Combinatorial formulation of Ising model revisited

In 1952, Kac and Ward developed a combinatorial formulation for the two dimensional Ising model which is another method of obtaining Onsager's formula for the free energy per site in the thermodynamic limit of the model. Feynman gave an important contribution to this formulation conjecturing a crucial mathematical relation which completed Kac and Ward ideas. In this paper, the method of Kac, Ward and Feynman for the free field Ising model in two dimensions is reviewed in a selfcontained way.Comment: 27 pages, 17 figure

The Electron Temperature Gradient in the Galactic Disk

We derive the electron temperature gradient in the Galactic disk using a sample of HII regions that spans Galactocentric distances 0--17 kpc. The electron temperature was calculated using high precision radio recombination line and continuum observations for more than 100 HII regions. Nebular Galactocentric distances were calculated in a consistent manner using the radial velocities measured by our radio recombination line survey. The large number of nebulae widely distributed over the Galactic disk together with the uniformity of our data provide a secure estimate of the present electron temperature gradient in the Milky Way. Because metals are the main coolants in the photoionized gas, the electron temperature along the Galactic disk should be directly related to the distribution of heavy elements in the Milky Way. Our best estimate of the electron temperature gradient is derived from a sample of 76 sources for which we have the highest quality data. The present gradient in electron temperature has a minimum at the Galactic Center and rises at a rate of 287 +/- 46 K/kpc. There are no significant variations in the value of the gradient as a function of Galactocentric radius or azimuth. The scatter we find in the HII region electron temperatures at a given Galactocentric radius is not due to observational error, but rather to intrinsic fluctuations in these temperatures which are almost certainly due to fluctuations in the nebular heavy element abundances. Comparing the HII region gradient with the much steeper gradient found for planetary nebulae suggests that the electron temperature gradient evolves with time, becoming flatter as a consequence of the chemical evolution of the Milky Way's disk.Comment: 43 pages, 9 figures (accepted for publication in the ApJ

Dynamical matrix for arbitrary quadratic fermionic bath Hamiltonians and non-Markovian dynamics of one and two qubits in an Ising model environment

We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, and derive simple functions to check the non-positivity of the intermediate map. These functions correspond to two different sufficient criteria for non-Markovianity. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions.Comment: 11 pages, 10 figure