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The spectral gap for some spin chains with discrete symmetry breaking

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Abstract

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and threfore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with a number of examples.

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References

  1. Affleck, I.: Large-n limit of SU(n) quantum “spin” chains. Phys. Rev. Lett.54, 966–969 (1985)

    Article  Google Scholar 

  2. Affleck, I.: Exact results on the dimerization transition in SU(n) antiferromagnetic chains. J. Phys.: Condens. Matter2, 405–415 (1990)

    Article  Google Scholar 

  3. Kennedy, T., Tasaki, H.: HiddenZ 2×Z 2 symmetry breaking in Haldane gap antiferromagnets. Phys. Rev.B45, 304–307 (1992)

    Article  Google Scholar 

  4. Dagotto, E.: The t-J and frustrated Heisenberg model: a status report on numerical studies. Int. J. Mod. PhysB5, 907–935 (1991)

    Article  Google Scholar 

  5. Wen, X.G., Wilczeck, F., Zee, A.: Chiral spin states and superconductivity. Phys. RevB39, 11413–11423 (1989)

    Article  Google Scholar 

  6. Long, M.W., Siak, S.: An exact solution to a spin-1 chain model. J. Phys.: Condens. Matter5, 5811–5828 (1993)

    Article  Google Scholar 

  7. Affleck, I., Arovas, D.P., Marston, J.B., Rabson, D.A.: SU(2n) Quantum Antiferromagnets with Exact C-Breaking Ground States. Nucl. Phys.B366, 467–506 (1991)

    Article  Google Scholar 

  8. Affleck, I., Haldane, F.D.M.: Critical theory of quantum spin chains. Phys. Rev.B36, 5291–5300 (1987)

    Article  Google Scholar 

  9. Affleck, I.: Quantum spin chains and the Haldane gap. J. Phys.: Condens. Matter1, 3047–3072 (1989)

    Article  Google Scholar 

  10. Sørensen, E.S., Affleck, I.: Large-Scale Numerical Evidence for Bose Condensation in theS=1 Antiferromagnetic Chain in a Strong Field. Phys. Rev. Lett.71, 1633–1636 (1993)

    Article  Google Scholar 

  11. Affleck, I., Lieb, E.H., Kennedy, T., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett.59, 799–802 (1987)

    Article  Google Scholar 

  12. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bongrground states in isotropic quantum antiferromagnets. Commun. Math. Phys.115, 477–528 (1988)

    Article  Google Scholar 

  13. Fannes, M., Nachtergaele, B., Werner, R.F.: Valence bond states on quantum spin chains as ground states with spectral gap. J. Phys. A: Math. Gen.24 L185-L190 (1991)

    Article  Google Scholar 

  14. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely Correlated States on Quantum Spin Chains. Commun. Math. Phys.144, 443–490 (1992)

    Google Scholar 

  15. Arovas, D.P., Auerbach, A., Haldane, F.D.M.: Extended Heisenberg models of antiferromagnetism: analogies to the fractional quantum Hall effect. Phys. Rev. Lett.60, 531–534 (1988)

    Article  Google Scholar 

  16. Knabe, S.: Energy gaps and elementary excitations for Certain VBS-Quantum Antiferromagnets. J. Stat. Phys.52, 627–638 (1988)

    Article  Google Scholar 

  17. Botet, R., Julien, R.: Ground-state properties of a spin-1 antiferromagnetic chain. Phys. Rev.B27, 613–615 (1983)

    Article  Google Scholar 

  18. Kolb, M., Botet, R., Julien, J.: Comparison of ground-state properties for odd half-integer and integer spin antiferromagnetic Heisenberg chains. J. Phys. A: Math. Gen.16, L673-L677 (1983)

    Article  Google Scholar 

  19. Parkinson, J.B., Bonner, J.C.: Spin chains in a field: Crossover from quantum to classical behavior. Phys. Rev.B32, 4703–4724 (1985)

    Article  Google Scholar 

  20. Nightingale, M.P., Blöte, H.W.: Gap of the linear spin-1 Heisenberg antiferromagnet: a Monte Carlo calculation. Phys. Rev.B33, 659–661 (1986)

