The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of c...

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Vydané v:Advances in physics Ročník 60; číslo 5; s. 679 - 798
Hlavní autori: Cobanera, Emilio, Ortiz, Gerardo, Nussinov, Zohar
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Abingdon Taylor & Francis 01.10.2011
Taylor & Francis Ltd
Predmet:
03.65.Fd Algebraic methods
05.30.-d Quantum statistical mechanics
05.50.+q Lattice theory and statistics
64.60.Cn Order-disorder transformations
Algebra
Algorithms
bond algebras, operator algebras
Bonding
Classical and quantum physics: mechanics and fields
dimensional reduction
Exact sciences and technology
fermionization
field theory
Gages
gauge theories
Gauges
High temperature
Lattice theory
Logic, set theory, and algebra
Low temperature
Mapping
Mathematical methods in physics
Mathematical models
Other topics in mathematical methods in physics
Phase boundaries
phase transitions
Physics
quantum and classical dualities
Quantum mechanics
Quantum physics
statistical mechanics
Transformations
Quantum system
Dimensionality
Symmetry property
Path integral
Heisenberg model
Strong coupling
High temperature
Quantum size effect
Structural models
Continuum
Local structure
Classical mechanics
Phase boundaries
Lattice field theory
Lattice model
Transfer matrix method
Statistical mechanics
Classical theory
Mathematical physics
Gauge field theory
Quantum theory
Crystal structure
ISSN:0001-8732, 1460-6976
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Popis
Shrnutí:An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ℤ 2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan-Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
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ISSN:0001-8732
1460-6976
DOI:10.1080/00018732.2011.619814