The volume of Gaussian states by information geometry

We formulate the problem of determining the volume of the set of Gaussian physical states in the framework of information geometry. That is, by considering phase space probability distributions parametrized by the covariances and supplying this resulting statistical manifold with the Fisher-Rao metric. We then evaluate the volume of classical, quantum and quantum entangled states for two-mode systems showing chains of strict inclusion

Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.Comment: 4 pages, 5 figure

Creation of multiple nanodots by single ions

In the challenging search for tools that are able to modify surfaces on the nanometer scale, heavy ions with energies of several 10 MeV are becoming more and more attractive. In contrast to slow ions where nuclear stopping is important and the energy is dissipated into a large volume in the crystal, in the high energy regime the stopping is due to electronic excitations only. Because of the extremely local (< 1 nm) energy deposition with densities of up to 10E19 W/cm^2, nanoscaled hillocks can be created under normal incidence. Usually, each nanodot is due to the impact of a single ion and the dots are randomly distributed. We demonstrate that multiple periodically spaced dots separated by a few 10 nanometers can be created by a single ion if the sample is irradiated under grazing angles of incidence. By varying this angle the number of dots can be controlled.Comment: 12 pages, 6 figure

Hyperbolic Deformation on Quantum Lattice Hamiltonians

A group of non-uniform quantum lattice Hamiltonians in one dimension is introduced, which is related to the hyperbolic $1 + 1$-dimensional space. The Hamiltonians contain only nearest neighbor interactions whose strength is proportional to $\cosh j \lambda$, where $j$ is the lattice index and where $\lambda \ge 0$ is a deformation parameter. In the limit $\lambda \to 0$ the Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians is induced by the corner Hamiltonians. As a simple example, we investigate the ground state of the deformed $S = 1/2$ Heisenberg spin chain by use of the density matrix renormalization group (DMRG) method. It is shown that the ground state is dimerized when $\lambda$ is finite. Spin correlation function show exponential decay, and the boundary effect decreases with increasing $\lambda$.Comment: 5 pages, 4 figure

Periodic boundary conditions on the pseudosphere

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.

Theory of Room Temperature Ferromagnet V(TCNE)_x (1.5 < x < 2): Role of Hidden Flat Bands

Theoretical studies on the possible origin of room temperature ferromagnetism (ferromagnetic once crystallized) in the molecular transition metal complex, V(TCNE)_x (1.5<x<2) have been carried out. For this family, there have been no definite understanding of crystal structure so far because of sample quality, though the effective valence of V is known to be close to +2. Proposing a new crystal structure for the stoichiometric case of x=2, where the valence of each TCNE molecule is -1 and resistivity shows insulating behavior, exchange interaction among d-electrons on adjacent V atoms has been estimated based on the cluster with 3 vanadium atoms and one TCNE molecule. It turns out that Hund's coupling among d orbitals within the same V atoms and antiferromagnetic coupling between d oribitals and LUMO of TCNE (bridging V atoms) due to hybridization result in overall ferromagnetism (to be precise, ferrimagnetism). This view based on localized electrons is supplemented by the band picture, which indicates the existence of a flat band expected to lead to ferromagnetism as well consistent with the localized view. The off-stoichiometric cases (x<2), which still show ferromagnetism but semiconducting transport properties, have been analyzed as due to Anderson localization.Comment: Accepted for publication in J. Phys. Soc. Jpn. Vol.79 (2010), No. 3 (March issue), in press; 6 pages, 8 figure

Boundary element method for resonances in dielectric microcavities

A boundary element method based on a Green's function technique is introduced to compute resonances with intermediate lifetimes in quasi-two-dimensional dielectric cavities. It can be applied to single or several optical resonators of arbitrary shape, including corners, for both TM and TE polarization. For cavities with symmetries a symmetry reduction is described. The existence of spurious solutions is discussed. The efficiency of the method is demonstrated by calculating resonances in two coupled hexagonal cavities.Comment: 9 pages, 7 figures (quality reduced

A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of K\"ahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an $n$-dimensional cone a rescaling of the canonical potential is an $n$-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde