Canonical decomposition of linear differential operators with selected differential Galois groups

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.Comment: 40 page

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their lambda-extensions. These integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, and even a remarkable weight-1 modular form emerging in the three-particle contribution $\chi^{(3)}$ of the magnetic susceptibility of the square Ising model. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $\infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.Comment: 55 page

0-π\pi quantum transition in a carbon nanotube Josephson junction: universal phase dependence and orbital degeneracy

We investigate experimentally the supercurrent in a clean carbon nanotube quantum dot, close to orbital degeneracy, connected to superconducting leads in a regime of strong competition between local electronic correlations and superconducting proximity effect. For an odd occupancy of the dot and intermediate coupling to the reservoir, the Kondo effect can develop in the normal state and screen the local magnetic moment of the dot. This leads to singlet-doublet transitions that strongly affect the Josephson effect in a single-level quantum dot: the sign of the supercurrent changes from positive to negative (0 to $\pi$-junction). In the regime of strongest competition between the Kondo effect and proximity effect, meaning that the Kondo temperature equals the superconducting gap, the magnetic state of the dot undergoes a first order quantum transition induced by the superconducting phase difference across the junction. This is revealed experimentally by anharmonic current-phase relations. In addition, the very specific electronic configuration of clean carbon nanotubes, with two nearly orbitally degenerated states, leads to different physics depending whether only one or both quasi-degenerate upper levels of the dots participate to transport, which is determined by their occupancy and relative widths. When the transport of Cooper pairs takes place through only one of these levels, we find that the phase diagram of the phase-dependent 0-$\pi$ transition is a universal characteristic of a discontinuous level-crossing quantum transition at zero temperature. In the case were two levels participate to transport, the nanotube Josephson current exhibits a continuous 0-$\pi$ transition, independent of the superconducting phase, revealing a different physical mechanism of the transition.Comment: 14 pages, 12 figure

Painleve versus Fuchs

The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Renormalization, isogenies and rational symmetries of differential equations

We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group.Comment: 36 page

Angular Momentum Profiles of Warm Dark Matter Halos

We compare the specific angular momentum profiles of virialized dark halos in cold dark matter (CDM) and warm dark matter (WDM) models using high-resolution dissipationless simulations. The simulations were initialized using the same set of modes, except on small scales, where the power was suppressed in WDM below the filtering length. Remarkably, WDM as well as CDM halos are well-described by the two-parameter angular momentum profile of Bullock et al. (2001), even though the halo masses are below the filtering scale of the WDM. Although the best-fit shape parameters change quantitatively for individual halos in the two simulations, we find no systematic variation in profile shapes as a function of the dark matter type. The scatter in shape parameters is significantly smaller for the WDM halos, suggesting that substructure and/or merging history plays a role producing scatter about the mean angular momentum distribution, but that the average angular momentum profiles of halos originate from larger-scale phenomena or a mechanism associated with the virialization process. The known mismatch between the angular momentum distributions of dark halos and disk galaxies is therefore present in WDM as well as CDM models. Our WDM halos tend to have a less coherent (more misaligned) angular momentum structure and smaller spin parameters than do their CDM counterparts, although we caution that this result is based on a small number of halos.Comment: 5 pages, 1 figure, Submitted to ApJ

Cosmological Evolution of Supergiant Star-Forming Clouds

In an exploration of the birthplaces of globular clusters, we present a careful examination of the formation of self-gravitating gas clouds within assembling dark matter haloes in a hierarchical cosmological model. Our high-resolution smoothed particle hydrodynamical simulations are designed to determine whether or not hypothesized supergiant molecular clouds (SGMCs) form and, if they do, to determine their physical properties and mass spectra. It was suggested in earlier work that clouds with a median mass of several 10^8 M_sun are expected to assemble during the formation of a galaxy, and that globular clusters form within these SGMCs. Our simulations show that clouds with the predicted properties are indeed produced as smaller clouds collide and agglomerate within the merging dark matter haloes of our cosmological model. We find that the mass spectrum of these clouds obeys the same power-law form observed for globular clusters, molecular clouds, and their internal clumps in galaxies, and predicted for the supergiant clouds in which globular clusters may form. We follow the evolution and physical properties of gas clouds within small dark matter haloes up to z = 1, after which prolific star formation is expected to occur. Finally, we discuss how our results may lead to more physically motivated "rules" for star formation in cosmological simulations of galaxy formation.Comment: Accepted to The Astrophysical Journal; 17 pages, 8 figure

