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A no-go theorem for the continuum limit of a periodic quantum spin chain. (English) Zbl 1397.82025
It is established (see Section 4) that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates a translation on the circle as a continuous limit of the rotations on the lattice.

MSC:
82B28 Renormalization group methods in equilibrium statistical mechanics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B81 Applications of Lie (super)algebras to physics, etc.
22E70 Applications of Lie groups to the sciences; explicit representations
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[1] Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, New York (1982) · Zbl 0538.60093
[2] Cannon, J.W.; Floyd, W.J.; Parry, W.R., Introductory notes on richard thompson’s groups, L’Enseign. Math., 42, 215-256, (1996) · Zbl 0880.20027
[3] Cirac, J.I.; Verstraete, F., Renormalization and tensor product states in spin chains and lattices, J.Phys. A Math. Theor., 42, 504004, (2009) · Zbl 1181.82010
[4] Dehornoy, P., Digne, F., Godelle, E., Krammer, D., Michel, J.: Foundations of Garside theory. arXiv:1309.0796 · Zbl 1370.20001
[5] Dehornoy, P., The group of parenthesized braids, Adv. Math., 205, 354-409, (2006) · Zbl 1160.20027
[6] Evenbly, G., Vidal, G.: Tensor network renormalization. Phys. Rev. Lett. 115, 180405 (2015) arXiv:1412.0732 · Zbl 1231.82021
[7] Ghosh, S.K., Jones, C.: Annular representation theory of rigid C\^{*}-tensor categories. J. Funct. Anal. 270, 1537-1584 (2016) · Zbl 1375.46039
[8] Graham, J.J.; Lehrer, G.I., The representation theory of affine Temperley-Lieb algebras, L’Enseign. Math., 44, 1-44, (1998) · Zbl 0964.20002
[9] Jones, V.F.R.: Planar algebras I, preprint. arXiv:math/9909027
[10] Jones, V.F.R., On knot invariants related to some statistical mechanical models, Pac. J. Math., 137, 311-334, (1989) · Zbl 0695.46029
[11] Jones, V.F.R., The annular structure of subfactors, in “essays on geometry and related topics”, Monogr. Enseign. Math., 38, 401-463, (2001)
[12] Jones, V.F.R.: In and around the origin of quantum groups. Prospects in math. Phys. Contemp. Math., 437 Am. Math. Soc. 101-126 (2007) arXiv:math.OA/0309199 · Zbl 0695.46029
[13] Jones, V.F.R.: Some unitary representations of Thompson’s groups F and T (2014). arXiv:1412.7740 · Zbl 0622.57004
[14] Jones, V.; Reznikoff, S., Hilbert space representations of the annular Temperley-Lieb algebra, Pac. Math. J., 228, 219-250, (2006) · Zbl 1131.46042
[15] Kauffman, L., State models and the Jones polynomial, Topology, 26, 395-407, (1987) · Zbl 0622.57004
[16] Morrison, S.; Peters, E.; Snyder, N., Categories generated by a trivalent vertex, Sel. Math. New Ser., 23, 817-868, (2017) · Zbl 06706083
[17] Pasquier, V.; Saleur, H., Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B, 330, 523-556, (1990)
[18] Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Applications of Combinatorial Mathematics, pp. 221-244. Academic Press, New York (1971) · Zbl 0216.43502
[19] Ren, Y.: From skein theory to presentations for Thompson group (2016). arXiv:1609.04077 · Zbl 06834827
[20] Thomas, R.: An update on the four-color theorem. Not. AMS 45, 848-859 (1998) · Zbl 0908.05040
[21] Temperley, H.N.V.; Lieb, E.H., Relations between the “percolation“ and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. R. Soc. A, 322, 251-280, (1971) · Zbl 0211.56703
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