# Symmetries and Interactions in Topological Matter 2015

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Item Evidence for the chiral anomaly in a Dirac Semi-metal(2015-05) Ong, N. P.After an introduction to Weyl metals and the chiral anomaly, I will discuss recent transport results showing evidence for the chiral anomaly in the Dirac semimetal Na3Bi. At low temperature, we detect a large conductivity enhancement when the magnetic field is aligned with the current. The enhanced conductivity corresponds to a collimated current plume that can be steered by the magnetic field. The importance of this field locking feature as a signature of the chiral anomaly is emphasized.|*With Jun Xiong, S. Kushwaha, Tian Liang, Wudi Wang, and R. J. CavaItem Progress in the materials science of hybrid nanowires for topological devices(2015-05) Nygard, JesperSemiconductor nanowires are a backbone in proposals for topological quantum computing based on manipulation of Majorana quasiparticles. Experimentally, various techniques exist for synthesis of semiconductor nanowires for quantum transport. For most applications, the fabrication method is not important once the semiconductor growth conditions and quality have been optimised. There are also different routes for forming heterostructures, branched geometries and other advanced nanowire materials. However, we will here focus on a unique process that is particularly promising for topological devices: epitaxial metal/semiconductor heterostructures based on Molecular Beam Epitaxy. Under vacuum conditions, an aluminum shell is grown in-situ onto InAs nanowires, leading to an epitaxially matched interface between the semiconducting core and the metal coating that acts as a superconductor at low temperatures. The perfect superconductor-semiconductor interface results in proximity induced superconductivity with a hard gap. The technique is compatible with formation of branched nanostructures and opens up for new directions in nanowire based quantum devices, e.g. transmon qubits and topological systems.Item Majoranamodes in Atomic Chains on the Surface of a Superconductor(2015-05) Nadj-Perge, StevanMajorana bound states are zero-energy excitations predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron atomic chains on the surface of superconducting lead [1]. Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which partly gaps the electronic density of states in the bulk of the chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides evidence, corroborated by other observations and theoretical modeling [2], for the formation of a topological phase and edge-bound Majorana states in this system. Our results demonstrates that atomic chains are viable platform for future experiments to manipulate Majorana bound states [3] and to realize other 1D and 2D topological superconducting phases.Item Symmetry Protected Topological Semimetals(2015-05) Kane, CharlesItem Effective field theory of the disordered Weyl semimetal(2015-05) Altland, AlexanderIn disordered Weyl semimetals, mechanisms of topological origin lead to the protection against Anderson localization, and at the same time to different types of transverse electromagnetic response -- the anomalous Hall, and chiral magnetic effect. In this talk, we will discuss the manifestation of these phenomena at length scales which are beyond the scope of diagrammatic perturbation theory. Specifically we show how an interplay of symmetry breaking and the chiral anomaly leads to a field theory containing two types of topological terms. Generating the unconventional response coefficients of the system, these terms remain largely unaffected by disorder, i.e. information on the chirality of the system remains visible even at large length scales.Item Time reversal invariant gapped boundaries of twisted Z2 gauge theory(2015-05) Chen, XieThe boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can exhibit different symmetry transformation properties, depending on the details of the condensation. In this talk, I discuss an explicit example of this kind - the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal doublets, depending on the coherent phase factor of the boson condensate. The existence of these two possibilities are demonstrated using both field theory argument and exactly solvable lattice models. Furthermore, we study the domain walls between these two types of gapped boundaries. We find that they carry symmetry protected degeneracies and applying time reversal symmetry tunnels a semion between them.Item The competing spin liquids and symmetry fractionalization for triangular lattice J1-J2 spin-1/2(2015-05) Sneng, DonnaWe study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first- (J1 ) and second-neighbor (J2 ) interactions by means of density matrix renormalization group (DMRG). Beside the three sublattice ordered Neel phase and a stripe antiferromagnetic phase at smaller (~0.07) and larger J2 (~0.16) sides, we find a quantum spin liquid in the intermediate range of the J2. We show that there are two topological sectors. The odd sector (by pinning spinons at cylinder boundaries) is very robust, while the even topological sector has higher energy for smaller systems possibly due to a competing chiral state. We demonstrate the stabilization of the possible Z2 spin liquid in both sectors with the increase of the system width. We analyze the quantum numbers of different near degenerating states, and discuss the possible characterization of the Z2 spin liquid.Item Detecting signatures of topological order from microscopic Hamiltonians(2015-05) Pollmann, FrankI will show that numerical investigations of a many-body ground state wavefunction using the density matrix renormalization group (DMRG) method can yield a remarkably complete characterization of different types of topological orders. A central tool is the entanglement which encodes many of the essential features. First, I will show how characteristic properties of the topological excitations in fractional quantum Hall states can be extracted directly from the ground state wave function. Second, I will consider symmetry protected topological phases for which the characterizing symmetry fractionalization can be determined.Item Crystalline topological phases and quantum anomalies(2015-05) Ryu, ShinseiIn this talk, I plan to discuss phases of matter with reflection symmetry (parity symmetry) with interactions. While a systematic analysis is possible for non-interacting fermions, an important challenge is to understand the effects