zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.