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Conformal Field Theories as Scaling Limit of Anyonic Chains

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Abstract

We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 4.3 on conditions when a chiral unitary rational (1+1)-conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4, 3) using Ising anyonic chains. Part of the conjecture is a precise relation between Temperley–Lieb generators {ei} and some finite stage operators of the Virasoro generators {LmLm} and {i (LmLm)} for unitary minimal models M(k + 2, k + 1) in Conjecture 5.5. A similar earlier relation is known as the Koo–Saleur formula in the physics literature (Koo and Saleur in Nucl Phys B 426(3):459–504, 1994). Assuming Conjecture 4.3, most of our main results for the Ising minimal model M(4, 3) hold for unitary minimal models \({M(k + 2, k + 1), k \geq 3}\) as well. Our approach is inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers, and supported by extensive numerical simulation and physical proofs in the physics literature.

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Acknowledgments

We would like to thank James Tener for many helpful discussions, in particular pointing out the relevance of Kaplansky’s theorem for the proof of Corollary 4.10. We thank P. Fendley, V. Jones, R. Koenig, H. Saleur, and anonymous referees for helpful comments to improve our paper and their encouragements. The second author is partially supported by NSF Grant DMS-1411212.

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Correspondence to Zhenghan Wang.

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Communicated by Y. Kawahigashi

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Zini, M.S., Wang, Z. Conformal Field Theories as Scaling Limit of Anyonic Chains. Commun. Math. Phys. 363, 877–953 (2018). https://doi.org/10.1007/s00220-018-3254-1

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