Ari M. Turner

Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates

Xiangang Wan, Ari Turner, Ashvin Vishwanath, Sergey Y. Savrasov National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, University of California, Berkeley, CA 94720 3 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley CA 94720. Department of Physics, University of California, Davis, One Shields Avenue, Davis, CA 95616.

Topological mechanics of gyroscopic metamaterials

Significance We have built a new type of mechanical metamaterial: a “gyroscopic metamaterial” composed of rapidly spinning objects that are coupled to each other. At the edges of these materials, we find sound waves that are topologically protected (i.e. they cannot be scattered backward or into the bulk). These waves, which propagate in one direction only, are directly analogous to edge currents in quantum Hall systems. Through a mathematical model, we interpret the robustness of these edge waves in light of the subtle topological character of the bulk material. Crucially, these edge motions can be controlled by distorting the metamaterial lattice, opening new avenues for the control of sound in matter. Topological mechanical metamaterials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. Here, we present an experimental and theoretical study of an active metamaterial—composed of coupled gyroscopes on a lattice—that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes that propagate in only one direction and are unaffected by disorder. We present a mathematical model that explains how the edge mode chirality can be switched via controlled distortions of the underlying lattice. This effect allows the direction of the edge current to be determined on demand. We demonstrate this functionality in experiment and envision applications of these edge modes to the design of one-way acoustic waveguides.

Symmetry protection of topological phases in one-dimensional quantum spin systems

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-S Haldane phase is a topologically nontrivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of π rotations about the x, y, and z axes, (ii) time-reversal symmetry Sx,y,z→−Sx,y,z, and (iii) link inversion symmetry (reflection about a bond center), consistent with previous results [ Phys. Rev. B 81 064439 (2010)]. On the other hand, an even-S Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by L. Fidkowski and A. Kitaev [Phys. Rev. B 81, 134509 (2010)], we find that in the presence of interactions there are only eight distinct phases which obey a

Quasiparticle statistics and braiding from ground state entanglement

{\mathbb{Z}}_{8}

Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions

group structure. This is in contrast to the

Shocks near Jamming

\mathbb{Z}

Vortices on curved surfaces

classification in the noninteracting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

Classifying Novel Phases of Spinor Atoms

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular S and U matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.

Anomalous coupling between topological defects and curvature.

We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ‘entropy density’ over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size, and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ‘zeroth Betti number’). In ????3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in ? ?? 3.