    Article  Google Scholar 

  21. Sólyom, J.: Competing bilinear and biquadratic exchange couplings in spin-1 Heisenberg chains. Phys. Rev.B36, 8642–8648 (1987)

    Article  Google Scholar 

  22. Chang, K., Affleck, I., Hayden, G.W., Soos, Z.G.: A study of the bilinear-biquadratic spin 1 antiferromagnetic chain using the valence-bond basis. J. Phys.C1, 153–167 (1989)

    Google Scholar 

  23. Kennedy, T.: Exact diagonalisations of open spin-1 chains. J. Phys.: Cond. Matter2, 5737–5745 (1990)

    Article  Google Scholar 

  24. White, S.R., Huse, D.A.: Numerical Renormalization Group Study of Low-lying Eigenstates of the AntiferromagneticS=1 Heisenberg Chain. Phys. Rev.B48, 3844–3852 (1993)

    Article  Google Scholar 

  25. White, S.R.: Density Matrix Formutation for Quantum Renormalization Groups. Phys. Rev. Lett.69, 2863–2866 (1992)

    Article  Google Scholar 

  26. White, S.R.: Density Matrix Formulation for Quantum Renormalization Groups. Phys. Rev.B48, 10345–10456 (1993)

    Article  Google Scholar 

  27. Kennedy, T.: Nonpositive matrix elements for Hamiltonians of spin 1 chains. J. Phys.: Cond. Matter6, 8015–8022 (1994)

    Article  Google Scholar 

  28. Caspers, W.J., Magnus, W.: Some exact excited states in a linear antiferromagnetic spin system. Phys. Lett.88A, 103–105 (1982)

    Google Scholar 

  29. Shastry, B.S., Sutherland, B.: Excitation Spectrum of a Dimerized Next-Neighbor Antiferromagnetic Chain. Phys. Rev. Lett.47 964–967 (1981)

    Article  Google Scholar 

  30. Holley, R.: Rapid convergence to equilibrium in one-dimensional stochastic Ising models. Ann. Prob.13, 72–89 (1985)

    Google Scholar 

  31. Aizenman, M., Holley R.: Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin-Shlosman regime. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems, H. Kesten (Ed.), Berlin-Heidelberg-New York: Springer Verlag, 1987, pp. 1–11

    Google Scholar 

  32. Lu, S.-L., Yau, H.T.: Spectral gap and logarithmic Sobolev Inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys.156, 399–433 (1993)

    Google Scholar 

  33. Martinelli, F., Olivieri, E.: Finite volume mixing conditions for lattice spin systems and exponential approach to equilibrium of Glauber dynamics. Parts I and II, Commun. Math. Phys.161, 447–486, 487–514 (1994)

    Google Scholar 

  34. Holley, R.A., Stroock, D.W.: Logarithmic Sobolev Inequalities and stochastic Ising models. J. Stat. Phys.46, 1159–1194 (1987)

    Article  Google Scholar 

  35. Holley, R.A., Stroock, D.A.: Uniform andL 2 convergence in one-dimensional stochastic Ising models. Commun. Math. Phys.123, 85–93 (1989)

    Article  Google Scholar 

  36. Zegarlinski, B.: Gibbsian description and description by stochastic dynamics in the statistical mechanics of lattice spin systems with finite range interactions. In: Proceedings of the Third International Conference on Stochastic Processes, Physics, and Geometry, S. Albeverio et al. (Eds), Singapore: World Scientific, 1993

    Google Scholar 

  37. Holley, R.: Rapid convergence to equilibrium in ferromagnetic stochastic Ising models. Resenhas IME-USP1, 131–149 (1993)

    Google Scholar 

  38. Zegarlinski, B.: Log-Sobolev inequlaities for infinite one-dimensional lattice systems. Commun. Math. Phys.133, 147–162 (1990)

    Article  Google Scholar 

  39. Stroock, D.W., Zegarlinski, B.: The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal.104, 299–326 (1992)

    Article  Google Scholar 

  40. Stroock, D.W., Zegarlinski, B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Commun. Math. Phys.144, 303–323 (1992)

    Google Scholar 

  41. Stroock, D.W., Zegarlinski, B.: The logarithmic Sobolev inequality for discrete spin systems on a lattice. Commun. Math. Phys.149, 175–193 (1992)

