Motivated by a recent experiment in which zero-bias peaks have been observed in scanning tunneling microscopy (STM) experiments performed on chains of magnetic atoms on a superconductor, we show, by generalizing earlier work, that a multichannel ferromagnetic wire deposited on a spin-orbit coupled superconducting substrate can realize a non-trivial chiral topological superconducting state with Majorana bound states localized at the wire ends. The non-trivial topological state occurs for generic parameters requiring no fine tuning, at least for very large exchange spin splitting in the wire. We theoretically obtain the signatures which appear in the presence of an arbitrary number of Majorana modes in multi-wire systems incorporating the role of finite temperature, finite potential barrier at the STM tip, and finite wire length. These signatures are presented in terms of spatial profiles of STM differential conductance which clearly reveal zero energy Majorana end modes and the prediction of a multiple Majorana based fractional Josephson effect. A substantial part of this work is devoted to a detailed critical comparison between our theory and the recent STM experiment claiming the observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. The conclusion of this detailed comparison is that although the experimental observations are not manifestly inconsistent with our theoretical findings, the very small topological superconducting gap and the very high temperature of the experiment make it impossible to decisively verify the existence of a localized Majorana zero mode, as the spectral weight of the Majorana mode is necessarily spread over a very broad energy regime exceeding the size of the gap. Such an extremely broad (and extremely weak) conductance peak could easily arise from any sub-gap states existing in the rather complex system studied experimentally and may or may not have anything to do with a putative Majorana zero mode as discussed in the first half of our paper. Thus, although the experimental findings are indeed consistent with a highly broadened and weakened Majorana zero bias peak, much lower experimental temperatures (and/or much larger experimental topological superconducting gaps) are necessary for any definitive conclusion.

We perform a numerical study of a spin-1/2 model with Z2 × Z2 symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by Landau-Ginzburg-Wilson symmetry-breaking theory. arXiv:1904.00010v1 [cond-mat.str-el]

The success of polynomial-time tensor network methods for computing ground states of certain quantum local Hamiltonians has recently been given a sound theoretical basis by Arad et al. [Math. Phys. 356, 65 (2017)]. The convergence proof, however, relies on "rigorous renormalization group" (RRG) techniques which differ fundamentally from existing algorithms. We introduce a practical adaptation of the RRG procedure which, while no longer theoretically guaranteed to converge, finds matrix product state ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians in realistic situations. In contrast to other schemes, RRG does not utilize variational methods on tensor networks. Rather, it operates on subsets of the system Hilbert space by constructing approximations to the global ground space in a treelike manner. We evaluate the algorithm numerically, finding similar performance to density matrix renormalization group (DMRG) in the case of a gapped nondegenerate Hamiltonian. Even in challenging situations of criticality, large ground-state degeneracy, or long-range entanglement, RRG remains able to identify candidate states having large overlap with ground and low-energy eigenstates, outperforming DMRG in some cases.

Trimethyl aluminum (TMA) trails released from sounding rockets have been used extensively as a tracer of the neutral atmospheric motions in the mesosphere/lower thermosphere region and serve as a tracer of the atmospheric structure in the altitude range where the trails are released. In addition to the bulk motion corresponding to the winds, smaller‐scale structure is also evident in the trail images, which can be an indication of turbulence or small‐scale wave behavior. In the 90 to 100 km altitude range, the trails are often visible for periods of 20 to 30 min or even longer, so that the horizontal scale sizes spanned by the trails increase significantly as the trails expand due to the combined effects of turbulent diffusion and horizontal displacements due to vertical shears in the winds. We describe a study involving data from high‐, middle‐, and low‐latitude TMA releases. In each case the structure function for the fluctuation structure within the trails was calculated for a sequence of trail images. Each image in the sequence corresponds to an outer horizontal spatial scale that increases with time. By analyzing the time evolution of the structure function, changes in the dynamical processes associated with the form of the structure function have been identified as a function of time and thus also of the increasing horizontal spatial scale size. A consistent pattern is found for all locations, namely a transition in fitted exponent for the structure function that suggests a change from more isotropic turbulence immediately after the trail is released to stratified turbulence at the later times when the horizontal scale size has increased to several hundred kilometers. The transition is found to occur near a horizontal spatial scale in the range between 100 and 200 km.