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Topological Defects on the Lattice I: The Ising model

Assen, David and Mong, Roger S. K. and Fendley, Paul (2016) Topological Defects on the Lattice I: The Ising model. Journal of Physics A: Mathematical and Theoretical, 49 (35). Art. No. 354001. ISSN 1751-8113. doi:10.1088/1751-8113/49/35/354001.

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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.

Item Type:Article
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Additional Information:© 2016 IOP Publishing Ltd. Received 25 January 2016; Accepted for publication 7 March 2016; Published 9 August 2016. We thank Jason Alicea for illuminating discussions. DA gratefully acknowledges the support of the NSERC PGSD program. RM is grateful for support from the Sherman Fairchild Foundation and the Institute for Quantum Information and Matter.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Sherman Fairchild FoundationUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Issue or Number:35
Record Number:CaltechAUTHORS:20160523-105542824
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Official Citation:David Aasen et al 2016 J. Phys. A: Math. Theor. 49 354001
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67253
Deposited By: Ruth Sustaita
Deposited On:23 May 2016 18:24
Last Modified:12 Jul 2022 19:46

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