Majorana Edge States in Interacting Two-chain Ladders of Fermions

In this work we study interacting spinless fermions on a two-chain ladder with inter-chain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a $\mathbb{Z}_2$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground state degeneracy against local perturbations. Lattice fermion models realizing such effective field theory are discussed.Comment: 6 pages, 1 figur

Analytical Potential Energy Function for the Ground State X^{1} Sigma^+ of LaCl

The equilibrium geometry, harmonic frequency and dissociation energy of lanthanum monochloride have been calculated at B3LYP, MP2, QCISD(T) levels with energy-consistent relativistic effective core potentials. The possible electronic state and reasonable dissociation limit for the ground state are determined based on atomic and molecular reaction statics. Potential energy curve scans for the ground state X^{1} Sigma^+ have been carried out with B3LYP and QCISD(T) methods due to their better performance in bond energy calculations. We find the potential energy calculated with QCISD(T) method is about 0.5 eV larger than dissociation energy when the diatomic distance is as large as 0.8 nm. The problem that single-reference ab initio methods don't meet dissociation limit during calculations of lanthanide heavy-metal elements is analyzed. We propose the calculation scheme to derive analytical Murrell-Sorbie potential energy function and Dunham expansion at equilibrium position. Spectroscopic constants got by standard Dunham treatment are in good agreement with results of rotational analyses on spectroscopic experiments. The analytical function is of much realistic importance since it is possible to be applied to predict fine transitional structure and study reaction dynamic process.Comment: 10 pages, 1 figure, 3 table

Asynchronous Distributed ADMM for Large-Scale Optimization- Part I: Algorithm and Convergence Analysis

Aiming at solving large-scale learning problems, this paper studies distributed optimization methods based on the alternating direction method of multipliers (ADMM). By formulating the learning problem as a consensus problem, the ADMM can be used to solve the consensus problem in a fully parallel fashion over a computer network with a star topology. However, traditional synchronized computation does not scale well with the problem size, as the speed of the algorithm is limited by the slowest workers. This is particularly true in a heterogeneous network where the computing nodes experience different computation and communication delays. In this paper, we propose an asynchronous distributed ADMM (AD-AMM) which can effectively improve the time efficiency of distributed optimization. Our main interest lies in analyzing the convergence conditions of the AD-ADMM, under the popular partially asynchronous model, which is defined based on a maximum tolerable delay of the network. Specifically, by considering general and possibly non-convex cost functions, we show that the AD-ADMM is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points as long as the algorithm parameters are chosen appropriately according to the network delay. We further illustrate that the asynchrony of the ADMM has to be handled with care, as slightly modifying the implementation of the AD-ADMM can jeopardize the algorithm convergence, even under a standard convex setting.Comment: 37 page