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Quantum spin chains with quantum group symmetry

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Abstract

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum groupC *-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.

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References

  • [AF] Accardi, L., Frigerio, A.: Markovian Cocycles. Proc. R. Ir. Acad.83A(2) 251–263 (1983)

    Google Scholar 

  • [AKLT] Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys.115, 477–528 (1988)

    Google Scholar 

  • [ASW] Alcaraz, F.C., Salinas, S.R., Wreszinski, W.F.: Anisotropic ferromagnetic quantum domains. Phys. Rev. Lett.75, 930–933 (1995)

    Google Scholar 

  • [BS] Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés deC *-algèbres. Ann. Ec. Norm. Sup.26, 425 (1993)

    Google Scholar 

  • [Bab] Babujian, H.M.: Exact solution of the isotropic Heisenberg chain with arbitrary spins: Thermodynamics of the model. Nucl. Phys. B215, 317–336 (1983)

    Google Scholar 

  • [BMNR] Batchelor, M.T., Mezincescu, L., Nepomechie, R., Rittenberg, V.:q-Deformations of the O(3)-symmetric spin-1 chain. J. Phys.A23, L141-L144 (1990)

    Google Scholar 

  • [BY] Batchelor, M.T., Yung, C.M.:q-Deformations of quantum spin chains with exact valence bond ground states. Preprint archived in cond-mat@babbage.sissa.it #9403080

  • [BF] Bernard, D., Felder, G.: Quantum group symmetries in two-dimensional lattice quantum field theory. Nucl. Phys. B365, 98–120 (1991)

    Google Scholar 

  • [Ber] Bernard, D.: Quantum group symmetries and non-local currents in 2D QFT. Commun. Math. Phys.142, 99–138 (1991)

    Google Scholar 

  • [Bie] Biedenharn, L.C.: Aq-Boson realization of the quantum group SU q (2), and the theory ofq-tensor operators. In: Doebner, H.-D., Hennig, J.-D. (eds.) Quantum groups. Springer Lect. Note: Phys370, Berlin, Heidelberg, New York: Springer Verlag, pp. 67–88 1990

    Google Scholar 

  • [BR] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 2 volumes, Berlin, Heidelberg, New York: Springer Verlag, 1979 and 1981

    Google Scholar 

  • [CE] Choi, M.D., Effros, E.G.: NuclearC *-algebras and the approximation property. Ann. Math.100, 61–79 (1978)

    Google Scholar 

  • [Cu1] Cuntz, J.: SimpleC *-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977)

    Google Scholar 

  • [Cu2] Cuntz, J.: Regular actions of Hopf algebras on theC *-algebra generated by a Hilbert space. In: Herman, R., Tanbay, B. (eds.). Operator algebras, mathematical physics, and low dimensional topology. Wellesley, MA: Peters, A.K. 1993

    Google Scholar 

  • [DC] Dasgupta, N., Chowdhury, A.R.: Algebraic Bethe ansatz with boundary condition for SU p, q (2) invariant spin chain. J. Phys. A26, 5427–5433 (1993)

    Google Scholar 

  • [DFJMN] Davies, B., Foda, O., Jimbo, M., Miwa, T., Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by Vertex Operators. Commun. Math. Phys.151, 89–153 (1993)

    Google Scholar 

  • [DHR] Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations, I. Commun. Math. Phys.13, 1–23 (1969); Part II in Commun. Math. Phys.15, 173–200 (1969)

    Google Scholar 

  • [DR1] Doplicher, S., Roberts, J.E.: Endomorphisms ofC *-algebras, cross products and duality for compact groups. Ann. Math.130, 75–119 (1989)

    Google Scholar 

  • [DR2] Doplicher, S., Roberts, J.E.:C *-algebras and duality for compact groups: Why there is a compact group of internal symmetries in particle physics. In: Sénéor, R., Mebkhout, M. (ed.) Proceedings of the International Conference on Mathematical Physics. Singapore: World Sicientific, 1986, pp. 489–498

    Google Scholar 

  • [Dri] Drinfel'd, V.G.: Quantum groups. In: Vol. I. of the Proceedings of the Int. Congr. Math. Berkeley 1986, New York: Academic Press 1987, pp. 798–820

    Google Scholar 

  • [DW] Duffield, N.G., Werner, R.F.: Local dynamics of mean-field quantum systems. Helv. Phys. Acta65, 1016–1054 (1992)

    Google Scholar 

  • [FNW1] Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states of quantum spin chains. Commun. Math. Phys.144, 443–490 (1992)

    Google Scholar 

  • [FNW2] Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated pure states. J. Funct. Anal.120, 511–534 (1994)

    Google Scholar 

  • [FSV] Fannes, M., Spohn, H., Verbeure, A.: Equilibrium states for mean field models. J. Math. Phys.21, 355–358 (1980)

    Google Scholar 

  • [GW] Gottstein, C.-T., Werner, R.F.: Ground states of the infiniteq-deformed Heisenberg ferromagnet. Archived in cond-mat@babbage.sissa.it#9501123

