Algebraic approach to interacting quantum systems

Batista, Cristian D.; Ortíz, Gerardo

doi:10.1080/00018730310001642086

Cited by 134 publications

(163 citation statements)

References 79 publications

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“…Key questions concern the physical organizing principles underlying such an order, how TQO does manifest, and how to mathematically characterize it. The symmetry based results apply also to systems displaying emergent symmetries (5). The latter are not actual invariances of the system but emerge as exact symmetries at low energies.…”

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confidence: 94%

Sufficient symmetry conditions for Topological Quantum Order

Nussinov

Ortíz

2009

Proc. Natl. Acad. Sci. U.S.A.

152

171

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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.T he role of invariance (symmetry) principles in accounting for observed regularities is well known (1). Invariances become particularly effective in quantum mechanics where the linear character of Hilbert space enables us to construct superpositions of states that transform as irreducible representations of various symmetry groups. First discovered were invariance principles of a geometric character relating to space-time displacements and uniform motion. The best known local invariance operations (i) relate different coordinate systems to one other by such geometric deformations that leave the (local) metric invariant and (ii) appear as local transformations that link different gauge theory representations. It was recently realized that probing nonlocal (topological) structures uncovers new invariance principles with physical (and experimental) consequences. Particle-wave type dualities link the seemingly different local and nonlocal structures.Understanding the thermodynamic phases of matter via symmetry principles enables characterization by universal behaviors as in Landau's theory of phase transitions (2). In this theory, the order parameter(s) of the system relate to probing its local structure. A non-vanishing order parameter, encoding the breaking of a symmetry, defines the ordered state while the restoration of that symmetry signals the transition to a disordered state. A new paradigm, Topological Quantum Order (TQO), extends the Landau symmetry-breaking framework (3). In essence, this new order is associated with robustness against local perturbations, and hence cannot be described in principle by local order parameters. Thus, the underlying order remains hidden to ordinary local probes. Indeed, this new order exhibits nonlocal correlations that potentially lead to novel physical consequenc...

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mentioning

confidence: 94%

Sufficient symmetry conditions for Topological Quantum Order

Nussinov

Ortíz

2009

Proc. Natl. Acad. Sci. U.S.A.

152

171

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“…But one is not limited to these oscillators realizations. Indeed, one can use, for instance, SU(N) or hard-core particles realizations to construct new exactly-solvable Hamiltonians [12].…”

Section: Introductionmentioning

confidence: 99%

Exactly-solvable models derived from a generalized Gaudin algebra

et al. 2005

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We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen-Cooper-Schrieffer, Suhl-Matthias-Walker, Lipkin-Meshkov-Glick, the generalized Dicke and atom-molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.

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“…The spin S = 1 system can be mapped into a gas of semi-hard-core bosons 13,14,15 with no more than two bosons per site. The |S z = 1 >, |S z = 0 > and |S z = −1 > states are mapped into the states with two, one and zero bosons respectively.…”

Section: Introductionmentioning

confidence: 99%

Magnetostriction in the Bose-Einstein condensate quantum magnet NiCl2–4SC(NH2)2 (Invited)

Zapf

Correa²,

Batista

et al. 2007

Journal of Applied Physics

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The quantum magnet NiCl2-4SC(NH2)2 is a candidate for observing Bose-Einstein Condensation of spin degrees of freedom in applied magnetic fields. An XY antiferromagnetic ordered state occurs in a dome-shaped region of the temperature-field phase diagram between Hc1 = 2.1 T and Hc2 = 12.6 T and below 1.2 K. BEC corresponds to the field-induced quantum phase transition into the ordered state. We investigate magnetostriction in single crystals of this compound at dilution refrigerator temperatures in magnetic fields up to 18 T, and as a function of magnetic field angle. We show that significant changes in the lattice parameters are induced by magnetic fields, and argue that these result from antiferromagnetic couplings between the Ni spins along the tetragonal c-axis. The magnetic phase diagram as a function of temperature, field, and field angle can be extracted from these data. We discuss the implications of these results to Bose-Einstein Condensation in this system.

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Algebraic approach to interacting quantum systems

Cited by 134 publications

References 79 publications

Sufficient symmetry conditions for Topological Quantum Order

Sufficient symmetry conditions for Topological Quantum Order

Exactly-solvable models derived from a generalized Gaudin algebra

Magnetostriction in the Bose-Einstein condensate quantum magnet NiCl2–4SC(NH2)2 (Invited)

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