Berry phases, quantum phase transitions and Chern numbers

Contreras, H.A.; Reyes-Lega, A. F.

doi:10.1016/j.physb.2007.10.131

Cited by 6 publications

(3 citation statements)

References 7 publications

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“…• Recently, new applications of geometric phases and topological invariants have appeared in the context of spin systems and in relation to quantum phase transitions. In particular, it has been shown in [62] that it is possible to relate certain topological invariants computed from the Hamiltonian and relate them to quantum criticality. The use of projective modules, as presented in this paper, can be very convenient in this context from the computational point of view.…”

Section: Discussionmentioning

confidence: 99%

On the geometry of quantum indistinguishability

Reyes-Lega¹

2011

J. Phys. A: Math. Theor.

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An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand–Naimark and Serre–Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wavefunctions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.

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Section: Discussionmentioning

confidence: 99%

On the geometry of quantum indistinguishability

Reyes-Lega¹

2011

J. Phys. A: Math. Theor.

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“…• Recently, new applications of geometric phases and topological invariants have appeared in the context of spin systems and in relation to quantum phase transitions. In particular, it has been shown in [63] that it is possible to relate certain topological invariants computed from the Hamiltonian and relate them to quantum criticality. The use of projective modules, as presented in the present paper, can be very convenient in this context from the computational point of view.…”

Section: Discussionmentioning

confidence: 99%

On the geometry of quantum indistinguishability

Reyes-Lega

2011

Preprint

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An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in nonrelativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wave functions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.

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“…Now, the relevance of this fact in the context of quantum phase transitions is that there are already some geometric characterizations of the critical point in terms of, e.g., Berry phases [44,45]. This has also been related to the behavior of certain Chern numbers associated to the parameter space of the spin chain [46]. Now, using Araki's selfdual formalism, it should also be possible to compute the cocycle outside the critical point.…”

Section: 3mentioning

confidence: 99%

Some Aspects of Operator Algebras in Quantum Physics

Reyes-Lega

2016

Geometric, Algebraic and Topological Methods for Quantum Field Theory

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Abstract. Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.

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Berry phases, quantum phase transitions and Chern numbers

Cited by 6 publications

References 7 publications

On the geometry of quantum indistinguishability

On the geometry of quantum indistinguishability

On the geometry of quantum indistinguishability

Some Aspects of Operator Algebras in Quantum Physics

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