum (which is also the minimum) of the distance of vector representatives of ? and ψ. In particular, $$||\xi (\varphi _1 ) - \xi (\varphi _2 )|| \leqq 2^{1/2} ||\xi _1 - \xi _2 ||$$ for any vector representatives ξ _j of ? _j ,j=1, 2.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Laser-Rf double-resonance studies of the hyperfine structure of metastable atomic states of55Mn

The hyperfine structure (hfs) of the metastable atomic states 3d⁶4s⁶ D _{1/2, 3/2, 5/2, 7/2, 9/2} of⁵⁵Mn was measured using theABMR- LIRF method (atomicbeammagneticresonance, detected bylaserinducedresonancefluorescence). The hfs constantsA andB, corrected for second order hfs perturbations, could be derived from these measurements. The theoretical interpretation of these correctedA- andB-factors was performed in the intermediate coupling scheme taking into account the configurations 3d ⁵4s ², 3d ⁶4s and 3d ⁷. Examining the influence of the composition of the eigenvectors on the hfs parameters \(\left\langle {r^{ - 3} } \right\rangle ^{k_s k_l } \) it was found, that for the configuration 3d ⁶4s the two-body magnetic interaction should be considered in the calculation of the eigenvectors. Investigating second order electrostatic configuration interactions and relativistic effects and using calculated relativistic correction factors we obtained for the nuclear quadrupole moment of the nucleus⁵⁵Mn a value ofQ=0.33(1) barn, which is not perturbed by a shielding or antishielding Sternheimer factor. The following hfs constants have been obtained: $$\begin{gathered} A\left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 882.056\left( {12} \right)MHz \hfill \\ A\left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 469.391\left( 7 \right)MHzB\left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = - 65.091\left( {50} \right)MHz \hfill \\ A\left( {{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 436.715\left( 3 \right)MHzB\left( {{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = - 46.769\left( {30} \right)MHz \hfill \\ A\left( {{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 458.930\left( 3 \right)MHzB\left( {{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 21.701\left( {40} \right)MHz \hfill \\ A\left( {{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 510.308\left( 8 \right)MHzB\left( {{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 132.200\left( {120} \right)MHz \hfill \\ \end{gathered} $$      

Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist

We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators \(|\left. {v_n } \right\rangle \left\langle {v_n |} \right.\) multiplied by g tan π[(α,n)+ω],g>0, α∈? ^d ,n∈? ^d , ω∈[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsv _n (x)=v(x-n),x∈? ^d ,n∈? ^d , is small then the spectrum ofH consists of a dense point component coinciding with? and an absolutely continuous component coinciding with [?, ∞), where ? is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=?) and zero temperature a.c. conductivityσ _λ (v), that also has a jump at λ=? and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠?.     

Resonance studies at STAR

Preliminary results from measurements of resonances (K ^*0(892), $\overline {K*^0 } (892)$ , Φ(1020), and ρ(770)) and weakly decaying particles (Λ(1116), $\bar \Lambda (1116)$ , and K _S ⁰ (498)) are presented. The measurements are performed at mid-rapidity by the STAR detector in $\sqrt {s_{NN} } = 130$ GeV Au?Au collisions at RHIC. The ratios K ^*0/h?, $\overline {K*^0 } /K$ , and $\bar \Lambda /\Lambda $ are compared to measurements at different energies and colliding systems. Estimates of thermal parameters, such as temperature and baryon chemical potential, are also presented.     

Gaussian limit for critical oriented percolation in high dimensions

In this paper, we consider the spread-out oriented bond percolation models inZ ^d ×Z withd>4 and the nearest-neighbor oriented bond percolation model in sufficiently high dimensions. Let η _n ,n=1, 2, ..., be the random measures defined onR ^d by $$\eta _n (A) = \sum\limits_{x \in Z^d } {1_A (x/\sqrt n )1_{\{ (0,0) \to (x,n)\} } } $$ The mean of η _n , denoted by $\bar \eta _n $ , is the measure defined by $$\bar \eta _n (A) = E_p [\eta _n (A)]$$ We use the lace expansion method to show that the sequence of probability measures $[\bar \eta _n (R^d )]^{ - 1} \bar \eta _n $ converges weakly to a Gaussian limit asn→∞ for everyp in the subcritical regime as well as the critical regime of these percolation models. Also we show that for these models the parallel correlation length $\xi (p)~|p_c - p|^{ - 1} $ asp?p_c     

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.     

Bounds on the density of states of random Schrödinger operators

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=?Δ + V acting onL ²(? ^d ) orl ²(? ^d ). In both cases the random potential is $$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$ in which the \(\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }\) areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the \(\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }\) are cells of unit dimensions centered ony∈? ^d . In the finite-difference case Λ(y) denotes the sitey∈? ^d itself. Under the assumptionf ∈ L ₀ ^1+? (?) it is proven that in the finitedifference casep ∈ L ^∞(?), and that in thed= 1 continuum casep ∈ L _loc ^∞ (?).     

