Anyons in three dimensions with geometric algebra
Alexander Soiguine2016, arXiv: General Physics
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Abstract
Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum phenomena.
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We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called "missing" states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum.
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2011
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Usach/96/05, IFUSP/P-1218 ANYONS IN 1+1 DIMENSIONS ARE ANOMALOUS
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We consider the analog in one spatial dimension of the Bose-Fermi transmutation for planar systems. That is, the construction of a purely bosonic effective local theory starting from a system of bosons and fermions upon integration over the fermionic variables. We consider a quantum mechanical system of a spin 1/2 particle coupled to an abelian gauge field, which is classically invariant under gauge transformations and charge conjugation. It is found that, unless the flux enclosed by the particle orbits is quantized, and the spin takes a value n + 1/2, at least one of the two symmetries would be anomalous. Thus, charge conjugation invariance and the existence of abelian instantons simultaneously avoid the anomaly and force the particles to be
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Annals of Physics, 2008
The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined ''anyons'', may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.
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Physics Letters B, 1991
We consider systems containing two or more distinct species of particles in two spatial dimensions. In quantizations of these systems, the statistics of composites containing more than one type of particle are not completely determined by the statistics of the constituents. In particular there exist quantum theories in which two bosons can combine to form an anyon with any desired statistical angle. We demonstrate these results using the topological approach to quantum kinematics, which in this case leads to a generalization of ordinary braid theory in two dimensions. Comparisons are made to three-dimensional systems.
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Anyons from three-body hard-core interactions in one dimension
Annals of Physics, 2019
Traditional anyons in two dimensions have generalized exchange statistics governed by the braid group. By analyzing the topology of configuration space, we discover that an alternate generalization of the symmetric group governs particle exchanges when there are hard-core three-body interactions in one-dimension. We call this new exchange symmetry the traid group and demonstrate that it has abelian and non-abelian representations that are neither bosonic nor fermionic, and which also transform differently under particle exchanges than braid group anyons. We show that generalized exchange statistics occur because, like hard-core two-body interactions in two dimensions, hard-core three-body interactions in one dimension create defects with co-dimension two that make configuration space no longer simply-connected. Ultracold atoms in effectively onedimensional optical traps provide a possible implementation for this alternate manifestation of anyonic physics.
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We study the dynamics of bosonic and fermionic anyons defined on a one-dimensional lattice, under the effect of Hamiltonians quadratic in creation and annihilation operators, commonly referred to as linear optics. These anyonic models are obtained from deformations of the standard bosonic or fermionic commutation relations via the introduction of a non-trivial exchange phase between different lattice sites. We study the effects of the anyonic exchange phase on the usual bosonic and fermionic bunching behaviors. We show how to exploit the inherent Aharonov-Bohm effect exhibited by these particles to build a deterministic, entangling two-qubit gate and prove quantum computational universality in these systems. We define coherent states for bosonic anyons and study their behavior under two-mode linear-optical devices. In particular we prove that, for a particular value of the exchange factor, an anyonic mirror can generate cat states, an important resource in quantum information proces...
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