Spin operator matrix elements in the superintegrable chiral Potts quantum chain

We derive spin operator matrix elements between general eigenstates of the superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by R.Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables Method.Comment: 24 pages, 1 figur

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/fnoise” by Bak et al. (1987), the idea has been applied to solar physics, in “Avalanches and the Distribution of Solar Flares” by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Fil: Aschwanden, Markus J.. Lockheed Martin Corporation; Estados UnidosFil: Crosby, Norma B.. Belgian Institute For Space Aeronomy; BélgicaFil: Dimitropoulou, Michaila. University Of Athens; GreciaFil: Georgoulis, Manolis K.. Academy Of Athens; GreciaFil: Hergarten, Stefan. Universitat Freiburg Im Breisgau; AlemaniaFil: McAteer, James. University Of New Mexico; Estados UnidosFil: Milovanov, Alexander V.. Max Planck Institute For The Physics Of Complex Systems; Alemania. Russian Academy Of Sciences. Space Research Institute; Rusia. Enea Centro Ricerche Frascati; ItaliaFil: Mineshige, Shin. Kyoto University; JapónFil: Morales, Laura Fernanda. Canadian Space Agency; Canadá. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Nishizuka, Naoto. Japan National Institute Of Information And Communications Technology; JapónFil: Pruessner, Gunnar. Imperial College London; Reino UnidoFil: Sanchez, Raul. Universidad Carlos Iii de Madrid. Instituto de Salud; EspañaFil: Sharma, A. Surja. University Of Maryland; Estados UnidosFil: Strugarek, Antoine. University Of Montreal; CanadáFil: Uritsky, Vadim. Nasa Goddard Space Flight Center; Estados Unido