Infinite differentiability for one-dimensional spin system with long range random interaction

We consider one-dimensional spin systems with Hamiltonian: $$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$ , where ? _tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α>3/2 andE(? _tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ^∞ both inh and β. Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ? _tt′ 's, where the quenched magnetization isC ^∞ but fails to be analytic inh for suitableh and β.     

OnΛ-hyperon(s) in the nuclear medium

Relativistic mean field theory is tested in reproducing the novel experimentalΛ single particle (π⁺, K⁺) spectra of Λ=12/90 hypernuclei (and extended to _Λ ²⁰⁹ Pb). The adjusted model is then applied to multistrange systems \({}_{n_\Lambda \Lambda }^{16 + n_\Lambda } O,_{n_\Lambda \Lambda }^{40 + n_\Lambda } Ca\) ; no anomalous behaviour of radii and densities in those multi-Λ hypernuclei is encountered.     

Spatial representation of groups of automorphisms of von Neumann algebras with properly infinite commutant

Theorem. Let a topological groupG be represented (a→φ _a ) by *-automorphisms of a von Neumann algebraR acting on a separable Hilbert spaceH. Suppose that

1. G is locally compact and separable,
2. R′ is properly infinite,
3. for anyT∈R,x,y∈H the function

$$a \to \left\langle {\phi _a (T)x,y} \right\rangle _H $$ is measurable onG. Then there exists a strongly continuous unitary representation ofG onH,a→U _a , such that forT∈R,a∈G, $$\phi _\alpha (T) = U_a TU_a *.$$ .     

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.     

A limit law for the ground state of Hill's equation

It is proved that the ground state Λ(L) of (?1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law $$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$      

One-range Addition Theorems for Noninteger n Slater Functions using Complete Orthonormal Sets of Exponential Type Orbitals in Standard Convention

By the use of complete orthonormal sets of ${\psi ^{(\alpha^{\ast})}}$ -exponential type orbitals ( ${\psi ^{(\alpha^{\ast})}}$ -ETOs) with integer (for α ^* = α) and noninteger self-frictional quantum number α ^*(for α ^* ≠ α) in standard convention introduced by the author, the one-range addition theorems for ${\chi }$ -noninteger n Slater type orbitals ${(\chi}$ -NISTOs) are established. These orbitals are defined as follows $$\begin{array}{ll}\psi _{nlm}^{(\alpha^*)} (\zeta ,\vec {r}) = \frac{(2\zeta )^{3/2}}{\Gamma (p_l ^* + 1)} \left[{\frac{\Gamma (q_l ^* + )}{(2n)^{\alpha ^*}(n - l - 1)!}} \right]^{1/2}e^{-\frac{x}{2}}x^{l}_1 F_1 ({-[ {n - l - 1}]; p_l ^* + 1; x})S_{lm} (\theta ,\varphi )\\ \chi _{n^*lm} (\zeta ,\vec {r}) = (2\zeta )^{3/2}\left[ {\Gamma(2n^* + 1)}\right]^{{-1}/2}x^{n^*-1}e^{-\frac{x}{2}}S_{lm}(\theta ,\varphi ),\end{array}$$ where ${x=2\zeta r, 0<\zeta <\infty , p_l ^{\ast}=2l+2-\alpha ^{\ast}, q_l ^{\ast}=n+l+1-\alpha ^{\ast}, -\infty <\alpha ^{\ast} <3 , -\infty <\alpha \leq 2,_1 F_1 }$ is the confluent hypergeometric function and ${S_{lm} (\theta ,\varphi )}$ are the complex or real spherical harmonics. The origin of the ${\psi ^{(\alpha ^{\ast})} }$ -ETOs, therefore, of the one-range addition theorems obtained in this work for ${\chi}$ -NISTOs is the self-frictional potential of the field produced by the particle itself. The obtained formulas can be useful especially in the electronic structure calculations of atoms, molecules and solids when Hartree–Fock–Roothan approximation is employed.     

The ϕ 2 4 quantum field as a limit of Sine-Gordon fields

We exhibit the λ? ₂ ⁴ quantum field theory as the limit of Sine-Gordon fields as suggested by the identity $$\varphi ^4 /4! = \mathop {\lim }\limits_{\varepsilon \to 0} (\varepsilon ^{ - 4} \cos \varepsilon \varphi - \varepsilon ^{ - 4} + \tfrac{1}{2}\varepsilon ^{ - 2} \varphi ^2 ).$$ The proofs of finite volume stability for the two models, due to Nelson and Fröhlich respectively, are unrelated. We find a generalized stability argument that incorporates ideas from both of the simpler cases. The above limit, for the Schwinger functions, then proceeds uniformly in ?. As a by-product, let (?,dμ) be a Gaussian random field, ? _K (1≦κ<∞) a regularization of ?, andV a function satisfying:

1. V(? _K )≧?ak ^α
2. ∥V(?) ?V(? _K )∥ _p≦bp ^β k ^?γ, 2≦p < ∞

Thene ^?V(?)∈L ¹(dμ) provided α(β?1)<γ.     

Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

The wave and scattering operators for the equation $$\left( {\square + m^2 } \right)\varphi + \lambda \varphi ^2 = 0$$ withm>0 and λ>0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e.L ₂ ¹ (R ³)⊕L ₂(R ³).