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Journal of Physics Conference Series

Hodge structures in topological quantum mechanics

Grzegorz Banaszak, Jan Milewski

Published Mar 1, 2010• 1 citation
Publisher

Abstract

We parametrize some quantum numbers in two-dimensional quantum mechanics by means of a Hodge structure. For an anyon system the statistical parameter leads to a deformation of the Hodge structure.
DOI
10.1088/1742-6596/213/1/012017

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An elementary introduction is given to the concept of anyons—quantum particles whose ‘statistics’ interpolate smoothly between those of bosons and fermions. Such particles—or quasi-particle excitations—can occur in two-dimensional systems in which the particles experience strong short-range mutual repulsions. These circumstances effectively mean that one has to apply quantum mechanics to a configuration space in which some points are excluded (i.e. it has ‘holes’)—namely, those points where two particles would coincide. The Aharonov—Bohm effect provides the simplest physical example of quantum mechanics in such a non-simply connected space, and is also fundamental to a simple model for anyons. A qualitative introduction is given to two currently proposed physical applications of anyons: high temperature superconductivity, and the (fractional) quantum Hall effect.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.
In this paper we investigate an entanglement measure, the Rényi entropy, in a 1+1D integrable quantum field theory known as the Federbush model. This is a deformation of the theory of two massive Dirac fermions by means of a bilinear term in the U(1)U(1)U(1) currents that couples the two fermion species. This deformation gives rise to SSS-matrix elements which are coupling-dependent phases, distinct from −1- 1−1. These non-trivial phases can be seen as encoding anyon-like statistics. From this viewpoint, the Federbush model is a toy model for topological features of entanglement in one space dimension. In this paper we show that, for an infinite system, these topological features play no role when computing many known measures of entanglement at equilibrium in the ground state. This conclusion applies also to the post-quench dynamics after a small quench of the topological parameter.