ar X iv : h ep - t h / 99 02 06 5 v 1 9 F eb 1 99 9 Dynamical Domain Wall Defects in 2 + 1 Dimensions

We study some dynamical properties of a Dirac field in 2+1 dimensions with spacetime dependent domain wall defects. We show that the Callan and Harvey mechanism applies even to the case of defects of arbitrary shape, and in a general state of motion. The resulting chiral zero modes are localized on the worldsheet of the defect, an embedded curved two dimensional manifold. The dynamics of these zero modes is governed by the corresponding induced metric and spin connection. Using known results about determinants and anomalies for fermions on surfaces embedded in higher dimensional spacetimes, we show that the chiral anomaly for this two dimensional theory is responsible for the generation of a current along the defect. We derive the general expression for such a current in terms of the geometry of the defect, and show that it may be interpreted as due to an “inertial” electric field, which can be expressed entirely in terms of the spacetime curvature of the defects. We discuss the application of this framework to fermionic systems with defects in condensed matter. [email protected] [email protected] [email protected] 1 The behavior of fermions in the background of defects is a problem of general interest in many areas of physics ranging from the behavior of textures in superfluid phases of He3 [1] to cosmic strings [2]. The prototype of these problems is the Callan-Harvey effect [3] in which a Fermi field (whose physical meaning depends on the specific problem) in an odd-dimensional spacetime couples to the defect through a mass term which takes a positive value on half of the space and a negative value on the other half. Under these circumstances, the matter (Fermi) field has a bound state (a zero mode) at the defect which is capable of carrying currents. Thus, the presence of the defect leads to an anomaly in the gauge current in the bulk which is exactly compensated by an equal and opposite anomaly at the defect. A similar phenomenon is also found at the edge of a droplet of a (non-relativistic) twodimensional electron gas in a magnetic field in the regime of the fractional and integer quantum Hall effects, where the anomaly in the bulk Hall current is cancelled by the gauge anomaly of the chiral edge states[4, 5]. While in most problems static defects are sufficient to describe the physics, in many others (particularly in condensed matter) it is of current interest to understand the possible effects due to the dynamics of the defects. In particular, it is interesting to investigate the interaction of the fermion zero modes with the fluctuating geometry induced by the dynamics of the defect. In this letter we shall generalize the results of [6], which dealt with a Dirac field in 2 + 1 dimensions having space dependent domain wall like defects in their mass, to the case of an arbitrary, space and time dependent defect. We shall show that the analog of the Callan and Harvey mechanism [3], namely, the existence of a chiral zero mode, still holds, with the zero mode now localized on the defect worldsheet. An important difference is that now, even in the absence of any external gauge field, there is an induced fermionic current on the defect, caused by the gravitational chiral anomaly for the fermions localized on the defect. This current can be thought of as due to the existence of an effective or “inertial” electric field, defined in terms of the geometry of the defect (the term “inertial” comes from the fact that it vanishes when the defect is not accelerated). The action for a Dirac fermion in three Euclidean dimensions, described by the Cartesian coordinates x, x, x (x denoting the Euclidean time), in the presence of a space-time dependent mass M(x), is given by S = ∫ dx Ψ̄(x) [γDα +M(x)] Ψ(x) (1)