The Fredholm Determinant for a Dirac Hamiltonian with a Topological Mass Term

Abstract

We consider the Fredholm determinant associated with two Hamiltonians H and H0. If these have discrete spectra with eigenvalues En and En0, respectively, then the Fredholm determinant, as a function of a complex variable z, is Det[(z − H)/(z − H0)] ≡ Πn [(z − En)/(z − En0)]. This object contains information on the spectra of the operators H and H0 which we take to be Dirac Hamiltonians appropriate to one spatial dimension. The Pauli matrices σ̂i (i = 1, 2, 3) have been used as a representation of the Dirac gamma matrices. An expression for the Fredholm determinant is derived when the term in H involving the mass has a variation with position, x, of the form Δ̂(x) ≡ Δ2(x) σ̂2 + Δ3(x) σ̂3. We consider the general case Δ̂( − ∞) ≠ Δ̂(∞) and when this holds, the Hamiltonian is said to possess a topological mass term. The mass term in H0 is taken to have the same asymptotic limits as Δ̂(x) but different local behaviour. We find that the Fredholm determinant can be compactly expressed in terms of the 2 × 2 matrices characterizing the asymptotic spatial properties of the Green′s function 1/(z − H). There are applications of this work to systems involving Fermions coupled to topological solitons. Examples of these have been studied in quantum field theory and condensed-matter physics. In the latter case, a Dirac-like equation may arise as an approximate description of non-relativistic Fermions; the mass term usually having the interpretation as an order-parameter field.