Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form \fracE₁^T(l) E₂(m)l+m{\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}, thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

NONCOMMUTATIVE HYPERGEOMETRIC AND BASIC HYPERGEOMETRIC EQUATIONS

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the ₂F₁ Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao's result, extended to the more general setting of hypergeometric functions over an abstract unital Banach algebra. We provide a similar (but more complicated-looking) result for a second type of noncommutative ₂F₁ Gauß hypergeometric function. We further give q-analogues for both types of noncommutative hypergeometric equations.     

Chain length probability distribution — equivalence of ASYNNNI and 1d Ising model

An expression for the chain length probability distribution p(l) of a one dimensional Ising chain was derived using the cluster variation method formalism, the p(l) being expressed through the pair cluster probabilities. It was shown numerically that the same expression also applies in the case of one dimensional chains formed along one of the next-nearest neighbor interactions included in the two dimensional ASYNNNI (Asymmetric Next-Nearest Neighbor Ising) model, widely used to describe the statistics of oxygen ordering in the basal CuO _x planes of the YBa₂Cu₃O_6+x type high-T _c superconducting materials. Equivalency between ASYNNNI and 1d Ising model is discussed.      

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

Principal Fibrations from Noncommutative Spheres

We construct noncommutative principal fibrations S_θ⁷→S_θ⁴ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. “The algebra inclusion is an example of a not-trivial quantum principal bundle.”     

The Skyrme model with a U*(1) gauged Wess‐Zumino term in a noncommutative spacetime

The Skyrme model is generalized for a noncommutative spacetime with the Weyl‐operators of SU(2) matrices and the corresponding star‐product. The unitary condition and the topological current can be extended to star‐exponential matrices. The Wess‐Zumino term which breaks unphysical symmetries of the Skyrme action is gauged with the U_*(1) group to allow for electromagnetic processes in a noncommutative spacetime. Apart from corrections to the anomalous decay γ→π⁰π⁺π^– in commuting spacetime, the additional anomalous process γ→π⁰π⁰π⁰ is found in the U_*(1) gauged Wess‐Zumino action for a noncommutative spacetime.     

Heterogenization of supercritical fluids and nonvariant critical equilibria in ternary systems with one volatile component (for water-salt systems as examples)

The phase equilibria in ternary systems with one volatile component and a homogeneous supercritical fluid (SCF) region in one of the binary subsystems (with two critical end-points p and Q) were analyzed. Heterogenization of SCFs, extending from the 2d′ boundary subsystem into a ternary system, starts with monovariant critical phenomena in saturated solutions (l₁ = l₂-s) or (g = l−s), which pass through the extremal parameters of the temperature or composition to form special critical equilibria such as double critical end-points DCEs (l₁ = l₂-s, g = l-s, l₁ = l₂-g) or double homogeneous points DHPs (l₁ = l₂-s ↔ g = l-s). Ternary nonvariant critical points (l₁ = l₂ = g, l₁ = l₂-g-s, and l₁ = g-l₂-s) appear at the intersections of the monovariant critical and noncritical curves.     

Examples of gauged Laplacians on noncommutative spaces

We present two (classes of) examples of gauged Laplacian operators. The first one is a model of spin-Hall effect on a noncommutative four-sphere S _ϑ ⁴ with isospin degrees of freedom, coming from a noncommutative instanton, and invariant under the quantum group SO _ϑ (5). The second one, a Hall effect on a quantum 2-dimensional sphere S _q ², describes ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole with symmetry coming from the quantum group SU _q (2). For both models, ample symmetries provide a complete diagonalization.     

Competing energy scales in high‐temperature superconductors: Ultrafast pump–probe studies

We present a review of photoexcited quasiparticle dynamics of cuprate and pnictide high‐temperature superconductors in regimes (temperature, doping) where different phases such as superconductivity, spin‐density‐wave (SDW) and pseudogap phases coexist or compete with one another. We start with the overdoped cuprate superconductor Y_1–xCa_x Ba₂Cu₃O_7–δ, where the superconducting gap and pseudogap coexist in the superconducting state. In another cuprate Tl₂Ba₂Ca₂Cu₃O_y, we ob‐ serve a competition between SDW and superconducting orders deep in the superconducting state. Finally, in the underdoped iron pnictide superconductor (Ba,K)Fe₂As₂, SDW order forms at 85 K, followed by superconductivity at 28 K. We also find the emergence of a normal‐state order that suppresses SDW at a temperature T * ～ 60 K and argue that this normal‐state order is a precursor to superconductivity. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crystal chemistry of the Tl2−xBa2Ca n Cu n+1O2n+6 (n=0, 1, 2) high-T c superconductors

We report on a comparative crystal chemical study of the Tl_2–xBa₂Ca _n Cu _n+1O_2n+6(n=0, 1, 2) high-T _c superconductors. Superconducting properties were determined by DC-conductivity and-susceptibility measurements. An X-ray diffraction analysis reveals the nonstoichiometry of the compounds as a link between structural and superconducting behavior. A comparison of the different structures gives evidence that T1 deficiency in the TlO double layers increases with increasingn; this deficiency causes apparent Cu valences larger than 2+in the superconducting phases thus maintaining the charge balance. By extracting appropriate chemical formulas and performing a bond strength calculation we obtained the mixed-valent oxidation state of copper for the superconducting tetragonal phases as being 2.17+(n=0), 2.25+(n=1) and 2.31+(n=2) respectively; for the non-superconducting orthorhombic phase Tl_2–xBa₂CuO_6–y a Cu valence of about 2+is suggested.     

