A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of K3 surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the K3 family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil L-function of the K3 surface. This result is shown to be a particular case of Deligne's conjectures relating values of L-functions inside the critical strip to periods.Comment: Latex. 70 pages. 3 figures. v3: minor changes and clarifications. Version to appear in Compositio Mathematic

Asymptotic Stability, Instability and Stabilization of Relative Equilibria

In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev

A theory of ferromagnetism by Ettore Majorana

We present and analyze in detail an unknown theory of ferromagnetism developed by Ettore Majorana as early as the beginnings of 1930s, substantially different in the methods employed from the well-known Heisenberg theory of 1928 (and from later formulations by Bloch and others). Similarly to this, however, it describes successfully the main features of ferromagnetism, although the key equation for the spontaneous mean magnetization and the expression for the Curie temperature are different from those deduced in the Heisenberg theory (and in the original phenomenological Weiss theory). The theory presented here contains also a peculiar prediction for the number of nearest neighbors required to realize ferromagnetism, which avoids the corresponding arbitrary assumption made by Heisenberg on the basis of known (at that time) experimental observations. Some applications of the theory (linear chain, triangular chain, etc.) are, as well, considered.Comment: Latex, amsart, 16 pages, 4 figure

On a conjecture by Boyd

The aim of this note is to prove the Mahler measure identity $m(x+x^{-1}+y+y^{-1}+5) = 6 m(x+x^{-1}+y+y^{-1}+1)$ which was conjectured by Boyd. The proof is achieved by proving relationships between regulators of both curves

Ordered and disordered dynamics in monolayers of rolling particles

We consider the ordered and disordered dynamics for monolayers of rolling self-interacting particles with an offset center of mass and a non-isotropic inertia tensor. The rolling constraint is considered as a simplified model of a very strong, but rapidly decaying bond with the surface, preventing application of the standard tools of statistical mechanics. We show the existence and nonlinear stability of ordered lattice states, as well as disturbance propagation through and chaotic vibrations of these states. We also investigate the dynamics of disordered gas states and show that there is a surprising and robust linear connection between distributions of angular and linear velocity for both lattice and gas states, allowing to define the concept of temperature

Direct Observation of Second Order Atom Tunnelling

Tunnelling of material particles through a classically impenetrable barrier constitutes one of the hallmark effects of quantum physics. When interactions between the particles compete with their mobility through a tunnel junction, intriguing novel dynamical behaviour can arise where particles do not tunnel independently. In single-electron or Bloch transistors, for example, the tunnelling of an electron or Cooper pair can be enabled or suppressed by the presence of a second charge carrier due to Coulomb blockade. Here we report on the first direct and time-resolved observation of correlated tunnelling of two interacting atoms through a barrier in a double well potential. We show that for weak interactions between the atoms and dominating tunnel coupling, individual atoms can tunnel independently, similar to the case in a normal Josephson junction. With strong repulsive interactions present, two atoms located on one side of the barrier cannot separate, but are observed to tunnel together as a pair in a second order co-tunnelling process. By recording both the atom position and phase coherence over time, we fully characterize the tunnelling process for a single atom as well as the correlated dynamics of a pair of atoms for weak and strong interactions. In addition, we identify a conditional tunnelling regime, where a single atom can only tunnel in the presence of a second particle, acting as a single atom switch. Our work constitutes the first direct observation of second order tunnelling events with ultracold atoms, which are the dominating dynamical effect in the strongly interacting regime. Similar second-order processes form the basis of superexchange interactions between atoms on neighbouring lattice sites of a periodic potential, a central component of quantum magnetism.Comment: 18 pages, 4 figures, accepted for publication in Natur

An almost Poisson structure for the generalized rigid body equations

In this paper we introduce almost Poisson structures on Lie groups which generalize Poisson structures based on the use of the classical Yang-Baxter identity. Almost Poisson structures fail to be Poisson structures in the sense that they do not satisfy the Jacobi identity.In the case of cross products of Lie groups, we show that an almost Poisson structure can be used to derive a system which is intimately related to a fundamental Hamiltonian integrable system — the generalized rigid body equations