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 {Lm + L−m} and {i (Lm − 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 \({M(k + 2, k + 1), k \geq 3}\) 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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Aasen D., Mong R.S., Fendley P.: Topological defects on the lattice: I. the Isingmodel. J. Phys. AMath. Theor. 49(35), 354001 (2016)
Aharonov,D.,Arad, I., Eban, E., Landau, Z.: Polynomial quantumalgorithms for additive approximations of the Potts model and other points of the Tutte plane (2007). arXiv preprint arXiv:quant-ph/0702008
Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Elsevier, Amsterdam (2016)
Bondesan R., Dubail J., Faribault A., Ikhlef Y.: Chiral SU(2)k currents as local operators in vertex models and spin chains. J. Phys. A Math. Theor. 48(6), 065205 (2015)
Bravyi S.B., Kitaev A.Y.: Fermionic quantum computation. Ann. Phys. 298(1), 210–226 (2002)
Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back (2015). arXiv preprint arXiv:1503.01260
Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Isingmodel (2012). arXiv preprint arXiv:1202.2838
Engel K.-J., Nagel R.: A Short Course on Operator Semigroups. Springer, Berlin (2006)
Feiguin A., Trebst S., Ludwig A.W., Troyer M., Kitaev A., Wang Z., Freedman M.H.: Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett. 98(16), 160409 (2007)
Fendley P.: Free parafermions. J. Phys. A Math. Theor. 47(7), 075001 (2014)
Feverati G., Pearce P.A.: Critical RSOS and minimal models: fermionic paths, Virasoro algebra and fields. Nucl. Phys. B 663(3), 409–442 (2003)
Fiedler L., Naaijkens P., Osborne T.J.: Jones index, secret sharing and total quantum dimension. New J. Phys. 19(2), 023039 (2017)
Francesco P., Mathieu P., Sénéchal D.: Conformal Field Theory. Springer, Berlin (2012)
Fredenhagen K., Hertel J.: Local algebras of observables and pointlike localized fields. Commun.Math. Phys. 80(4), 555–561 (1981)
Freedman M., Kitaev A., Larsen M., Wang Z.: Topological quantum computation. Bull. Am. Math. Soc. 40(1), 31–38 (2003)
Freedman M.H., Kitaev A., Wang Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227(3), 587–603 (2002)
Frenkel I., Huang Y.-Z., Lepowsky J.: On Axiomatic Approaches to Vertex Operator Algebras and Modules, vol. 494. American Mathematical Society, Providence (1993)
Gainutdinov A., Jacobsen J., Read N., Saleur H., Vasseur R.: Logarithmic conformal field theory: a lattice approach. J. Phys. A Math. Theor. 46(49), 494012 (2013a)
Gainutdinov A., Read N., Saleur H.: Bimodule structure in the periodic \({\mathfrak{g} \mathfrak{l}(1|1)}\) spin chain. Nucl. Phys. B 871(2), 289–329 (2013b)
Gainutdinov, A., Read, N., Saleur, H.: Continuumlimit and symmetries of the periodic \({\mathfrak{g} \mathfrak{l}(1|1)}\) spin chain. Nucl. Phys. B 871(2), 245–288 (2013)
Gainutdinov A., Read N., Saleur H.: Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the \({\mathfrak{g} \mathfrak{l}(1|1)}\) periodic spin chain, Howe duality and the interchiral algebra. Commun. Math. Phys. 341(1), 35–103 (2016)
Gainutdinov A., Read N., Saleur H., Vasseur R.: The periodic s l(2|1) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at c = 0. J. High Energy Phys. 2015(5), 114 (2015)
Gainutdinov A., Saleur H., Tipunin I.Y.: Lattice W-algebras and logarithmic CFTs. J. Phys. A Math. Theor. 47(49), 495401 (2014)
Gils C., Ardonne E., Trebst S., Huse D.A., Ludwig A.W., Troyer M., Wang Z.: Anyonic quantum spin chains: spin-1 generalizations and topological stability. Phys. Rev. B 87(23), 235120 (2013)
Ginsparg, P.: Applied conformal field theory (1988). arXiv preprint arXiv:hep-th/9108028
Goodman R., Wallach N.R.: Projective unitary positive-energy representations of Diff(S 1). J. Funct. Anal. 63(3), 299–321 (1985)
