Sufficient symmetry conditions for Topological Quantum Order

We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, 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) 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 suggested quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.

[1]  A. Niemi,et al.  Fermion number fractionization in quantum field theory , 1986 .

[2]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[3]  S. Stringari,et al.  Uncertainty principle, quantum fluctuations, and broken symmetries , 1991 .

[4]  S. Coleman.  There are no Goldstone bosons in two dimensions , 1973 .

[5]  F. Wilczek,et al.  Geometric Phases in Physics , 1989 .

[6]  D. Hamann,et al.  Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution, and Some New Results on One-Dimensional Classical Statistical Models , 1970 .

[7]  Z. Nussinov,et al.  2 2 D ec 2 00 8 Bond Algebras and Exact Solvability of Hamiltonians : Spin S = 1 / 2 Multilayer Systems and Other Curiosities , 2008 .

[8]  Zohar Nussinov,et al.  Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems , 2007, 0709.2717.

[9]  A. Auerbach.  Interacting electrons and quantum magnetism , 1994 .

[10]  Spectral gap and decay of correlations in U(1)-symmetric lattice systems in dimensions D<2 , 2005, math-ph/0505022.

[11]  Singular or Non-Fermi Liquids , 2001, cond-mat/0103393.

[12]  D. Rokhsar,et al.  Superconductivity and the quantum hard-core dimer gas. , 1988, Physical review letters.

[13]  Quantum fluctuations in quantum lattice systems with continuous symmetry , 1996, cond-mat/9602138.

[14]  Xiao-Gang Wen.  Quantum orders in an exact soluble model. , 2003, Physical review letters.

[15]  Hal Tasaki,et al.  Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains , 1992 .

[16]  Quantum Mechanical Position Operator and Localization in Extended Systems , 1998, cond-mat/9810348.

[17]  Zohar Nussinov,et al.  A symmetry principle for topological quantum order , 2007, cond-mat/0702377.

[18]  John B. Kogut,et al.  An introduction to lattice gauge theory and spin systems , 1979 .

[19]  Z. Nussinov,et al.  Renormalization, duality, and phase transitions in two- and three-dimensional quantum dimer models , 2008, 0812.3903.

[20]  Carlos M. da Fonseca,et al.  The γ-Spectrum of a Graph , 2011, Ars Comb..

[21]  C. Batista,et al.  Algebraic approach to interacting quantum systems , 2002, cond-mat/0207106.

[22]  S. Elitzur,et al.  Impossibility of spontaneously breaking local symmetries , 1975 .

[23]  Zohar Nussinov,et al.  Exact results of the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations , 2007, cond-mat/0703633.

[24]  Discrete sliding symmetries, dualities, and self-dualities of quantum orbital compass models and p+ip superconducting arrays , 2004, cond-mat/0410720.

[25]  D. Bohm,et al.  Significance of Electromagnetic Potentials in the Quantum Theory , 1959 .

[26]  R. Martin,et al.  Macroscopic polarization as a geometric quantum phase: Many-body formulation. , 1994, Physical review. B, Condensed matter.

[27]  F. Wegner.  Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters , 1971 .

[28]  E. Lieb,et al.  Valence bond ground states in isotropic quantum antiferromagnets , 1988 .

[29]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[30]  S. Roth,et al.  Solitons in polyacetylene , 1987 .

[31]  N. Mermin,et al.  Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models , 1966 .

[32]  X. Wen.  Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons , 2004 .

[33]  K. Kugel,et al.  The Jahn-Teller effect and magnetism: transition metal compounds , 1982 .

[34]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[35]  Quantum phase transitions involving a change in polarization. , 1996, Physical review. B, Condensed matter.

[36]  D. Thouless.  Topological Quantum Numbers in Nonrelativistic Physics , 1998 .