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Journal of Physics A Mathematical and General

Conformal invariance and line defects in the two-dimensional Ising model

Malte Henkel, A. Patkós

Published Feb 21, 1988• 37 citations
Publisher

Abstract

The quantum Ising chain with p equidistant defects is studied. The exact form of the Hamiltonian is found for an infinite number of sites. Using conformal invariance, generalised corner exponents are computed and found to depend continuously on the defect strengths. Realisations of extended conformal algebras in the Hamiltonian spectrum are obtained.

DOI
10.1088/0305-4470/21/4/008

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The quantum Ising chain with p equidistant defects is studied. The exact form of the Hamiltonian is found for an infinite number of sites. Using conformal invariance, generalised corner exponents are computed and found to depend continuously on the defect strengths. Realisations of extended conformal algebras in the Hamiltonian spectrum are obtained.
The Ising model with an infinite line of defects is mapped onto a strip with two defect lines. The Hamiltonian spectrum is studied at the bulk critical point. Its exact diagonal form is found for an infinite number of sites. The spectrum of physical excitations contains an infinite number of primary fields, while the leading ground-state energy correction is independent of the defect strength. A novel algebraic structure interpolating between those belonging to periodic and free boundary conditions is signalled.
Abstract In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations , cousins of the Yang–Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers–Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.
Critical phenomena in the two-dimensional Ising model with a defect line are studied using boundary conformal field theory on the c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 orbifold. Novel features of the boundary states arising from the orbifold structure, including continuously varying boundary critical exponents, are elucidated. New features of the Ising defect problem are obtained including a novel universality class of defect lines and the universal boundary to bulk crossover of the spin correlation function.