A Note on braid statistics and the nonAbelian Aharonov-Bohm effect

It is widely believed that the braiding statistics of the quasiparticles of the fractional quantum Hall effect is a robust, topological property, independent of the details of the Hamiltonian or the wave function. However, for the quasiparticles of the 1/3 state, an explicit evaluation of the braiding phases using Laughlin's wave function has not produced a well-defined braiding statistics. We revisit this issue and demonstrate that the expected braiding statistics is recovered in the thermodynamic limit for exchange paths that are of finite extent but not for macroscopically large exchange loops that encircle a finite fraction of electrons. We argue that the difference between the two kinds of paths arises due to tiny (order $1/N$ ) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models.It is widely believed that the braiding statistics of the quasiparticles of the fractional quantum Hall effect is a robust, topological property, independent of the details of the Hamiltonian or the wave function. However, for the quasiparticles of the 1/3 state, an explicit evaluation of the braiding phases using Laughlin's wave function has not produced a well-defined braiding statistics. We revisit this issue and demonstrate that the expected braiding statistics is recovered in the thermodynamic limit for exchange paths that are of finite extent but not for macroscopically large exchange loops that encircle a finite fraction of electrons. We argue that the difference between the two kinds of paths arises due to tiny (order $1/N$) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models.

Physical Review BJan 12, 201013 citations

Effective field theory of dipolar braiding statistics in two dimensions

Yun-Tak Oh, Jintae Kim, Jung Hoon Han

A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.

arXiv:2204.01279Physical review. B./Physical review. BOct 26, 202223 citations

1/N

) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models.