The Fredholm determinant for a Dirac Hamiltonian with a topological mass term

Journal Article · Tue Aug 01 04:00:00 EDT 1995 · Annals of Physics (New York)

DOI:https://doi.org/10.1006/aphy.1995.1064· OSTI ID:113770

Waxman, D ^[1]

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1. Univ. of Sussex (United Kingdom)

We consider the Fredholm determinant associated with two Hamiltonians H and H{sub 0}. If these have discrete spectra with eigenvalues E{sub n} and E{sub n0}, respectively, then the Fredholm determinant, as a function of a complex variable z, is Det(z-H)/(z-H{sub 0}) {triple_bond} {Pi}{sub n}[(z-E{sub n})/(z-E{sub n0}]. This object contains information on the spectra of the operators H and H{sub 0} which we take to be Drac Hamiltonians appropriate to one spatial dimension. The Pauli matrices {sigma}{sup i} (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 {delta}(x) {triple_bond} {delta}{sub 2}(x) {sigma}{sup 2} + {delta}{sub 3}(x) {sigma}{sup 3}. We consider the general case {delta}(-{infinity}) = {delta}({infinity}) and when this holds, the Hamiltonian is said to possess a {delta}(x) but different local behaviour. We find that the Fredholm determinant can be compactly expressed in terms of the 2x2 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. 8 refs.

• OSTI ID: 113770
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 2 Vol. 241; ISSN APNYA6; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
DIRAC OPERATORS
FREDHOLM EQUATION
GREEN FUNCTION
HAMILTONIANS
SOLITONS