Quantum Topological Order and Extended Quantum Symmetries

In noncrystalline systems and certain quantum (Hall) liquids with long-range coupling symmetry-breaking descriptions of phase transitions were suggested to be insufficient, and alternative theories in terms of topological order were proposed to replace previous Landau symmetry-breaking models. Such glassy systems with short-range structural order, but long-range correlations in magnetic and/or electrical properties may thus exhibit several topological orders both in lower and higher dimensions. Quantum states with different topological orders can be interchanged only through a (quantum) phase transition– a result that should be provable by means of Quantum Algebraic Topology means in terms of quantum operator algebras and locally compact quantum groupoid representations [15]. On the other hand, spontaneous symmetry-breaking is known to involve the generation of Goldstone bosons according to the well-known Goldstone theorem [21]; thus, low temperature superconductivity is known to occur via the generation of (electron) Cooper pairs (phonon-coupled to the lattice ions)–that are Goldstone bosons of spin-0– with long-range correlations and quantum coherence throughout the quantum superconductor. Thus, the superconductivity phase transition involves a change from Fermi statistics for electrons in the metal at temperature above to Bose-Einstein statsistics for the Cooper pairs that are responsible for superconductivity at the lower temperatures . Despite considerable theoretical and experimental efforts, prior to 2009 the appropriate algebraic topological structures responsible for quantum topological orders have not been either classified or identified. What has become however abundantly clear since the work of Sir Neville F. Mott, P.W. Anderson, Steve Weinberg, John Van Vleck, Sir Michael Atyiah, van 't Hooft, E. Witten, Turaev, K. Porter, L. Vainerman, Isham, M. Levin, X-G. Wen , Y.K. Levine, A. Tijon, and many other quantum theoretical physicists, is that quantum algebraic topology approaches are key to understanding long-range order correlations, the effects of partial disorder in solids and the emergence of extended quantum symmetry in many-body systems and quantum field theories (QFT), including quantum gravity (QG).

A basic concept in topological order theories is that of an ordered, entangled ground state for a many-body system with long-range coupling(s) (as for example magnetic dipole-dipole coupled ferromagnets, high or low temperature superconductors, and so on). Therefore, quantum topological order (QTO) can be described as a pattern of long-range quantum entanglement in quantum states, and it can be classified as an extended quantum symmetry in terms of categorical representations, categorical groups, locally compact quantum groupoid representations, braided tensor categories/ categories, quantum algebroids or quantum double groupoid representations.

Potential Applications

Bibliography

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