Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90641/1/AIAA-53965-537.pd

Universal transport signatures of Majorana fermions in superconductor-Luttinger liquid junctions

One of the most promising proposals for engineering topological superconductivity and Majorana fermions employs a spin-orbit coupled nanowire subjected to a magnetic field and proximate to an s-wave superconductor. When only part of the wire's length contacts to the superconductor, the remaining conducting portion serves as a natural lead that can be used to probe these Majorana modes via tunneling. The enhanced role of interactions in one dimension dictates that this configuration should be viewed as a superconductor-Luttinger liquid junction. We investigate such junctions between both helical and spinful Luttinger liquids, and topological as well as non-topological superconductors. We determine the phase diagram for each case and show that universal low-energy transport in these systems is governed by fixed points describing either perfect normal reflection or perfect Andreev reflection. In addition to capturing (in some instances) the familiar Majorana-mediated `zero-bias anomaly' in a new framework, we show that interactions yield dramatic consequences in certain regimes. Indeed, we establish that strong repulsion removes this conductance anomaly altogether while strong attraction produces dynamically generated effective Majorana modes even in a junction with a trivial superconductor. Interactions further lead to striking signatures in the local density of states and the line-shape of the conductance peak at finite voltage, and also are essential for establishing smoking-gun transport signatures of Majorana fermions in spinful Luttinger liquid junctions.Comment: 25 pages, 6 figures, v

Scattering in the adjoint sector of the c = 1 Matrix Model

Closed string tachyon emission from a traveling long string in Liouville string theory is studied. The exact collective field Hamiltonian in the adjoint sector of the c=1 matrix model is computed to capture the interaction between the tip of the long string and the closed string tachyon field. The amplitude for emission of a single tachyon quantum is obtained in a closed form using the chiral formalism.Comment: 22 pages, 2 figure

Permutation Symmetric Critical Phases in Disordered Non-Abelian Anyonic Chains

Topological phases supporting non-abelian anyonic excitations have been proposed as candidates for topological quantum computation. In this paper, we study disordered non-abelian anyonic chains based on the quantum groups $SU(2)_k$, a hierarchy that includes the $\nu=5/2$ FQH state and the proposed $\nu=12/5$ Fibonacci state, among others. We find that for odd $k$ these anyonic chains realize infinite randomness critical {\it phases} in the same universality class as the $S_k$ permutation symmetric multi-critical points of Damle and Huse (Phys. Rev. Lett. 89, 277203 (2002)). Indeed, we show that the pertinent subspace of these anyonic chains actually sits inside the ${\mathbb Z}_k \subset S_k$ symmetric sector of the Damle-Huse model, and this ${\mathbb Z}_k$ symmetry stabilizes the phase.Comment: 13 page

Interacting invariants for Floquet phases of fermions in two dimensions

We construct a many-body quantized invariant that sharply distinguishes among two-dimensional nonequilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasienergy winding number and describes chiral pumping of quantum information along the edge. In particular, our invariant sharply distinguishes between a trivial and anomalous Floquet Anderson insulator in the interacting, many-body localized setting. It also applies more generally to models where only fermion parity is conserved, where it differentiates between trivial models and ones that pump Kitaev Majorana chains to the boundary, such as ones recently introduced in the context of emergent fermions arising from eigenstate Z2 topological order. We evaluate our invariant for the edge of such a system with eigenstate Z2 topological order, and show that it is necessarily nonzero when the Floquet unitary exchanges electric and magnetic excitations, proving a connection between bulk anyonic symmetry and edge chirality that was recently conjectured

An Unsteady Entropy Adjoint Approach for Adaptive Solution of the Shallow-Water Equations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90693/1/AIAA-2011-3694-887.pd

Output-based Adaptive Meshing Using Triangular Cut Cells

This report presents a mesh adaptation method for higher-order (p > 1) discontinuous Galerkin (DG) discretizations of the two-dimensional, compressible Navier-Stokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cut-cell approach permits anisotropic adaptation without the difficulty of constructing meshes that conform to potentially complex geometries. A quadrature technique is presented for accurately integrating on general cut cells. In addition, an output-based error estimator and adaptive method are presented, with emphasis on appropriately accounting for high-order solution spaces in optimizing local mesh anisotropy. Accuracy on cut-cell meshes is demonstrated by comparing solutions to those on standard boundary-conforming meshes. Adaptation results show that, for all test cases considered, p = 2 and p = 3 discretizations meet desired error tolerances using fewer degrees of freedom than p = 1. Furthermore, an initial-mesh dependence study demonstrates that, for sufficiently low error tolerances, the final adapted mesh is relatively insensitive to the starting mesh

Excursions beyond the horizon: Black hole singularities in Yang-Mills theories (I)

We study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.Comment: 34 page, 10 figures, uses harvmac, references adde

Majorana Zero Modes in 1D Quantum Wires Without Long-Ranged Superconducting Order

We show that long-ranged superconducting order is not necessary to guarantee the existence of Majorana fermion zero modes at the ends of a quantum wire. We formulate a concrete model which applies, for instance, to a semiconducting quantum wire with strong spin-orbit coupling and Zeeman splitting coupled to a wire with algebraically-decaying superconducting fluctuations. We solve this model by bosonization and show that it supports Majorana fermion zero modes. We argue that a large class of models will also show the same phenomenon. We discuss the implications for experiments on spin-orbit coupled nanowires coated with superconducting film and for LaAlO3/SrTiO3 interfaces.Comment: 14 pages. Figures added and a discussion of the effects of quantum phase slips. References Added. Fourth author adde

Infinite Randomness Phases and Entanglement Entropy of the Disordered Golden Chain

Topological insulators supporting non-abelian anyonic excitations are at the center of attention as candidates for topological quantum computation. In this paper, we analyze the ground-state properties of disordered non-abelian anyonic chains. The resemblance of fusion rules of non-abelian anyons and real space decimation strongly suggests that disordered chains of such anyons generically exhibit infinite-randomness phases. Concentrating on the disordered golden chain model with nearest-neighbor coupling, we show that Fibonacci anyons with the fusion rule $\tau\otimes\tau={\bf 1}\oplus \tau$ exhibit two infinite-randomness phases: a random-singlet phase when all bonds prefer the trivial fusion channel, and a mixed phase which occurs whenever a finite density of bonds prefers the $\tau$ fusion channel. Real space RG analysis shows that the random-singlet fixed point is unstable to the mixed fixed point. By analyzing the entanglement entropy of the mixed phase, we find its effective central charge, and find that it increases along the RG flow from the random singlet point, thus ruling out a c-theorem for the effective central charge.Comment: 16 page