nLab fictitious gauge field

Idea

In general the term fictitious gauge field could refer to any auxiliary gauge field. Specifically the term is used:

• in anyon statistics for a model of anyons as fermions which each source and each couple to a flat U(1)-gauge field causing mutual Aharonov-Bohm phases;

• in solid state physics for effects analogous to Berry curvatures caused by crystal strain or dislocations.

For anyon statistics

A model for anyon statistics via otherwise free fermions in 2d, mutually interacting through a flat “fictitious gauge field” has been proposed in Arovas, Schrieffer, Wilczek & Zee 1985, developed in Chen, Wilczek, Witten & Halperin 1989. This model has been advertized in several early reviews (e.g. Wilczek 1990, §I.3, Wilczek 1991, but seems not to have been developed much since:

The model regards anyons as a priori free fermions, but equipped now with a non-local mutual interaction via a “fictitious gauge field” (CWWH89, §2), in that each of the particles is modeled as the singular source of a flat circle connection (a vector potential with vanishing field strength), which hence exerts no Lorentz force but has the effect that globally each particle is subject to the same Aharonov-Bohm effect as would be caused by a tuple of infinite solenoids piercing through each of the other particle’s positions.

For emphasis, from CWWH89, p. 359:

Here the particles are to be regarded (in the absence of interactions) as fermions; the interaction then makes them anyons with statistical parameter $\theta = \pi(1 - 1/n)$.

It follows that (quoting from Fröhlich, Gabbiani & Marchetti 1990, p. 20):

If $\theta \in\!\!\!\!\!/ \frac{1}{2}\mathbb{Z}$ the Hilbert space of anyon wave functions must be chosen to be a space of multi-valued functions with half-monodromies given by the phase factors $exp(2 \pi \mathrm{i} \theta)$. Such wave functions can be viewed as single-valued functions on the universal cover $\widetilde M_n$ of $M_n$ $[$the configuration space of points$]$.

Incidentally, the quasi-particle-excitations of (or in) a gas of such Aharonov-Bohm phased anyons are argued to be vortices (CWWH89, p. 457):

we are led to conclude that in anyon superconductivity, charged quasi-particles and vortices do not constitute two separate sorts of elementary excitations - they are one and the same.

References

For anyon statistics

• Daniel P. Arovas, Robert Schrieffer, Frank Wilczek, Anthony Zee, Statistical mechanics of anyons, Nuclear Physics B 251 (1985) 117-126 (reprinted in Wilczek 1990, p. 173-182) $[$doi:10.1016/0550-3213(85)90252-4$]$

• Yi-Hong Chen, Frank Wilczek, Edward Witten, Bertrand Halperin, On Anyon Superconductivity, International Journal of Modern Physics B 03 07 (1989) 1001-1067 (reprinted in Wilczek 1990, p. 342-408) $[$doi:10.1142/S0217979289000725, pdf$]$

Review in:

• Jürg Fröhlich, Fabrizio Gabbiani, Pieralberto Marchetti, Braid statistics in three-dimensional local quantum field theory, in: H.C. Lee (ed.) Physics, Geometry and Topology NATO ASI Series, 238 Springer (1990) $[$doi:10.1007/978-1-4615-3802-8_2, pdf$]$

• Frank Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific (1990) $[$doi:10.1142/0961$]$

• Frank Wilczek, States of Anyon Matter, International Journal of Modern Physics B 05 09 (1991) 1273-1312 $[$doi:10.1142/S0217979291000626$]$

In solid state physics

Fictitious gauge fields:

• Mircea Trif, Pramey Upadhyaya, Yaroslav Tserkovnyak, Theory of electromechanical coupling in dynamical graphene $[$arXiv:1210.7384, doi:10.1103/PhysRevB.88.245423$]$