Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems

A criterion to determine the existence of zero-energy edge states is discussed for a class of particle-hole symmetric Hamiltonians. A ``loop'' in a parameter space is assigned for each one-dimensional bulk Hamiltonian, and its topological properties, combined with the chiral symmetry, play an essential role. It provides a unified framework to discuss zero-energy edge modes for several systems such as fully gapped superconductors, two-dimensional d-wave superconductors, and graphite ribbons. A variants of the Peierls instability caused by the presence of edges is also discussed.Comment: Completely rewritten. Discussions on coexistence of is- or id_{xy}-wave order parameter near edges in d_{x^{2}-y^{2}}-wave superconductors are added; 4 pages, 3 figure

Cosmic ray acceleration at supergalactic accretion shocks: a new upper energy limit due to a finite shock extension

Accretion flows onto supergalactic-scale structures are accompanied with large spatial scale shock waves. These shocks were postulated as possible sources of ultra-high energy cosmic rays. The highest particle energies were expected for perpendicular shock configuration in the so-called "Jokipii diffusion limit", involving weakly turbulent conditions in the large-scale magnetic field imbedded in the accreting plasma. For such configuration we discuss the process limiting the highest energy that particles can obtain in the first-order Fermi acceleration process due to finite shock extensions to the sides, along and across the mean magnetic field. Cosmic ray outflow along the shock structure can substantially lower (below ~10^18 eV for protons) the upper particle energy limit for conditions considered for supergalactic shocks.Comment: A&A, accepte

Entanglement entropy and the Berry phase in solid states

The entanglement entropy (von Neumann entropy) has been used to characterize the complexity of many-body ground states in strongly correlated systems. In this paper, we try to establish a connection between the lower bound of the von Neumann entropy and the Berry phase defined for quantum ground states. As an example, a family of translational invariant lattice free fermion systems with two bands separated by a finite gap is investigated. We argue that, for one dimensional (1D) cases, when the Berry phase (Zak's phase) of the occupied band is equal to $\pi \times ({odd integer})$ and when the ground state respects a discrete unitary particle-hole symmetry (chiral symmetry), the entanglement entropy in the thermodynamic limit is at least larger than $\ln 2$ (per boundary), i.e., the entanglement entropy that corresponds to a maximally entangled pair of two qubits. We also discuss this lower bound is related to vanishing of the expectation value of a certain non-local operator which creates a kink in 1D systems.Comment: 11 pages, 4 figures, new references adde

Superconductivity and Abelian Chiral Anomalies

Motivated by the geometric character of spin Hall conductance, the topological invariants of generic superconductivity are discussed based on the Bogoliuvov-de Gennes equation on lattices. They are given by the Chern numbers of degenerate condensate bands for unitary order, which are realizations of Abelian chiral anomalies for non-Abelian connections. The three types of Chern numbers for the $x,y$ and $z$-directions are given by covering degrees of some doubled surfaces around the Dirac monopoles. For nonunitary states, several topological invariants are defined by analyzing the so-called $q$-helicity. Topological origins of the nodal structures of superconducting gaps are also discussed.Comment: An example with a figure and discussions are supplemente