Gui, B.:Unitarity of themodular tensor categories associated to unitary vertex operator algebras, I (2017). arXiv preprint arXiv:1711.02840
Haag R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (2012)
Huang Y.-Z.: Differential equations and intertwining operators. Commun. Contemp. Math. 7(03), 375–400 (2005a)
Huang Y.-Z.: Differential equations, duality and modular invariance. Commun. Contemp. Math. 7(05), 649–706 (2005b)
Jones, V.F.: In and around the origin of quantum groups (2003). arXiv preprint arXiv:math/0309199
Jones, V.F.: Some unitary representations of Thompson’s groups F and T (2014). arXiv preprint arXiv:1412.7740
Jones, V.F.: A no-go theorem for the continuum limit of a periodic quantum spin chain (2016). arXiv preprint arXiv:1607.08769
Jones, V.F.: Scale invariant transfer matrices and Hamiltionians (2017). arXiv preprint arXiv:1706.00515
Jordan S.P., Lee K.S., Preskill J.: Quantum algorithms for quantum field theories. Science 336(6085), 1130–1133 (2012)
Kaufmann R.M.: Path space decompositions for the Virasoro algebra and its Vermamodules. Int. J.Mod. Phys. A 10(07), 943–961 (1995)
Kawahigashi Y., Longo R.: Classification of local conformal nets. Case c < 1 Ann.Math. 160, 493–522 (2004)
König R., Scholz V.B.: Matrix product approximations to conformal field theories.Nucl. Phys.B 920, 32–121 (2017)
Koo W., Saleur H.: Representations of the Virasoro algebra from lattice models. Nucl. Phys. B 426(3), 459–504 (1994)
Lloyd S.: Universal quantum simulators. Science 273(5278), 1073 (1996)
Milsted A., Vidal G.: Extraction of conformal data in critical quantum spin chains using the Koo–Saleur formula. Phys. Rev. B 96(24), 245105 (2017)
Mong R.S., Clarke D.J., Alicea J., Lindner N.H., Fendley P.: Parafermionic conformal field theory on the lattice. J. Phys. A Math. Theor. 47(45), 452001 (2014)
Murg V., Cirac J.I.: Adiabatic time evolution in spin systems. Phys. Rev. A 69(4), 042320 (2004)
Neretin Y.A.: Categories of Symmetries and Infinite-Dimensional Groups.vol. 16. Oxford University Press, Oxford (1996)
Pasquier V., Saleur H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330(2-3), 523–556 (1990)
Read N., Saleur H.: Enlarged symmetry algebras of spin chains, loop models, and S-matrices. Nucl. Phys. B 777(3), 263–315 (2007)
Rowell E.C., Wang Z.: Localization of unitary braid group representations. Commun. Math. Phys. 311(3), 595–615 (2012)
Rowell, E.C., Wang, Z.: Mathematics of topological quantum computing (2017). arXiv preprint arXiv:1705.06206
Schweigert, C., Fuchs, J., Runkel, I.: Categorification and correlation functions in conformal field theory. In: Proceedings of the ICM, pp. 443–458. European Mathematical Society Zürich (2006)
Seiberg, N.: What is a quantum field theory? (2015). Simons center talk: http://scgp.stonybrook.edu/video_portal/video.php?id=389
Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Proceedings of the International Congress of Mathematicians Madrid, 22–30 Aug 2006, pp. 1421–1451 (2007)
Tener, J.E.: Geometric realization of algebraic conformal field theories (2016). arXiv preprint arXiv:1611.01176
Tener, J.E., Wang, Z.: On classification of extremal non-holomorphic conformal field theories (2016). arXiv preprint arXiv:1611.04071
Weiner, M.: Conformal covariance and related properties of chiral QFT (2007). arXiv preprint arXiv:math/0703336
Witten E.: Quantumfield theory and the jones polynomial. Commun.Math.Phys. 121(3), 351–399 (1989)
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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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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DOI: https://doi.org/10.1007/s00220-018-3254-1