Unified Approach to Quantum and Classical Dualities

Cobanera, Emilio; Ortíz, Gerardo; Nussinov, Zohar

doi:10.1103/physrevlett.104.020402

Cited by 56 publications

(139 citation statements)

References 20 publications

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“…A simple variant of the Feynman mapping invokes duality considerations (Cobanera et al, 2010Nussinov and Fradkin, 2005) to another quantum system Moore, 2004, 2005) prior to the use of the standard transfer matrix technique. Here we merely quote the results.…”

Section: Ising Model Representationsmentioning

confidence: 99%

“…The solution relies on an exact gauge type symmetry which results from dilution and the use of a bond algebra mapping (Cobanera et al, 2010Nussinov and Ortiz, 2008;Nussinov and Ortiz, 2009b;Nussinov et al, 2012b; wherein the system is mapped onto decoupled one dimensional transverse field Ising chains (Nussinov and Ortiz, 2008) that exhibit quantum criticality at their isotropic point. The symmetries associated with the dilution increase the degeneracy of the system.…”

Section: E Effect Of Dilutionmentioning

confidence: 99%

“…This global Ising reflection symmetry is related to a duality (J x ↔ J y ) between the couplings. Along the self-dual line, J x = J y , the duality between the x and y bonds becomes a symmetry [as in general self-dual systems (Cobanera et al, 2010. As a d = 2 dimensional Ising type symmetry can be broken at finite temperature, this reflection symmetry can (and indeed is) broken at finite temperatures.…”

Section: Classical Square Latticementioning

confidence: 99%

“…(107). As was discussed in section VI, any generally non-zero quantity (as such, involving any number of bonds (Cobanera et al, 2010;Nussinov and Ortiz, 2009b)) that is invariant under all low dimensional gauge like symmetries can serve as an order parameter. That is, general composites of such bonds can serve as order parameters.…”

Section: Classical Square Latticementioning

confidence: 99%

“…• compass model is, in fact, dual to the Moore-Lee model describing Josephson coupling between superconducting grains in a square lattice (Cobanera et al, 2010;Moore and Lee, 2004;Nussinov and Fradkin, 2005;Moore, 2004, 2005).…”

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confidence: 99%

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Compass models: Theory and physical motivations

2015

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Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. A simple illustrative example is furnished by the 90 • compass model on a square lattice in which only couplings of the form τ x i τ x j (where {τ a i }a denote Pauli operators at site i) are associated with nearest neighbor sites i and j separated along the x axis of the lattice while τ y i τ y j couplings appear for sites separated by a lattice constant along the y axis. Such compass-type interactions can appear in diverse physical systems. This includes Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved, the low energy effective theories of frustrated quantum magnets, vacancy centers and cold atomic gases. The fundamental inter-dependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors including the frustration of (semi-)classical ordered states on non-frustrated lattices and to enhanced quantum effects prompting, in certain cases, the appearance of zero temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. We review compass models in a unified manner, paying close attention to exact consequences of these symmetries, and to thermal and quantum fluctuations that stabilize orders via order out of disorder effects. This is complemented by a survey of numerical results.

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Section: Ising Model Representationsmentioning

confidence: 99%

Section: E Effect Of Dilutionmentioning

confidence: 99%

Section: Classical Square Latticementioning

confidence: 99%

Section: Classical Square Latticementioning

confidence: 99%

mentioning

confidence: 99%

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Compass models: Theory and physical motivations

2015

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The Role of Pairing in Dynamically Stable QBHs

Flynn

2024

Springer Theses

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Effective and exact holographies from symmetries and dualities

2012

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The theoretical basis of the phenomenon of effective and exact dimensional reduction, or holographic correspondence, is investigated in a wide variety of physical systems. We first derive general inequalities linking quantum systems of different spatial (or spatio-temporal) dimensionality, thus establishing bounds on arbitrary correlation functions. These bounds enforce an {\em effective} dimensional reduction and become most potent in the presence of certain symmetries. {\em Exact} dimensional reduction can stem from a duality that (i) follows from properties of the local density of states, and/or (ii) from properties of Hamiltonian-dependent algebras of interactions. Dualities of the first type (i) are illustrated with large-$n$ vector theories whose local density of states may remain invariant under transformations that change the dimension. We argue that a broad class of examples of dimensional reduction may be understood in terms of the functional dependence of observables on the local density of states. Dualities of the second type (ii) are obtained via {\em bond algebras}, a recently developed algebraic tool. We apply this technique to systems displaying topological quantum order, and also discuss the implications of dimensional reduction for the storage of quantum information.Comment: 53 pgs, 6 figs. Submitted to Nuclear Physics

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Unified Approach to Quantum and Classical Dualities

Cited by 56 publications

References 20 publications

Compass models: Theory and physical motivations

Compass models: Theory and physical motivations

The Role of Pairing in Dynamically Stable QBHs

Effective and exact holographies from symmetries and dualities

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