Communications in Mathematical Physics

, Volume 174, Issue 3, pp 477–507 | Cite as

Quantum spin chains with quantum group symmetry

  • M. Fannes
  • B. Nachtergaele
  • R. F. Werner


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.


Quantum Group Unitary Representation Spin Chain Lattice Spin Weak Locality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AF] Accardi, L., Frigerio, A.: Markovian Cocycles. Proc. R. Ir. Acad.83A(2) 251–263 (1983)Google Scholar
  2. [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
  3. [ASW] Alcaraz, F.C., Salinas, S.R., Wreszinski, W.F.: Anisotropic ferromagnetic quantum domains. Phys. Rev. Lett.75, 930–933 (1995)Google Scholar
  4. [BS] Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés deC *-algèbres. Ann. Ec. Norm. Sup.26, 425 (1993)Google Scholar
  5. [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
  6. [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
  7. [BY] Batchelor, M.T., Yung, C.M.:q-Deformations of quantum spin chains with exact valence bond ground states. Preprint archived in #9403080Google Scholar
  8. [BF] Bernard, D., Felder, G.: Quantum group symmetries in two-dimensional lattice quantum field theory. Nucl. Phys. B365, 98–120 (1991)Google Scholar
  9. [Ber] Bernard, D.: Quantum group symmetries and non-local currents in 2D QFT. Commun. Math. Phys.142, 99–138 (1991)Google Scholar
  10. [Bie] Biedenharn, L.C.: Aq-Boson realization of the quantum group SUq(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 1990Google Scholar
  11. [BR] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 2 volumes, Berlin, Heidelberg, New York: Springer Verlag, 1979 and 1981Google Scholar
  12. [CE] Choi, M.D., Effros, E.G.: NuclearC *-algebras and the approximation property. Ann. Math.100, 61–79 (1978)Google Scholar
  13. [Cu1] Cuntz, J.: SimpleC *-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977)Google Scholar
  14. [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. 1993Google Scholar
  15. [DC] Dasgupta, N., Chowdhury, A.R.: Algebraic Bethe ansatz with boundary condition for SUp, q(2) invariant spin chain. J. Phys. A26, 5427–5433 (1993)Google Scholar
  16. [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
  17. [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
  18. [DR1] Doplicher, S., Roberts, J.E.: Endomorphisms ofC *-algebras, cross products and duality for compact groups. Ann. Math.130, 75–119 (1989)Google Scholar
  19. [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. 489–498Google Scholar
  20. [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. 798–820Google Scholar
  21. [DW] Duffield, N.G., Werner, R.F.: Local dynamics of mean-field quantum systems. Helv. Phys. Acta65, 1016–1054 (1992)Google Scholar
  22. [FNW1] Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states of quantum spin chains. Commun. Math. Phys.144, 443–490 (1992)Google Scholar
  23. [FNW2] Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated pure states. J. Funct. Anal.120, 511–534 (1994)Google Scholar
  24. [FSV] Fannes, M., Spohn, H., Verbeure, A.: Equilibrium states for mean field models. J. Math. Phys.21, 355–358 (1980)Google Scholar
  25. [GW] Gottstein, C.-T., Werner, R.F.: Ground states of the infiniteq-deformed Heisenberg ferromagnet. Archived in Scholar
  26. [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. 385–392Google Scholar
  27. [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
  28. [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, 1994Google Scholar
  29. [Jim] Jimbo, M.: Aq-Difference Analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985)Google Scholar
  30. [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 Scholar
  31. [KSZ] Klümper, A., Schadschneider, A., Zittartz, J.: Groundstate properties of a generalized VBS-model. Zeits. für Phys. B87, 281–287 (1992)Google Scholar
  32. [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
  33. [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
  34. [LB] Lienert, C.R., Butler, P.H.: Racah-Wigner algebra forq-deformed algebras. J. Phys. A25, 1223–1235 (1992)Google Scholar
  35. [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
  36. [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
  37. [Mat] Wolfram Research, Inc.: Mathematica 2.2, Wolfram Research, Inc., Champaign, Illinois, 1992Google Scholar
  38. [MMP] Meljanac, S., Mileković, M., Pallua, S.: Deformed SU(2) Heisenberg chain. J. Phys. A24, 581–591 (1991)Google Scholar
  39. [MN] Mezincescu, L., Nepomechie, R.I.: Analytical Bethe Ansatz for quantum-algebrainvariant spin chains. Nucl. Phys. B372, 597–621 (1992)Google Scholar
  40. [RW] Raggio, G.A., Werner, R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta62, 980–1003 (1989)Google Scholar
  41. [Rue] Ruegg, H.: A simple derivation of the quantum Clebsch-Gordan coefficients for SU(2). J. Math. Phys.31, 1085–1087 (1991)Google Scholar
  42. [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
  43. [SV] Szlachányi, K., Vecsernyés, P.: Quantum symmetry and braid group statistics inG-spin models. Commun. Math. Phys.156, 127–168 (1993)Google Scholar
  44. [Tak] Takesaki, M.: Theory of operator algebras I. Berlin, Heidelberg New York: Springer 1979Google Scholar
  45. [Vec] Vecsernyés, P.: On the quantum symmetry of the chiral Ising model. Princeton U. preprint PUPT-1406, archived in #9306118Google Scholar
  46. [Wo1] Woronowicz, S.L.: Twisted SU(2) group. An example of a non-commutative differential calculus. Publ. RIMS, Kyoto23, 117–181 (1987)Google Scholar
  47. [Wo2] Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613–665 (1987)Google Scholar
  48. [Wo3] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125–170 (1989)Google Scholar
  49. [Wo4] Woronowicz, S.L.: Compact quantum groups. In preparationGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • M. Fannes
    • 1
    • 2
  • B. Nachtergaele
    • 3
  • R. F. Werner
    • 4
  1. 1.Inst. Theor. FysicaUniversiteit LeuvenHeverleeBelgium
  2. 2.OnderzoeksleiderN.F.W.O.Belgium
  3. 3.Dept. of PhysicsPrinceton UniversityPrincetonUSA
  4. 4.Fachbereich PhysikUniversität OsnabrückOsnabrückGermany

Personalised recommendations