Exploration of Resonant Continuum and Giant Resonance in the Relativistic Approach

Single-particle resonant-states in the continuum are determined by solving scattering states of the Dirac equation with proper asymptotic conditions in the relativistic mean field theory (RMF). The regular and irregular solutions of the Dirac equation at a large radius where the nuclear potentials vanish are relativistic Coulomb wave functions, which are calculated numerically. Energies, widths and wave functions of single-particle resonance states in the continuum for ^{120}Sn are studied in the RMF with the parameter set of NL3. The isoscalar giant octupole resonance of ^{120}Sn is investigated in a fully consistent relativistic random phase approximation. Comparing the results with including full continuum states and only those single-particle resonances we find that the contributions from those resonant-states dominate in the nuclear giant resonant processes.Comment: 16 pages, 2 figure

Fast Low-Rank Matrix Learning with Nonconvex Regularization

Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better recovery performance. However, the resultant optimization problem is much more challenging. A very recent state-of-the-art is based on the proximal gradient algorithm. However, it requires an expensive full SVD in each proximal step. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows the use of power method to approximate the SVD efficiently. Besides, the proximal operator can be reduced to that of a much smaller matrix projected onto this leading subspace. Convergence, with a rate of O(1/T) where T is the number of iterations, can be guaranteed. Extensive experiments are performed on matrix completion and robust principal component analysis. The proposed method achieves significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the traditional nuclear norm regularizer.Comment: Long version of conference paper appeared ICDM 201

Critical Dynamical Exponent of the Two-Dimensional Scalar ϕ4\phi^4 Model with Local Moves

We study the scalar one-component two-dimensional (2D) $\phi^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D $\phi^4$ model $\gamma= 1.75$ and $\nu= 1$. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent $z_c$ is not settled. In this paper, we obtain $z_c$ for the 2D $\phi^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\tau$ for the correlation function $\langle \phi(t)\phi(0)\rangle$, and thereafter fitting the data as $\tau \sim L^{z_c}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) $\langle \Delta \phi^2(t)\rangle\sim t^c$ as $c=\gamma/(\nu z_c)$, and from the numerically obtained value $c\approx 0.80$, we calculate $z_c$. For different values of the coupling constant $\lambda$, we report that $z_c=2.17\pm0.03$ and $z_c=2.19\pm0.03$ for the two methods respectively. Our results indicate that $z_c$ is independent of $\lambda$, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the Generalised Langevin Equation (GLE) formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently capture the observed anomalous diffusion behavior.Comment: 14 pages, 4 figures, 6 figure files, to appear in Phys. Rev.

Spin-dependent resonant tunneling through quantum-well states in magnetic metallic thin films

Quantum-well (QW) states in {\it nonmagnetic} metal layers contained in magnetic multilayers are known to be important in spin-dependent transport, but the role of QW states in {\it magnetic} layers remains elusive. Here we identify the conditions and mechanisms for resonant tunneling through QW states in magnetic layers and determine candidate structures. We report first-principles calculations of spin-dependent transport in epitaxial Fe/MgO/FeO/Fe/Cr and Co/MgO/Fe/Cr tunnel junctions. We demonstrate the formation of sharp QW states in the Fe layer and show discrete conductance jumps as the QW states enter the transport window with increasing bias. At resonance, the current increases by one to two orders of magnitude. The tunneling magnetoresistance ratio is several times larger than in simple spin tunnel junctions and is positive (negative) for majority- (minority-) spin resonances, with a large asymmetry between positive and negative biases. The results can serve as the basis for novel spintronic devices.Comment: 4 figures in 5 eps file

Some ground-state expectation values for the free parafermion Z(N) spin chain

We consider the calculation of ground-state expectation values for the non-Hermitian Z(N) spin chain described by free parafermions. For N=2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann-Feynman theorem to obtain exact results for particular single site and nearest-neighbour ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N=3, along with how they change as a function of distance from the boundary.Comment: 17 pages, 4 figures; extra reference