Study of the local field distribution on a single-molecule magnet-by a single paramagnetic crystal; a DPPH crystal on the surface of an Mn12-acetate crystal

The local magnetic field distribution on the subsurface of a single-molecule magnet crystal, SMM, above blocking temperature (T >> Tb) detected for a very short time interval (~ 10-10 s), has been investigated. Electron Paramagnetic Resonance (EPR) spectroscopy using a local paramagnetic probe was employed as a simple alternative detection method. An SMM crystal of [Mn12O12(CH3COO)16(H2O)4].2CH3COOH.4H2O (Mn12-acetate) and a crystal of 2,2- diphenyl-1-picrylhydrazyl (DPPH) as the paramagnetic probe were chosen for this study. The EPR spectra of DPPH deposited on Mn12-acetate show additional broadening and shifting in the magnetic field in comparison to the spectra of the DPPH in the absence of the SMM crystal. The additional broadening of the DPPH linewidth was considered in terms of the two dominant electron spin interactions (dipolar and exchange) and the local magnetic field distribution on the crystal surface. The temperature dependence of the linewidth of the Gaussian distribution of local fields at the SMM surface was extrapolated for the low temperature interval (70-5 K)

Star Formation, Supernovae Feedback and the Angular Momentum Problem in Numerical CDM Cosmogony: Half Way There?

We present a smoothed particle hydrodynamic (SPH) simulation that reproduces a galaxy that is a moderate facsimile of those observed. The primary failing point of previous simulations of disk formation, namely excessive transport of angular momentum from gas to dark matter, is ameliorated by the inclusion of a supernova feedback algorithm that allows energy to persist in the model ISM for a period corresponding to the lifetime of stellar associations. The inclusion of feedback leads to a disk at a redshift $z=0.52$, with a specific angular momentum content within 10% of the value required to fit observations. An exponential fit to the disk baryon surface density gives a scale length within 17% of the theoretical value. Runs without feedback, with or without star formation, exhibit the drastic angular momentum transport observed elsewhere.Comment: 4 pages, 3 figures, accepted for publication in ApJ Letter

Mg II Absorption Systems in SDSS QSO Spectra

We present the results of a MgII absorption-line survey using QSO spectra from the SDSS EDR. Over 1,300 doublets with rest equivalent widths greater than 0.3\AA and redshifts $0.366 \le z \le 2.269$ were identified and measured. We find that the $\lambda2796$ rest equivalent width ($W_0^{\lambda2796}$) distribution is described very well by an exponential function $\partial N/\partial W_0^{\lambda2796} = \frac{N^*}{W^*} e^{-\frac{W_0}{W^*}}$, with $N^*=1.187\pm0.052$ and $W^*=0.702\pm0.017$\AA. Previously reported power law fits drastically over-predict the number of strong lines. Extrapolating our exponential fit under-predicts the number of $W_0 \le 0.3$\AA systems, indicating a transition in $dN/dW_0$ near $W_0 \simeq 0.3$\AA. A combination of two exponentials reproduces the observed distribution well, suggesting that MgII absorbers are the superposition of at least two physically distinct populations of absorbing clouds. We also derive a new redshift parameterization for the number density of $W_0^{\lambda2796} \ge 0.3$\AA lines: $N^*=1.001\pm0.132(1+z)^{0.226\pm0.170}$ and $W^*=0.443\pm0.032(1+z)^{0.634\pm 0.097}$\AA. We find that the distribution steepens with decreasing redshift, with $W^*$ decreasing from $0.80\pm0.04$\AA at $z=1.6$ to $0.59\pm0.02$\AA at $z=0.7$. The incidence of moderately strong MgII $\lambda2796$ lines does not show evidence for evolution with redshift. However, lines stronger than $\approx 2$\AA show a decrease relative to the no-evolution prediction with decreasing redshift for $z \lesssim 1$. The evolution is stronger for increasingly stronger lines. Since $W_0$ in saturated absorption lines is an indicator of the velocity spread of the absorbing clouds, we interpret this as an evolution in the kinematic properties of galaxies from moderate to low z.Comment: 50 pages, 26 figures, accepted for publication in Ap