of strong electron correlations. To get some insight into this problem, I will discuss an example where by the effects of interactions the non-interacting classification breaks down. I will also propose a generalization of Laughlin’ s thought experiment, a theoretical method which is powerful enough to diagnose topological phases with U(1) symmetry but no other symmetries, to the cases of various symmetry protected topological phases. For the case of parity symmetry, the proposed generalization consists of putting they boundary theories of a SPT phase on an unoriented surfaces, and hence is related to the so-called orientifold quantum field theories.Item Metal-Insulator Transition and Beyond in the Pyrochlore Iridates(2015-05) Balents, LeonIridates are interesting materials in which Coulomb repulsion, kinetic energy, and spin-orbit coupling all are comparable. In particular the latter suggests they may be good candidates to observe topological phenomena. The pyrochlore family, with chemical formula A2Ir2O7 (A is a trivalent rare earth), displays both magnetic ordering and a metal-insulator transition. I will discuss theoretical and experimental studies of these materials, focusing on aspects related to topology and correlations, highlighting recent results.Item TOPOLOGY + QUANTUM CRITICALITY 1D TOPOLOGICAL ANDERSON INSULATORS(2015-05) Kamanev, AlexItem Topological Insulators2D Topological Superconductors&WeylSemimetalsFine(2015-05) Hasan, M. Z.In this talk I plan to present realization of 2D topological superconductors (TSC) with Helical Pairing [1,2] based on a Bi-based topological material and a route to SUSY critical point and then present our experimental discovery (and theory of TaAs) of Weyl semimetal state with Fermi arc surface states in TaAs and other related systems [3-5]. I discuss the progress in realizing exotic Cooper pairing in these systems. ||[1] Hasan & Kane ; RMP 82, 3045 (2010) and Qi & Zhang, RMP 83, 1057 (2011); Fu & Kane, PRL 100, 096407 (2008).|[2] S.-Y. Xu, N. Aldoust et al., Nature Physics 10, 943 (2014).|[3] T. Grover et.al., Science 344, 280 (2014).|[4] S.-Y. Xu, C. Liu, S. Kushwaha et al., Science 347, 294 (2015).|[5] S.-Y. Xu, Belopolski et.al., arXiv:1502.03807 (2015).Item Fractionalization of Faraday lines in generalized compact quantum electrodynamics and SPT- and SET-like phases of quantum lines and particles(2015-05) Motrunich, OlexeiMotivated by ideas of fractionalization and topological order in bosonic models with short-range interactions, we consider similar phenomena in formal lattice gauge theory models, which are models where basic constituents are quantum lines. In the first example, we show that a compact quantum electrodynamics (CQED) can have, besides familiar Coulomb and confined phases, additional unusual confined phases where excitations are quantum lines carrying fractions of the elementary unit of electric field strength; specifically, we construct a model that has $N$-tupled monopole condensation and realizes 1/N fractionalization of the quantum Faraday lines. In the second example, we consider a system consisting of two copies of CQED in (4+1)D and engineer condensation of bound states of monopoles (which are quantum lines in four spatial dimensions) and U(1) electric field lines. When the bound states contain a single monopole, we find lattice gauge theory analogs of Symmetry Protected Topological phases, while when the bound states contain multiple monopoles, we find analogs of Symmetry-Enriched Topological phases, where we also have fractionalization of Faraday lines. The distinct character of these “topological” phases of quantum lines is revealed by unusual response properties and physics at a boundary of a spatial region in such a phase.Item Surfaces of 3d symmetry protected phases(2015-05) Fidkowski, LukaszI will consider gapped Hamiltonians of generalized spin models, which are invariant under a certain unbroken onsite unitary symmetry group. It is well known that such Hamiltonians can realize topologically ordered phases, which in (2+1)d can be studied with modular tensor categories. When a symmetry is included, the corresponding `symmetry enriched’ phases correspond to a richer mathematical structure - e.g. braided G-crossed categories in (2+1) d. However, in systematically constructing such braided G-crossed categories by extending ordinary modular ones one sometimes encounters obstructions. Here we give a physical interpretation for such obstruction, and show that the corresponding topologically ordered theory, though it cannot be realized in 2d in a G-symmetric way, can be realized at the surface of a 3d ‘symmetry protected’ phase. I will try to emphasize the physical interpretation of the various mathematical concepts involved, and I will explain a specific example in detail.Item Composite Dirac liquids(2015-05) Alicea, JasonTopological phases of matter often feature boundary physics that naively seems impossible from the viewpoint of systems in one lower dimension. In this talk I will introduce a new class of exotic boundary states known as `composite Dirac liquids’ that can appear at a strongly interacting surface of a 3D electronic topological insulator. Composite Dirac liquids exhibit a gap to all charge excitations but nevertheless feature a single massless Dirac cone built from emergent electrically neutral fermions. These states thus comprise electrical insulators that, interestingly, retain thermal properties similar to those of the non-interacting topological insulator surface. I will show how gapping the neutral fermions via Cooper pairing naturally recovers symmetric non-Abelian surface topological orders captured recently in several works.Item Braiding statistics and symmetry-protected topological phases(2015-05) Michael, LevinSymmetry-protected topological (SPT) phases can be thought of as generalizations of topological insulators. Just as topological insulators have robust boundary modes protected by time reversal and charge conservation symmetry, SPT phases have boundary modes protected by more general symmetries. In this talk, I will describe a method for analyzing 2D and 3D SPTphases using braiding statistics. More specifically, I will show that 2D and 3D SPT phases can be characterized by gauging their symmetries and studying the braiding statistics of their gauge flux excitations. The 3D case is of particular interest as it involves a generalization of quasiparticle braiding statistics to three dimensions.Item Geometric responses of Quantum Hall systems(2015-05) Abanov, Alexander