Factorization of the Non-Stationary Schrodinger Operator

We consider a factorization of the non-stationary Schrodinger operator based on the parabolic Dirac operator introduced by Cerejeiras/ Kahler/ Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct appropriated Teodorescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinear Schrodinger equation using Banach fixed point theorem.Comment: Accepted for publication in Advances in Applied Clifford Algebra

A theorem regarding families of topologically non-trivial fermionic systems

We introduce a Hamiltonian for fermions on a lattice and prove a theorem regarding its topological properties. We identify the topological criterion as a $\mathbb{Z}_2-$ topological invariant $p(\textbf{k})$ (the Pfaffian polynomial). The topological invariant is not only the first Chern number, but also the sign of the Pfaffian polynomial coming from a notion of duality. Such Hamiltonian can describe non-trivial Chern insulators, single band superconductors or multiorbital superconductors. The topological features of these families are completely determined as a consequence of our theorem. Some specific model examples are explicitly worked out, with the computation of different possible topological invariants.Comment: 6 page

Volume change of bulk metals and metal clusters due to spin-polarization

The stabilized jellium model (SJM) provides us a method to calculate the volume changes of different simple metals as a function of the spin polarization, $\zeta$, of the delocalized valence electrons. Our calculations show that for bulk metals, the equilibrium Wigner-Seitz (WS) radius, $\bar r_s(\zeta)$, is always a n increasing function of the polarization i.e., the volume of a bulk metal always increases as $\zeta$ increases, and the rate of increasing is higher for higher electron density metals. Using the SJM along with the local spin density approximation, we have also calculated the equilibrium WS radius, $\bar r_s(N,\zeta)$, of spherical jellium clusters, at which the pressure on the cluster with given numbers of total electrons, $N$, and their spin configuration $\zeta$ vanishes. Our calculations f or Cs, Na, and Al clusters show that $\bar r_s(N,\zeta)$ as a function of $\zeta$ behaves differently depending on whether $N$ corresponds to a closed-shell or an open-shell cluster. For a closed-shell cluster, it is an increasing function of $\zeta$ over the whole range $0\le\zeta\le 1$, whereas in open-shell clusters it has a decreasing behavior over the range $0\le\zeta\le\zeta_0$, where $\zeta_0$ is a polarization that the cluster has a configuration consistent with Hund's first rule. The resu lts show that for all neutral clusters with ground state spin configuration, $\zeta_0$, the inequality $\bar r_s(N,\zeta_0)\le\bar r_s(0)$ always holds (self-compression) but, at some polarization $\zeta_1>\zeta_0$, the inequality changes the direction (self-expansion). However, the inequality $\bar r_s(N,\zeta)\le\bar r_s(\zeta)$ always holds and the equality is achieved in the limit $N\to\infty$.Comment: 7 pages, RevTex, 10 figure

Bicomplex neural networks with hypergeometric activation functions

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colorized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.publishe

Analytical and numerical studies of disordered spin-1 Heisenberg chains with aperiodic couplings

We investigate the low-temperature properties of the one-dimensional spin-1 Heisenberg model with geometric fluctuations induced by aperiodic but deterministic coupling distributions, involving two parameters. We focus on two aperiodic sequences, the Fibonacci sequence and the 6-3 sequence. Our goal is to understand how these geometric fluctuations modify the physics of the (gapped) Haldane phase, which corresponds to the ground state of the uniform spin-1 chain. We make use of different adaptations of the strong-disorder renormalization-group (SDRG) scheme of Ma, Dasgupta and Hu, widely employed in the study of random spin chains, supplemented by quantum Monte Carlo and density-matrix renormalization-group numerical calculations, to study the nature of the ground state as the coupling modulation is increased. We find no phase transition for the Fibonacci chain, while we show that the 6-3 chain exhibits a phase transition to a gapless, aperiodicity-dominated phase similar to the one found for the aperiodic spin-1/2 XXZ chain. Contrary to what is verified for random spin-1 chains, we show that different adaptations of the SDRG scheme may lead to different qualitative conclusions about the nature of the ground state in the presence of aperiodic coupling modulations.Comment: Accepted for publication in Physical Review

The Anomalous Hall effect in re-entrant AuFe alloys and the real space Berry phase

The Hall effect has been studied in a series of AuFe samples in the re-entrant concentration range, as well as in the spin glass range. The data demonstrate that the degree of canting of the local spins strongly modifies the anomalous Hall effect, in agreement with theoretical predictions associating canting, chirality and a real space Berry phase. The canonical parametrization of the Hall signal for magnetic conductors becomes inappropriate when local spins are canted.Comment: 4 pages, 1 eps figur