Abstract
Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.
Highlights
The collective excitations of two dimensional electron gas in strong magnetic field carry a serial of fractional fractional charges, which are measured by the fractional Hall conductance [1] and explained by Laughlin wavefunction [2] as well as composite fermion theory [3]
Topological order inspired by Fractional quantum Hall effect (FQHE) have attracted longstanding research interest on fractionally charged quasiparticles in condensed matter physics [4][5][6][7]
A topological path fusion theory is developed to generate serial of fractional charges that is similar to those in FQHE and beyond. This physical theory is rooted in the path integral theory of quantum mechanics, topology theory of train tracks and knot lattice model of anyon, providing a mathematical insight on the physical origin of fractional charges in quantum system in magnetic flux lattice
Summary
The collective excitations of two dimensional electron gas in strong magnetic field carry a serial of fractional fractional charges, which are measured by the fractional Hall conductance [1] and explained by Laughlin wavefunction [2] as well as composite fermion theory (i.e., one electron binding together with a pair of magnetic flux) [3]. Mapping the unbroken electric current into a simple closed curve and the flux into a genus under the mathematical constraint that the curve avoid crossing itself everywhere, the braiding operations of fluxes enclosed by a loop current can be well quantified by Thurston’s train track theory [16][17], which is applied to design the optimal mixing strategy of two fluids with low Reynolds number [18][19], study the topological fluid mechanics of point vortex [20] and topological chaos in dynamics systems [21][22] This topological path fusion model can be implemented by topological mixing of two quantum fluids, one is charged superfluid which is experimentally realizable by charged superfluid helium [23], the other is normal viscous fluid helium.
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