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

Abstract

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 andmore » 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.« less

Authors:
 [1]
  1. Univ. of Sussex (United Kingdom)
Publication Date:
OSTI Identifier:
113770
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 241; Journal Issue: 2; Other Information: PBD: 1 Aug 1995
Country of Publication:
United States
Language:
English
Subject:
66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIRAC OPERATORS; FREDHOLM EQUATION; HAMILTONIANS; GREEN FUNCTION; SOLITONS

Citation Formats

Waxman, D. The Fredholm determinant for a Dirac Hamiltonian with a topological mass term. United States: N. p., 1995. Web. doi:10.1006/aphy.1995.1064.
Waxman, D. The Fredholm determinant for a Dirac Hamiltonian with a topological mass term. United States. https://doi.org/10.1006/aphy.1995.1064
Waxman, D. 1995. "The Fredholm determinant for a Dirac Hamiltonian with a topological mass term". United States. https://doi.org/10.1006/aphy.1995.1064.
@article{osti_113770,
title = {The Fredholm determinant for a Dirac Hamiltonian with a topological mass term},
author = {Waxman, D},
abstractNote = {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.},
doi = {10.1006/aphy.1995.1064},
url = {https://www.osti.gov/biblio/113770}, journal = {Annals of Physics (New York)},
number = 2,
volume = 241,
place = {United States},
year = {1995},
month = {8}
}