Holographic Symmetries and Generalized Order Parameters for Topological Matter

We introduce a universally applicable method, based on the bond-algebraic theory of dualities, to search for generalized order parameters in disparate systems including non-Landau systems with topological order. A key notion that we advance is that of {\em holographic symmetry}. It reflects situations wherein global symmetries become, under a duality mapping, symmetries that act solely on the system's boundary. Holographic symmetries are naturally related to edge modes and localization. The utility of our approach is illustrated by systematically deriving generalized order parameters for pure and matter-coupled Abelian gauge theories, and for some models of topological matter.Comment: v2, 10 pages, 3 figures. Accepted for publication in Physical Review B Rapid Communication

Symmetry and Topological Order

We prove sufficient conditions for Topological Quantum Order at both zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (that notably extend and differ from standard local gauge symmetries) and their associated defects, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges), discuss associated braiding, and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on certain suggested quantum computing schemes and lead to "thermal fragility". Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.Comment: 10 pages, 2 figures. Minimal changes relative to published version- most notably the above shortened title (which was too late to change upon request in the galley proofs). An elaborate description of all of the results in this article appeared in subsequent works, principally in arXiv:cond-mat/0702377 which was published in the Annals of Physics 324, 977- 1057 (2009

The "glass transition'' as a topological defect driven transition in a distribution of crystals and a prediction of a universal viscosity collapse

Topological defects are typically quantified relative to ordered backgrounds. The importance of these defects to the understanding of physical phenomena including diverse equilibrium melting transitions from low temperature ordered to higher temperatures disordered systems (and vice versa) can hardly be overstated. Amorphous materials such as glasses seem to constitute a fundamental challenge to this paradigm. A long held dogma is that transitions into and out of an amorphous glassy state are distinctly different from typical equilibrium phase transitions and must call for radically different concepts. In this work, we critique this belief. We examine systems that may be viewed as simultaneous distribution of different ordinary equilibrium structures. In particular, we focus on the analogs of melting (or freezing) transitions in such distributed systems. The theory that we arrive at yields dynamical, structural, and thermodynamic behaviors of glasses and supercooled fluids that, for the properties tested thus far, are in qualitative and quantitative agreement with experiment. We arrive at a prediction for the viscosity and dielectric relaxations that is universally satisfied for all experimentally measured supercooled liquids and glasses over 15 decades.Comment: 21 pages, 2 figure