    Article  Google Scholar 

  42. Haldane, F.D.M.: The Hierarchy of Fractional States and Numerical Studies. In R.E. Prange, S.M. Girvin (eds.). The Quantum Hall Effect New York: Springer Verlag, 1987, pp. 303–352

    Google Scholar 

  43. Fröhlich, J., Studer, U.M.: U(1)×SU(2) gauge invariance of non-relativistic quantum mechanics, and generalized Hall effects. Commun. Math. Phys.148, 553–600 (1992)

    Google Scholar 

  44. Fannes, M., Nachtergaele, B., Werner, B.F.: Exact Ground States of Quantum Spin Chains Europhys. Lett.10, 633–637 (1989)

    Google Scholar 

  45. Werner, R.F.: Remarks on a quantum state extension problem. Lett. Math. Phys.19, 319–326 (1990)

    Article  Google Scholar 

  46. Haldane, F.D.M.:Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model. Phys. Lett.93A, 464–468 (1983)

    Google Scholar 

  47. Affleck, I., Lieb, E.H.: A proof of part of Haldane's conjecture on quantum spin chains. Lett. Math. Phys.12, 57–69 (1986)

    Article  Google Scholar 

  48. Majumdar, C.K.: Antiferromagnetic model with known ground state. J. Phys. C: Cond Matt.3, 911–915 (1970)

    Google Scholar 

  49. Majumdar, C.K., Ghosh, D.K.: On next nearest-neighbor interaction in linear chain, I and II. J. Math. Phys.10, 1388–1398, and 1399–1402 (1969)

    Article  Google Scholar 

  50. Klein, D.J.: Variational localized-site cluster expansions. IX. Many-body valence-bond theory. Phys. Rev.B19, 870–876 (1979)

    Article  Google Scholar 

  51. van den Broeck, P.M.: Exact value of the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest and next-nearest neighbor interactions. Phys. Lett.77A, 261–262 (1980)

    Google Scholar 

  52. Klein, D.J.: Exact ground states for a class of antiferromagnetic Heisenberg models with short range interactions. J. Phys. A: Math. Gen.15, 661–671 (1982)

    Article  Google Scholar 

  53. Caspers, W.J.: Exact ground states for a class of linear antiferromagnetic spin systems. Physica115A, 275–280 (1982)

    Google Scholar 

  54. Caspers, W.J., Magnus, W.: Exact ground states for a class of linear quantum spin systems. Physica119A, 291–294 (1983)

    Google Scholar 

  55. Fannes, M., Nachtergaele, B., Werner, B.F.: Entropy Estimates for Finitely Correlated States. Ann. Inst. H. Poincaré57, 259–277 (1992)

    Google Scholar 

  56. Bose, I.: Exact ground and excited states of an antiferromagnetic quantum spin model. J. Phys.: Condens. Matt.1, 9267–9271 (1989)

    Article  Google Scholar 

  57. Klümper, A., Schadschneider, A., Zittartz, J.: Ground state properties of a generalized VBS-model. Z. Phys. B-Condensed Matter87, 281–287 (1992)

    Article  Google Scholar 

  58. Kennedy, T., Lieb, E.H., Tasaki, H.: A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state. J. Stat. Phys.53, 383–415 (1988)

    Article  Google Scholar 

  59. Chayes, J., Chayes, L., Kivelson, S.: Valence bond ground states in a frustrated two-dimensional spin 1/2 Heisenberg antiferromagnet. Commun. Math. Phys.123, 53–83 (1989)

    Article  Google Scholar 

  60. Kirillov, A.N., Korepin, V.E.: The resonating valence bond in quasicrystals. Leningrad Math. J.1, 343–377 (1990)

    Google Scholar 

  61. Long, M.W., Siak, S.: An exactly soluble two-dimensional quantum mechanical Heisenberg model: quantum fluctuations versus magnetic order. J. Phys.: Condens. Matter2, 10321–10341 (1990)

    Article  Google Scholar 

  62. Bose, I.: Two-dimensional spin models with resonating valence bond ground states. J. Phys.: Consdens. Matter2, 5479–5482 (1990)

