Conformal Field Theories as Scaling Limit of Anyonic Chains

• Published in: Communications in mathematical physics 2018-11, Vol.363 (3), p.877-953
• Main Authors: Zini, Modjtaba Shokrian, Wang, Zhenghan
• Format: Article
• Language: English
• Subjects: Chains; Classical and Quantum Gravitation; Complex Systems; Computer simulation; Field theory; Generators; Ising model; Mathematical and Computational Physics; Mathematical models; Mathematical Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum computers; Quantum Physics; Relativistic theory; Relativity Theory; Scaling; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-018-3254-1

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 { e i } and some finite stage operators of the Virasoro generators {L m +  L − m } and { i ( L m − L − m )} 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 ≥ 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.