  • [GS] Grosse, H., Raschhofer, E.: Bethe-Ansatz solution of a modified SU(2)-XXZ model. In: Fannes, M., Maes, C., Verbeure, A. (eds.) On three levels; micro-, meso-, and macro-approaches in physics. New York: Plenum Press 1994, pp. 385–392

    Google Scholar 

  • [Ha1] Haldane, F.D.M.: Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r 2 exchange. Phys. Rev. Lett.60, 635–638 (1988)

    Google Scholar 

  • [Ha2] Haldane, F.D.M.: Physics of the ideal semion gas: spinons and quantum symmetries of the integrable Haldane-Shastry spin chain. To appear in the Proceedings of the 16th Taniguchi Symposium, Kashikojima, Japan. Okiji, A., Kawakami, N. (eds.) Berlin, Heidelberg, New York: Springer, 1994

    Google Scholar 

  • [Jim] Jimbo, M.: Aq-Difference Analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985)

    Google Scholar 

  • [JSW] Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: Positive representations of general commutation relations allowing Wick ordering. Preprint Osnabrück and lowa, 1993 archived in funct-an@babbage.sissa.it#9312004

  • [KSZ] Klümper, A., Schadschneider, A., Zittartz, J.: Groundstate properties of a generalized VBS-model. Zeits. für Phys. B87, 281–287 (1992)

    Google Scholar 

  • [KNW] Konishi, Y., Nagisa, M., Watatani, Y.: Some remarks on actions of compact matrix quantum groups onC *-algebras. Pacific. J. Math.153, 119–128 (1992)

    Google Scholar 

  • [KS] Kulish, P.P., Sklyanin, E.K.: The generalU q [sl(2)] invariant XXZ integrable quantum spin chain. J. Phys. A24, L435-L439 (1991)

    Google Scholar 

  • [LB] Lienert, C.R., Butler, P.H.: Racah-Wigner algebra forq-deformed algebras. J. Phys. A25, 1223–1235 (1992)

    Google Scholar 

  • [MS] Mack, G., Schomerus, V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys.134, 139–196 (1990)

    Google Scholar 

  • [Maj] Majid, S.: Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group. Commun. Math. Phys.156, 607–638 (1993)

    Google Scholar 

  • [Mat] Wolfram Research, Inc.: Mathematica 2.2, Wolfram Research, Inc., Champaign, Illinois, 1992

    Google Scholar 

  • [MMP] Meljanac, S., Mileković, M., Pallua, S.: Deformed SU(2) Heisenberg chain. J. Phys. A24, 581–591 (1991)

    Google Scholar 

  • [MN] Mezincescu, L., Nepomechie, R.I.: Analytical Bethe Ansatz for quantum-algebrainvariant spin chains. Nucl. Phys. B372, 597–621 (1992)

    Google Scholar 

  • [RW] Raggio, G.A., Werner, R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta62, 980–1003 (1989)

    Google Scholar 

  • [Rue] Ruegg, H.: A simple derivation of the quantum Clebsch-Gordan coefficients for SU(2). J. Math. Phys.31, 1085–1087 (1991)

    Google Scholar 

  • [Sha] Shastri, B.S.: Exact solution of anS=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions. Phys. Rev. Lett.60, 639–642 (1988)

    Google Scholar 

  • [SV] Szlachányi, K., Vecsernyés, P.: Quantum symmetry and braid group statistics inG-spin models. Commun. Math. Phys.156, 127–168 (1993)

    Google Scholar 

  • [Tak] Takesaki, M.: Theory of operator algebras I. Berlin, Heidelberg New York: Springer 1979

    Google Scholar 

  • [Vec] Vecsernyés, P.: On the quantum symmetry of the chiral Ising model. Princeton U. preprint PUPT-1406, archived in hepth@xxx.lanl.gov #9306118

  • [Wo1] Woronowicz, S.L.: Twisted SU(2) group. An example of a non-commutative differential calculus. Publ. RIMS, Kyoto23, 117–181 (1987)

    Google Scholar 

  • [Wo2] Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613–665 (1987)

    Google Scholar 

  • [Wo3] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125–170 (1989)

    Google Scholar 

  • [Wo4] Woronowicz, S.L.: Compact quantum groups. In preparation

Download references

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Authors and Affiliations

  1. Inst. Theor. Fysica, Universiteit Leuven, B-3001, Heverlee, Belgium

    M. Fannes

  2. Onderzoeksleider, N.F.W.O., Belgium

    M. Fannes

  3. Dept. of Physics, Princeton University, 08544-0708, Princeton, NJ, USA

    B. Nachtergaele

  4. Fachbereich Physik, Universität Osnabrück, D-49069, Osnabrück, Germany

    R. F. Werner

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  1. M. Fannes
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  2. B. Nachtergaele
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  3. R. F. Werner
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Communicated by M. Jimbo

Supported in part by NSF Grant # PHY90-19433 A02

Copyright © 1995 by the authors. Faithful reproduction of this article by any means is permitted for non-commercial purposes.

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Fannes, M., Nachtergaele, B. & Werner, R.F. Quantum spin chains with quantum group symmetry. Commun.Math. Phys. 174, 477–507 (1996). https://doi.org/10.1007/BF02101525

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