Scale invariant transfer matrices and Hamiltionians. (English) Zbl 1387.82010

Summary: Given a direct system of Hilbert spaces \(s\mapsto\mathcal{H}_s\) (with isometric inclusion maps \(\iota_s^t:\mathcal{H}_s\to\mathcal{H}_t\) for \(s\leqslant t\)) corresponding to quantum systems on scales \(s\), we define notions of scale invariant and weakly scale invariant operators. In some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.


82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
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[1] Andrews, G. E.; Baxter, R. J.; Forrester, P. J., Eight vertex SOS model and generalized Rogers-Ramanujan type identities, J. Stat. Phys., 35, 193-266, (1984) · Zbl 0589.60093
[2] Araki, H., Gibbs states of a one dimensional quantum lattice, Commun. Math Phys., 14, 120-157, (1969) · Zbl 0199.28001
[3] Asaeda, M.; Haagerup, U., Exotic subfactors of finite depth with Jones indices \({(5+\sqrt{13})}/{2}\) and \({(5+\sqrt{17})}/{2} \), Commun. Math. Phys., 202, 1-63, (1999) · Zbl 1014.46042
[4] Baxter, R. J., Exactly Solved Models in Statistical Mechanics, (1982), New York: Academic, New York · Zbl 0538.60093
[5] Birman, J. S.; Wenzl, H., Braids, link polynomials and a new algebra, Trans. Am. Math. Soc., 313, 249-273, (1989) · Zbl 0684.57004
[6] Cannon, J. W.; Floyd, W. J.; Parry, W. R., Introductory notes on Richard Thompson’s groups, L’Enseignement Math., 42, 215-256, (1996) · Zbl 0880.20027
[7] Cirac, J. I.; Verstraete, F., Renormalization and tensor product states in spin chains and lattices, J. Phys. A: Math. Theor., 42, (2009) · Zbl 1181.82010
[8] Evans, D. E.; Gannon, T., The exoticness and realisability of twisted Haagerup-Izumi modular data, Commun. Math. Phys., 307, 463-512, (2011) · Zbl 1236.46055
[9] Evenbly, G.; Vidal, G., Tensor network renormalization, Phys. Rev. Lett., 115, (2015) · Zbl 1231.82021
[10] Fadeev, L., Instructive history of the quantum inverse scattering method, Quantum Field Theory: Perspective and Prospective, 161-177, (1999)
[11] Graham, J. J.; Lehrer, G. I., The representation theory of affine temperley Lieb algebras, L’Enseignement Math., 44, 1-44, (1998) · Zbl 0964.20002
[12] Izergin, A. G.; Korepin, V. E., The inverse scattering method approach to the quantum Shabat-Mikhailov model, Commun. Math. Phys., 79, 303-316, (1981)
[13] Jones, V. F R., Index for subfactors, Inventory Math., 72, 1-25, (1983) · Zbl 0508.46040
[14] Jones, V. F R., Planar algebras I, (1999) · Zbl 1328.46049
[15] Jones, F. R V., In and around the origin of quantum groups, Prospects in Mathematical Physics, 101-126, (2007), Providence, RI: American Mathematical Society, Providence, RI
[16] Jones, V. F R., The annular structure of subfactors, Essays on Geometry and Related Topics, 401-463, (2001), Genève: Kundig, Genève · Zbl 1019.46036
[17] Jones, V. F R., A no-go theorem for the continuum limit of a periodic quantum spin chain, Commun. Math. Phys., (2017)
[18] Jones, V. F R., Some unitary representations of Thompson’s groups F and T, J. Comb. Algebra, 1, 1-44, (2017) · Zbl 1472.57014
[19] Jones, V.; Morrison, S.; Snyder, N., The classification of subfactors of index  ⩽5, Bull. Amer. Math. Soc., 51, 277-327, (2014) · Zbl 1301.46039
[20] Jones, V.; Reznikoff, S., Hilbert space representations of the annular Temperley-Lieb algebra, Pac. Math. J., 228, 219-250, (2006) · Zbl 1131.46042
[21] Kauffman, L., State models and the Jones polynomial, Topology, 26, 395-407, (1987) · Zbl 0622.57004
[22] Kulish, P. P.; Manojlovic, N.; Nagy, Z., Symmetries of spin systems, the Birman Wenzl Murakami algebra, J. Math. Phys., 51, 489-493, (2010) · Zbl 1310.82016
[23] Morrison, S.; Peters, E.; Snyder, N., Categories generated by a trivalent vertex, Sel. Math. New Ser., 23, 817-868, (2017) · Zbl 1475.18025
[24] Murakami, J., The Kauffman polynomial of lins and representation theory, Osaka J. Math., 24, 745-758, (1987)
[25] Nielsen, M.; Chuang, I., Quantum Computation and Quantum Information, (2011), Cambridge: Cambridge University Press, Cambridge
[26] Penrose, R., Applications of negative dimensional tensors, Applications of Combinatorial Mathematics, 221-244, (1971), New York: Academic, New York
[27] Simon, B., Hamiltonians defined as quadratic forms, Commun. Math. Phys., 21, 192-210, (1970) · Zbl 0208.38804
[28] Temperley, N. V H.; 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
[29] von Neumann, J., On infinite direct products, Compos. Math., 6, 1-77, (1939) · Zbl 0019.31103
[30] Wassermann, A., Operator algebras and conformal field theory III: fusion of positive energy representations of \(LSU(N)\) using bounded operators, Inventiones Math., 133, 467-538, (1998) · Zbl 0944.46059
[31] Yang, C. N., S matrix for the one-dimensional N-body problem with repulsive or attractive {\it δ}-function interaction, Phys. Rev., 167, 1920-1923, (1968)
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