Tunneling between fermionic vacua and the overlap formalism

TL;DR

This paper investigates tunneling between different fermionic vacua in 2+1 dimensions, connecting the probability amplitudes to the overlap formalism and evaluating specific cases with boundary conditions and magnetic fields.

Contribution

It establishes a link between tunneling probabilities in fermionic vacua and the overlap formalism for chiral determinants, providing explicit calculations for particular geometries.

Findings

Transition probability approaches the squared chiral determinant in the large mass limit.

Explicit evaluation of tunneling probabilities on a torus with twisted boundary conditions.

Calculation of tunneling probabilities on a disk with an external magnetic field.

Abstract

The probability amplitude for tunneling between the Dirac vacua corresponding to different signs of a parity breaking fermionic mass $M$ in $2+1$ dimensions is studied, making contact with the continuum overlap formulation for chiral determinants. It is shown that the transition probability in the limit when $M \to \infty$ corresponds, via the overlap formalism, to the squared modulus of a chiral determinant in two Euclidean dimensions. The transition probabilities corresponding to two particular examples: fermions on a torus with twisted boundary conditions, and fermions on a disk in the presence of an external constant magnetic field are evaluated.