Noncommutative theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer index

Fujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)_A anomaly for noncommutative SU(N). We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θ^μν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.     

An apparatus for the measurement of thermal expansion of solids at low temperatures

A dilatometer, using the three terminal capacitance technique, suitable for measurement of linear thermal expansion of solids in the temperature range 1.3–300 K is described. The dialtometer is designed such that the mounting system for the specimen does not undergo any significant changes in dimensions when the specimen is heated. The apparatus, therefore, yields in principle absolute values of α, the coefficient of linear thermal expansion. The performance of the apparatus has been checked by measurements on copper in the temperature range of 77–300 K. Some preliminary results on the behaviour of α for Y₁Ba₂Cu₃O_6.9 compound in the vicinity of superconducting transition temperature,T _c are also described. The system can detect relative changes in length Δl/l ₀ of about 10⁻⁸. Attempts are being made to improve the sensitivity.     

Bound states and critical behavior of the Yukawa potential

We investigate the bound states of the Yukawa potential V (r)=−λexp(−αr)/r, using different algorithms: solving the Schr?dinger equation numerically and our Monte Carlo Hamiltonian approach. There is a critical α = α_C, above which no bound state exists. We study the relation between α_C and λ for various angular momentum quantum number l, and find in atomic units, α_C(l) = λ[A ₁ exp(−l/B ₁) + A ₂ exp(−l/B ₂)], with A ₁ = 1.020(18), B ₁ = 0.443(14), A ₂ = 0.170(17), and B ₂ = 2.490(180).     

Tunneling conductance of the graphene SNS junction with a single localized defect

Using the Dirac-Bogoliubov-de Gennes equation, we study the electron transport in a graphene-based superconductor-normal(graphene)-superconductor (SNS) junction. We consider the properties of tunneling conductance through an undoped strip of graphene with heavily doped superconducting electrodes in the dirty limit l _def ≪ L ≪ ξ. We find that the spectrum of Andreev bound states is modified in the presence of a single localized defect in the bulk. The minimum tunneling conductance remains the same, and this result is independent of the actual location of the imperfection.     

Scalar Field in the Bianchi I: Noncommutative Classical and Quantum Cosmology

Using the ADM formalism in the minisuperspace, we obtain the commutative and noncommutative exact classical solutions and exact wave function to the Wheeler-DeWitt equation with an arbitrary factor ordering, for the anisotropic Bianchi type I cosmological model, coupled to a scalar field, cosmological term and barotropic perfect fluid. We introduce noncommutative scale factors, considering that all minisuperspace variables q ⁱ do not commute, so the symplectic structure was modified. In the classical regime, it is shown that the anisotropic parameter β _±nc and the field φ, for some value in the λ _eff cosmological term and noncommutative θ parameter, present a dynamical isotropization up to a critical cosmic time t _c ; after this time, the effects of isotropization in the noncommutative minisuperspace seems to disappear. In the quantum regimen, the probability density presents a new structure that corresponds to the value of the noncommutativity parameter.     

Relations between nonlinear kernels of the collision integral

Previously, to solve the Boltzmann equation by the moments method with expansion of the distribution function by spherical Hermit polynomials, a new computational method was suggested which allowed to construct nonlinear matrix elements of the collision integral with very large indices. This made it possible to substantially advance in construction of the distribution function. Limitations to convergence of the distribution function that appear in moment method are eliminated if we come to expansion by spherical harmonics from expansion by spherical Hermit polynomials. In this case, a complex five-fold collision integral is replaced by a set of comparatively simple integral operators, and kernels G _l1,l2 ^l (c, c ₁, c ₂) of these operators become the analog of matrix elements. We found the relations between expansions of the distribution function in the reference frames with various velocities of motion along marked axis. Starting from the invariance condition of the collision integral with respect to selection of such reference frames, we derived recurrent relations between the kernels with various indices. These relations allow us to construct any nonlinear kernel G _{l1, l2} ^l (c, c ₁, c ₂), if the kernel G _0,0⁰(c, c ₁, c ₂) is known.     

Instantons on the Quantum 4-pheres S4q

We introduce noncommutative algebras A _q of quantum 4-spheres S ⁴ _q , with q∈ℝ, defined via a suspension of the quantum group SU _q (2), and a quantum instanton bundle described by a selfadjoint idempotent e∈ Mat₄(A _q ), e ²=e=e ^*. Contrary to what happens for the classical case or for the noncommutative instanton constructed in [8], the first Chern–Connes class ch ₁(e) does not vanish thus signaling a dimension drop. The second Chern–Connes class ch ₂(e) does not vanish as well and the couple (ch ₁(e), ch ₂(e) defines a cycle in the (b,B) bicomplex of cyclic homology. Received: 12 December 2000 / Accepted: 10 March 2001