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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| Veröffentlicht in: | Advances in physics Jg. 60; H. 5; S. 679 - 798 |
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Taylor & Francis
01.10.2011
Taylor & Francis Ltd |
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| ISSN: | 0001-8732, 1460-6976 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 ... 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. (ProQuest: ... denotes formulae/symbols omitted.) 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 '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. |
| Author | Nussinov, Zohar Ortiz, Gerardo Cobanera, Emilio |
| Author_xml | – sequence: 1 givenname: Emilio surname: Cobanera fullname: Cobanera, Emilio email: ecobaner@indiana.edu organization: Department of Physics , Indiana University – sequence: 2 givenname: Gerardo surname: Ortiz fullname: Ortiz, Gerardo organization: Department of Physics , Indiana University – sequence: 3 givenname: Zohar surname: Nussinov fullname: Nussinov, Zohar organization: Department of Physics , Washington University |
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| Copyright | Copyright Taylor & Francis Group, LLC 2011 2015 INIST-CNRS Copyright Taylor & Francis Ltd. 2011 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | The bond-algebraic approach to dualities |
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