States of Local Moment Induced by Nonmagnetic Impurities in Cuprate Superconductors

By using a model Hamiltonian with d-wave superconductivity and competing antiferromagnetic (AF) orders, the local staggered magnetization distribution due to nonmagnetic impurities in cuprate superconductors is investigated. From this, the net magnetic moment induced by a single or double impurities can be obtained. We show that the net moment induced by a single impurity corresponds to a local spin with S_z=0, or 1/2 depending on the strength of the AF interaction and the impurity scattering. When two impurities are placed at the nearest neighboring sites, the net moment is always zero. For two unitary impurities at the next nearest neighboring sites, and at sites separated by a Cu-ion site, the induced net moment has S_z=0, or 1/2, or 1. The consequence of these results on experiments will be discussed.Comment: 4 pages, 4 figure

Absence of broken inversion symmetry phase of electrons in bilayer graphene under charge density fluctuations

On a lattice model, we study the possibility of existence of gapped broken inversion symmetry phase (GBISP) of electrons with long-range Coulomb interaction in bilayer graphene using both self-consistent Hartree-Fock approximation (SCHFA) and the renormalized-ring-diagram approximation (RRDA). RRDA takes into account the charge-density fluctuations beyond the mean field. While GBISP at low temperature and low carrier concentration is predicted by SCHFA, we show here the state can be destroyed by the charge-density fluctuations. We also present a numerical algorithm for calculating the self-energy of electrons with the singular long-range Coulomb interaction on the lattice model.Comment: 8 pages, 6 figure

Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org