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     

Automatic Regularization by Quantization in Reducible Representations of CCR: Point-Form Quantum Optics with Classical Sources

Electromagnetic fields are quantized in a manifestly covariant way by means of a class of reducible “center-of-mass N-representations” of the algebra of canonical commutation relations (CCR). The four-potential A _a (x) transforms in these representations as a Hermitian four-vector field in Minkowski four-position space (without change of gauge), but in momentum space it splits into spin-1 massless photons and two massless scalars. What we call quantum optics is the spin-1 sector of the theory. The scalar fields have physical status similar to that of dark matter (spin-1 and spin-0 particle numbers are separately conserved). There are no negative-norm or zero-norm states. Unitary dynamics is given by the point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary of the Milne universe. SL(2,C) transformations as well as the dynamics are represented unitarily in the Hilbert space corresponding to N four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the N-oscillator gas and is given by any N-oscillator product state annihilated by all annihilation operators. The form of A _a (x) is determined by an analogue of the twistor equation. The same equation guarantees that the set of vacuum states is Poincaré invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined even for pointlike charges, with ultraviolet and infrared regularizations occurring automatically as a consequence of the formalism. The probabilities are not Poissonian but of a Rényi type with α=1−1/N; the Shannon limit N→∞ is an ultraviolet/infrared-regularized Poisson distribution. The average number of photons occurring in Bremsstrahlung splits into two parts: The one due to acceleration, and the one that remains nonvanishing even for inertially moving charges. Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. We show in particular that static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is sensitive to both the choice of representation of CCR and the corresponding vacuum state.     

Spin observables of the reactions NN → Δ N and pd → Δ( pp )(1 S 0) in collinear kinematics

A general formalism for double and triple spin correlations of the reaction is developed for the case of collinear kinematics. A complete polarization experiment allowing one to reconstruct all four amplitudes describing this process is suggested. Furthermore, the spin observables of the inelastic charge-exchange reaction (pp)(¹ S ₀ are analyzed in collinear kinematics within the single pN-scattering mechanism involving the subprocess pn → Δ⁰ p. The full set of spin observables, related to the polarization of one or two initial particles and one final particle, is obtained in terms of three invariant amplitudes of the reaction pd → Δ(pp)(¹ S ₀) and the transition form factor d → (pp)(¹ S ₀). A complete polarization experiment for the reaction (pp)(¹ S ₀) is suggested which allows one to determine three independent combinations of the four amplitudes of the elementary subprocess . The text was submitted by the authors in English.     

On the Rate of Convergence to Equilibrium of the Andersen Thermostat in Molecular Dynamics

It has been shown in E and Li (Comm. Pure. Appl. Math., 2007, in press) that the Andersen dynamics is uniformly ergodic. Exponential convergence to the invariant measure is established with an error bound of the form

Neutral current cross-sections for neutrinos on protons

Defining the ratiosr _p =σ(vp →vx)/σ(vp →μ ⁻ x) and we obtain the bounds 0.28 ⩽r _p ⩽ 0.61 and using only the parton model and the data of CDHS group with iron target. We also give the complete set of parton-model relations which would allow the determination of all the neutral-current coupling constants from inclusive cross sections alone.     

Modification of Drude model on the free carrier property of IV-VI compound semiconductor in magnetic field

A general analytical expression is reported for the Drude model's dielectric function of free carrier in multi-valley IV-VI compound semiconductor in the magnetic field. The modifications of the energy band non-parabolicity and free carrier re-population are introduced to the dielectric function in combination with the model. The difference of the dielectric function between the modified Drude model and the classical Drude model is demonstrated by the calculation for the typical IV-VI compound semiconductor material PbTe.Hai Da Fellow     

Quantum Electrodynamics of Atomic Resonances

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass m, finitely many excited states and an electric dipole moment, \({\vec{d}_0 = -\lambda_{0} \vec{d}}\), where \({\| d^{i}\| = 1, i = 1, 2, 3,}\) and \({\lambda_0}\) is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, \({-\vec{d}_0 \cdot \vec{E}}\), where \({\vec{E}}\) is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum \({\vec{p}}\) of the atom and of the coupling constant \({\lambda_0}\), provided \({\vert\vec{p} \vert < mc}\) and \({\vert \Im \vec{p} \vert}\) and \({\vert \lambda_{0} \vert}\) are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of ‘smooth Feshbach–Schur maps’ applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.     

Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles

We consider an m-dimensional analytic cocycle \({\mathbb{T} \times \mathbb{R}^m \ni (x, \vec{\psi}) \mapsto (x + \omega, A (x) \cdot \vec{\psi}) \in \mathbb{T} \times \mathbb{R}^m}\) , where \({\omega \notin \mathbb{Q}}\) and \({A \in C^\omega (\mathbb{T}, \mathrm{Mat}_m (\mathbb{R}))}\) . Assuming that the d × d upper left corner block of A is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets–Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrödinger operators.     

Discussion About the Magnetic Field Dimensionality,Invariant Axis Condition,and Coulomb Gauge to Solve the Grad-Shafranov Equation

We discuss the relationship between the Coulomb gauge, the existence of an invariant axis, and the dimensionality (2-D or 2\(\frac {1}{2}\)-D) of the magnetic field in a mathematical-physical formalism that leads us to the Grad-Shafranov (GS) equation. In the literature, we found that a 2-D magnetic structure is used as a prerequisite to derive the GS equation from the Vlasov equation. However, other consulted works are based on a 2\(\frac {1}{2}\)-D (two-and-a-half) magnetic structure as a prerequisite to derive the GS equation from the balance of forces between the pressure gradient and the magnetic force, respectively. We replaced the magnetic vector potential on Ampère’s equation and used the Coulomb gauge to obtain a system of three Poisson equations, one for each component. We also used the same procedure explained above, but without the Coulomb gauge. Comparing z-component in both equation systems, we concluded that there are two possible solutions. We suggest using a 2\(\frac {1}{2}\)-D magnetic field configuration instead of a 2-D, when working with kinetic theory or magnetostatic equilibrium to derive the GS equation. We clarified that there is no relationship between the Coulomb gauge and the magnetic field dimensionality. In this problem, the invariant axis condition is imposed, which means that \(\vec {\nabla }\cdot \vec {A}\) is independent of z, i.e., \(\vec {\nabla }\cdot \vec {A}\) could have any value in which an invariant axis is a sufficient condition to obtain the GS equation.     

Toward a theory of the third-order elastic modulus of crystals with A2 structure: The case of α-Fe

The third-order elastic modulus of α-Fe were calculated based on the computation of lattice sums. The lattice sums were determined using an integer rational basis of invariants composed by vectors connecting equilibrium atomic positions in the crystal lattice. Irreducible interactions within clusters consisting of atomic pairs and triplets were taken into account in performing the calculations. Comparison with experimental data showed that the potential can be written in the form of e₉ = - ?_i,k A₁₉ r_ik ^{- 6} + ?_i,k A₂₉ r_ik ^{- 12} + ?_i,k,l Q₉ I₉ ^{- 1}\varepsilon _9 = - \sum\nolimits_{i,k} {A_{19} r_{ik}^{ - 6} } + \sum\nolimits_{i,k} {A_{29} r_{ik}^{ - 12} + \sum\nolimits_{i,k,l} {Q_9 I_9^{ - 1} } }, where I₉ = [(r)\vec]_ik² [ ( [(r)\vec]_ik [(r)\vec]_kl )² + ( [(r)\vec]_li [(r)\vec]_ik )² ] + [(r)\vec]_kl² [ ( [(r)\vec]_ik [(r)\vec]_kl )² + ( [(r)\vec]_kl [(r)\vec]_li )² ] + [(r)\vec]_li² [ ( [(r)\vec]_li [(r)\vec]_ik )² + ( [(r)\vec]_kl [(r)\vec]_li )² ]I_9 = \vec r_{ik}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{li} \vec r_{ik} } \right)^2 } \right] + \vec r_{kl}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right] + \vec r_{li}^2 \left[ {\left( {\vec r_{li} \vec r_{ik} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right]. If the values of [(r)\vec]_ik\vec r_{ik} are scaled in half-lattice constant units, then A₁₉ = 1.22 ë t⁹ û GPa, A₂₉ = 5.07 ×10² ë t¹⁵ û GPa, Q₉ = 5.31 ë t⁹ û GPaA_{19} = 1.22\left\lfloor {\tau ^9 } \right\rfloor GPa, A_{29} = 5.07 \times 10^2 \left\lfloor {\tau ^{15} } \right\rfloor GPa, Q_9 = 5.31\left\lfloor {\tau ^9 } \right\rfloor GPa, and τ = 1.26 ?. It is shown that the condition of thermodynamic stability of a crystal requires that we allow for irreducible interactions in atom triplets in at least four coordination spheres. The analytical expressions for the lattice sums determining the contributions from irreducible interactions in the atom triplets to the second- and third-order elastic moduli of cubic crystals in the case of interactions determined by I ₉ are presented in the appendix.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Hidden Grassmann Structure in the XXZ Model II: Creation Operators

