Thermal Entanglement for Interacting Spin-1/2 Systems and its Quantum Criticality

In this work, we investigate the thermal entanglement for interacting spin systems , by varying the parameters of temperature T, direction and magnetic field B. PACS numbers: 03.67.Mn, 03.65.Ud, 05.30.Cd, 73.43.Nq     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

Generalized parton distributions in impact parameter space

The kinematical factor in the positivity bound (36) is incorrect. The bound correctly reads Our corrected result agrees with inequality (25) in [1], taking into account the different normalization conventions here and there.Published online: 9 October 2003Erratum published online: 10 October 2003     

Why is \mathcal{CPT} Fundamental?

Lüders and Pauli proved the theorem based on Lagrangian quantum field theory almost half a century ago. Jost gave a more general proof based on “axiomatic” field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why is fundamental—because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the transform of any relativistic field without calculating , and separately and then multiplying them. The purpose of this pedagogical paper is to “deaxiomatize” the theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.     

Dirac Equation in an Axial Coulomb Potential

We consider solutions to the Dirac equation in the presence of an external axial vector potential coupled to the spinor field psi through the interaction term . There turn out to be no bound-state energies in this system consistent with a normalizable wave function.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

For an N-body Stark Hamiltonian , the resolvent estimate holds uniformly in with Re and Im , where , and is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization in the configuration space, a refined resolvent estimate holds uniformly in with Re and Im . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday     

Three Classes of Orthomodular Lattices

Let denote the class of all orthomodular lattices and denote the class of those that are commutator-finite. Also, let denote the class of orthomodular lattices that satisfy the block extension property, those that satisfy the weak block extension property, and those that are locally finite. We show that the following strict containments hold: Dedicated to the memory of Günter Bruns.     

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field and static magnetic field for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as , where , , and . Here m^* and e denote the effective mass and charge of electron, respectively. _g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear v₀ vs E₀ characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields

The twist two contribution in the operator product expansion of for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space–time dimensions is a field V ₁ (x₁, x₂) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V ₁ can be equivalently characterized by a “single–pole property” concerning the pole structure of the (rational) correlation functions involving the product . This property is established for the dimension d = 2 of . As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space–time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields.     

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

Dynamical Differential Equations Compatible with Rational <Emphasis Type="Italic">qKZ</Emphasis> Equations

For the Lie algebra _N we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the _N rational quantized Knizhnik–Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the _N Weyl group.Mathematics Subject Classifications (2000). 17B37, 17B80, 81R10.     

Terahertz generation by nonlinear mixing of laser and its second harmonic in a rippled density plasma

Terahertz radiation generation by second-order nonlinear mixing of laser $ (\omega_{1} ,\,\vec{k}_{1} ) $ and its frequency shifted second harmonic $ \omega_{2} = 2\omega_{1} - \omega ,\,\,\vec{k}_{2} \, $ $ (\omega \ll \omega_{1} ) $ in a plasma, in the presence of an obliquely inclined density ripple of wave number $ \vec{q} $ , are investigated. The lasers exert ponderomotive force on electrons and drive density perturbations at $ (2\omega_{1} ,\,2\vec{k}_{1} - \vec{q}) $ and $ (\omega_{1} - \omega_{2} ,\,\vec{k}_{1} - \vec{k}_{2} - \vec{q}) $ . These perturbations beat with the electron oscillatory velocities due to the lasers to produce a nonlinear current at $ \omega ,\,\vec{k} = 2\vec{k}_{1} - \vec{k}_{2} - \vec{q} $ , resonantly driving the terahertz radiation when $ \vec{q} $ satisfies the phase matching condition. The radiated THz intensity depends on the relative polarization of the lasers and scales as the square of intensity of the fundamental laser and linearly with the square root of the intensity of the second harmonic. The THz emission is maximized when the polarization of the lasers is aligned. These results are consistent with the recent experimental results.     

TYZ Expansion for the Kepler Manifold

The main goal of the paper is to address the issue of the existence of Kempf’s distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non-compact manifold. Motivated by the recent results for compact manifolds we construct Kempf’s distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of n − 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter ( standing for Planck’s constant and , ). Moreover, the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We show that our estimates are sharp by analyzing the nonharmonic behaviour of T _m for . The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces. The first author was supported in part by the project PRIN (Cofin) n. 2006019457 with M.I.U.R., Italy. The second author was supported in part by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

Geometric Quantization, Parallel Transport and the Fourier Transform

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle over the space of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.