    Article  Google Scholar 

  63. Bose, L.: Frustrated spin 1/2-model in two dimensions with a known ground state. Phys. Rev.B44, 443–445 (1991)

    Article  Google Scholar 

  64. Bose, I.: Antiferromagnetic spin models in two dimensions with known ground states. Phys. Rev.B45, 13072–13075 (1992)

    Article  Google Scholar 

  65. Freitag, W.-D., Müller-Hartmann, E.: Complete analysis of two-spin correlations of valence bond solid chains for all integer spins. Z. Phys. B Condensed Matter83, 381–390 (1991)

    Article  Google Scholar 

  66. Freitag, W.-D., Müller-Hartmann, E.; Spin correlations of inhomogeneous valence bond solid chains. Z. Phys. B Condensed Matter88, 279–282 (1992)

    Article  Google Scholar 

  67. Accardi, L.: Topics in Quantum Probability. Physics Rep.77, 169–192 (1981)

    Article  Google Scholar 

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

    Google Scholar 

  69. Fannes, M., Nachtergaele, B., Werner, R.F.: Abundance of Translation Invariant Pure States on Quantum Spin Chains Lett. Math. Phys.25, 249–258 (1992)

    Article  Google Scholar 

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

    Article  Google Scholar 

  71. Monti, F., Sütő, A.: Spin 1/2 Heisenberg model on Δ trees. Phys. Lett.156, 197–200 (1991)

    Article  Google Scholar 

  72. Monti, F., Sütő, A.: Heisenberg Antiferromagnet on Triangulated Trees. Helv. Phys. Acta65, 560–595 (1992)

    Google Scholar 

  73. Gottstein, C.-T., Werner, R.F.: Ground states of the infinite q-deformed Heisenberg ferromagnet. Preprint archived as cond-mat/9501123

  74. Albanese, C.: Unitary Dressing Transformations and Exponential Decay Below Threshold for Quantum Spin Systems. Part I–II. Commun. Math. Phys.134, 1–27 (1990)

    Article  Google Scholar 

  75. Albanese, C.: Unitary Dressing Transformations and Exponential Decay Below Threshold for Quantum Spin Systems. Part III–IV. Commun. Math. Phys.134, 237–272 (1990)

    Article  Google Scholar 

  76. Kennedy, T., Tasaki, H.: Hidden Symmetry Breaking and the Haldane Phase in S=1 Quantum Spin Chains. Commun. Math. Phys.147, 431–484 (1992)

    Google Scholar 

  77. Matsui, T.: Purification and Uniqueness of Quantum Gibbs States. Commun. Math. Phys.162, 321–332 (1994)

    Google Scholar 

  78. Datta, N., Fernández, R., Fröhlich, J.: Low temperature phase diagrams of quantum lattice systems I. Texas archive 95–288

  79. Koma, T., Tasaki, H.: Symmetry Breaking and Finite-Size Effects in Quantum Many-Body Systems. J. Stat. Phys.76, 745–803 (1994)

    Google Scholar 

  80. Werner, R.F.: Finitely correlated pure states. In: M. Fannes, C. Maes, and A. Verbeure (eds), On three levels; micro-, meso, and macro-approaches in physics. New York: Plenum, 1994

    Google Scholar 

  81. Hiai, F., Petz, D.: Entropy Density for Algebraic States. J. Funct. Analysis125, 287–308 (1994)

    Article  Google Scholar 

  82. Olivieri, E., Picco, P.: Cluster Expansion ford-Dimensional Lattice Systems and Finite-Volume Factorization Properties. J. Stat. Phys.59, 221–256 (1990)

    Article  Google Scholar 

  83. Lange, C., Klümper, A., Zittartz, J.: Exact ground states for antiferromagnetic spin-one chains with nearest and next-nearest neighbour interactions. Preprint

  84. den Nijs, M., Rommelse, K.: Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. RevB40, 4709 (1989)

    Article  Google Scholar 

  85. Chen, J.-Q.: Group Representation Theory for Physicists, Singapore: World Scientific, 1989

    Google Scholar 

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  1. Department of Physics, Princeton University, 08544-0708, Princeton, NJ, USA

    Bruno Nachtergaele

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Communicated by D. Brydges

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Nachtergaele, B. The spectral gap for some spin chains with discrete symmetry breaking. Commun.Math. Phys. 175, 565–606 (1996). https://doi.org/10.1007/BF02099509

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