In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space of all quasi-local operators, which are products of the disorder field with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants. Membre du CNRS On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992 Moscow, Russia Dedicated to the memory of Alexei Zamolodchikov     

Parity nonconservation in deuteron photoreactions

We calculate the asymmetries in parity-nonconserving deuteron photodisintegration due to circularly polarized photons $ \vec \gamma d \to np $ \vec \gamma d \to np with the photon laboratory energy ranging from the threshold up to 10MeV and the radiative capture of thermal polarized neutrons by protons $ \vec np \to \gamma d $ \vec np \to \gamma d . We use the leading-order electromagnetic Hamiltonian neglecting the smaller nuclear exchange currents. Comparative calculations are done by using the Reid93 and Argonne v₁₈ potentials for the strong interaction and the DDH and FCDH “best” values for the weak couplings in a weak one-meson exchange potential. A weak NΔ transition potential is used to incorporate also the Δ(1232) -isobar excitation in the coupled-channels formalism.     

Γ andX bandgap hydrostatic deformation potentials for epitaxial In0.52Al0.48As on InP(001)

The pressure dependence of the direct and indirect bandgap of epitaxial In_0.52Al_0.48As on InP(001) substrate has been measured using photoluminescence up to 92 kbar hydrostatic pressure. The bandgap changes from Γ toX at an applied pressure of ∼ 43 kbar. Hydrostatic deformation potentials for both the Γ andX bandgaps are deduced, after correcting for the elastic constant (bulk modulus) mismatch between the epilayer and the substrate. For the epilayer we obtain and+(2.81±0.15)eV for the Γ andX bandgaps respectively. From the pressure dependence of the normalized Γ-bandgap photoluminescence intensity a Γ-X lifetime ratio, (τ_Γ/τ _X ), of 4.1×10⁻³ is deduced.     

Limiting Absorption Principle at Critical Values for the Dirac Operator

We prove estimates for the resolvent H ₀ – z)^-1 of the Dirac operator , valid, even for z close to the critical points ±m. In particular, it is shown that the operator -smooth. As a by-product, the absence of the singular spectrum as well as the existence and unitarity of the wave operators are obtained for a class of perturbations .     

The Elliptic Algebra and the Drinfeld Realization of the Elliptic Quantum Group

By using the elliptic analogue of the Drinfeld currents in the elliptic algebra , we construct a L-operator, which satisfies the RLL-relations characterizing the face type elliptic quantum group . For this purpose, we introduce a set of new currents in . As in the N=2 case, we find a structure of as a certain tensor product of and a Heisenberg algebra. In the level-one representation, we give a free field realization of the currents in . Using the coalgebra structure of and the above tensor structure, we derive a free field realization of the -analogue of -intertwining operators. The resultant operators coincide with those of the vertex operators in the -type face model.     

On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, , , with any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of H(x,u) with x at infinity determines many properties of geodesics. Essentially, a subquadratic growth of H ensures geodesic completeness and connectedness, while the critical situation appears when H(x,u) behaves in some direction as , as in the classical model of exact gravitational waves.     

Spectral Extension of the Quantum Group Cotangent Bundle

The structure of a cotangent bundle is investigated for quantum linear groups GL _q (n) and SL _q (n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SL _q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL _q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.     

Θ+-Pentaquark parity from associative baryon production in πd collisions, π ± + d → Θ+ + Σ±

We derive, in model-independent way, the spin structure of the matrix element for the reaction of associative Θ⁺-pentaquark production, π ^± + d → Θ⁺ + Σ^±, in the threshold region and in collinear kinematics. The expressions for the polarization observables in this reaction are found assuming spin 1/2 and different parities for Θ⁺. We have proved that such a reaction can be used for a model-independent determination of the P parity of Θ⁺ only by measuring the Θ⁺ polarization. Other polarization observables, such as the dependence of the Σ^± polarization on the vector and tensor deuteron polarizations, are insensitive to the Θ⁺ parity under the considered kinematical conditions. The linear and quadratic relations between polarization observables in (Θ⁺ is unpolarized) do not depend on the parity of the Θ⁺ pentaquark. The analysis performed for this reaction is also applicable without any modification to the reaction K ⁺ + d → Θ⁺ + p. Using the pole model for the reaction mechanism, we estimated the value of the cross section for the reaction K ⁺ + d → Θ⁺ + p near the threshold. The text was submitted by the authors in English.