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Physical review. B, Condensed matter

Elementary derivation of one-dimensional fermion-number fractionalization

Michael Stone

Published May 1, 1985• 35 citations
Publisher

Abstract

An elementary explanation of the origin of topological or fractional charges in one dimension is presented. Attention is paid to where the balance of the charge is located and the influence of the choice of boundary conditions on the problem.
DOI
10.1103/physrevb.31.6112

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Fermion number fractionization in quantum field theory on a finite interval is studied for a (1 + 1) dimensional fermion-soliton system with explicit charge conjugation symmetry breaking. The effects of boundary conditions on the fractional fermion number and the connection with the corresponding open space problem are investigated. It is argued that the open space fractional charges can be correctly reproduced from the finite interval results only through a careful definition of what is meant by the soliton charge. This definition of the charge distinguishes between the fermionic and boundary induced charges in the system, and isolates the soliton from possibly other charged topological objects in the system. It therefore gives a true measure of the localized fractional fermion number induced on the soliton of interest. It is then rigorously proven that the corresponding charge fluctuations vanish, and hence that the induced fractional charge on the soliton is a quantum observable.
Invertible fermionic topological (IFT) phases are gapped phases of matter with nondegenerate ground states on any closed spatial manifold. When open boundary conditions are imposed, nontrivial IFT phases support gapless boundary degrees of freedom. Distinct IFT phases in one-dimensional space with an internal symmetry group Gf{G}_{f}Gf​ have been characterized by a triplet of indices $([(\ensuremath{\nu},\ensuremath{\rho})],[\ensuremath{\mu}])$. Our main result is an elementary derivation of the fermionic stacking rules of one-dimensional IFT phases for any given internal symmetry group Gf{G}_{f}Gf​ from the perspective of the boundary, i.e., we give an explicit operational definition for the boundary representation $([({\ensuremath{\nu}}_{\ensuremath{\wedge}},{\ensuremath{\rho}}_{\ensuremath{\wedge}})],[{\ensuremath{\mu}}_{\ensuremath{\wedge}}])$ obtained from stacking two IFT phases characterized by the triplets of boundary indices $([({\ensuremath{\nu}}_{1},{\ensuremath{\rho}}_{1})],[{\ensuremath{\mu}}_{1}])$ and $([({\ensuremath{\nu}}_{2},{\ensuremath{\rho}}_{2})],[{\ensuremath{\mu}}_{2}])$, respectively.
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