Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 081, 18 pages      arXiv:1206.3653

Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry

Chikashi Arita a and Kohei Motegi b
a) Institut de Physique Théorique CEA, F-91191 Gif-sur-Yvette, France
b) Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Okayama 700-0015, Japan

Received July 06, 2012, in final form October 23, 2012; Published online October 27, 2012

We study the entanglement properties of a higher-integer-spin Affleck-Kennedy-Lieb-Tasaki model with quantum group symmetry in the periodic boundary condition. We exactly calculate the finite size correction terms of the entanglement entropies from the double scaling limit. We also evaluate the geometric entanglement, which serves as another measure for entanglement. We find the geometric entanglement reaches its maximum at the isotropic point, and decreases with the increase of the anisotropy. This behavior is similar to that of the entanglement entropies.

Key words: valence-bond-solid state; entanglement; quantum group.

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  1. Affleck I., Kennedy T., Lieb E.H., Tasaki H., Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys. 115 (1988), 477-528.
  2. Amico L., Fazio R., Osterloh A., Vedral V., Entanglement in many-body systems, Rev. Modern Phys. 80 (2008), 517-576, quant-ph/0703044.
  3. Arita C., Motegi K., Spin-spin correlation functions of the q-valence-bond-solid state of an integer spin model, J. Math. Phys. 52 (2011), 063303, 15 pages, arXiv:1009.4018.
  4. 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 (1988), 531-534.
  5. Batchelor M.T., Mezincescu L., Nepomechie R.I., Rittenberg V., q-deformations of the O(3) symmetric spin-1 Heisenberg chain, J. Phys. A: Math. Gen. 23 (1990), L141-L144.
  6. Bennett C.H., DiVincenzo D.P., Quantum information and computation, Nature 404 (2000), 247-255.
  7. Calabrese P., Cardy J., Entanglement entropy and quantum field theory, J. Stat. Mech. Theory Exp. 2004 (2004), P06002, 27 pages, hep-th/0405152.
  8. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 32 (1985), 254-258.
  9. Fang H., Korepin V.E., Roychowdhury V., Entanglement in a valence-bond-solid state, Phys. Rev. Lett. 93 (2004), 227203, 4 pages, quant-ph/0406067.
  10. Fannes M., Nachtergaele B., Werner R.F., Exact antiferromagnetic ground-states of quantum spin chains, Europhys. Lett. 10 (1989), 633-637.
  11. Freitag W.D., Müller-Hartmann E., Complete analysis of two spin correlations of valence bond solid chains for all integer spins, Z. Phys. B 83 (1991), 381-390.
  12. García-Ripoll J.J., Martín-Delgado M.A., Cirac J.I., Implementation of spin Hamiltonians in optical lattices, Phys. Rev. Lett. 93 (2004), 250405, 4 pages, cond-mat/0404566.
  13. Haldane F.D.M., Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model, Phys. Lett. A 93 (1983), 464-468.
  14. Haldane F.D.M., Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50 (1983), 1153-1156.
  15. Jimbo M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  16. Katsura H., Hirano T., Hatsugai Y., Exact analysis of entanglement in gapped quantum spin chains, Phys. Rev. B 76 (2007), 012401, 4 pages, cond-mat/0702196.
  17. Katsura H., Hirano T., Korepin V.E., Entanglement in an SU(n) valence-bond-solid state, J. Phys. A: Math. Theor. 41 (2008), 135304, 13 pages, arXiv:0711.3882.
  18. Katsura H., Kawashima N., Kirillov A.N., Korepin V.E., Tanaka S., Entanglement in valence-bond-solid states on symmetric graphs, J. Phys. A: Math. Theor. 43 (2010), 255303, 28 pages, arXiv:1003.2007.
  19. Klümper A., Schadschneider A., Zittartz J., Equivalence and solution of anisotropic spin-1 models and generalized t-J fermion models in one dimension, J. Phys. A: Math. Gen. 24 (1991), L955-L959.
  20. Klümper A., Schadschneider A., Zittartz J., Groundstate properties of a generalized VBS-model, Z. Phys. B 87 (1992), 281-287.
  21. Klümper A., Schadschneider A., Zittartz J., Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets, Europhys. Lett. 24 (1993), 293-297, cond-mat/9307028.
  22. Korepin V.E., Xu Y., Entanglement in valence-bond-solid states, Internat. J. Modern Phys. B 24 (2010), 1361-1440, arXiv:0908.2345.
  23. Lyoyd S., A potentially realizable quantum computer, Science 261 (1993), 1569-1571.
  24. Motegi K., The matrix product representation for the q-VBS state of one-dimensional higher integer spin model, Phys. Lett. A 374 (2010), 3112-3115, arXiv:1003.0050.
  25. Orús R., Geometric entanglement in a one-dimensional valence bond solid state, Phys. Rev. A 78 (2008), 062332, 4 pages, arXiv:0808.0938.
  26. Orús R., Universal geometric entanglement close to quantum phase transitions, Phys. Rev. Lett. 100 (2008), 130502, 4 pages, arXiv:0711.2556.
  27. Orús R., Dusuel S., Vidal J., Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model, Phys. Rev. Lett. 101 (2008), 025701, 4 pages, arXiv:0803.3151.
  28. Orús R., Tu H.H., Entanglement and SU(n) symmetry in one-dimensional valence-bond solid states, Phys. Rev. B 83 (2011), 201101(R), 4 pages, arXiv:1103.3994.
  29. Orús R., Wei T.C., Geometric entanglement of one-dimensional systems: bounds and scalings in the thermodynamic limit, Quantum Inf. Comput. 11 (2011), 563-573, arXiv:1006.5584.
  30. Orús R., Wei T.C., Tu H.H., Phase diagram of the SO(n) blinear-biquadratic chain from many-body entanglement, Phys. Rev. B 84 (2011), 064409, 7 pages, arXiv:1010.5029.
  31. Santos R.A., Korepin V.E., Entanglement of disjoint blocks in the one-dimensional spin-1 VBS, J. Phys. A: Math. Theor. 45 (2012), 125307, 19 pages, arXiv:1110.3300.
  32. Santos R.A., Paraan F.N.C., Korepin V.E., Klümper A., Entanglement spectra of q-deformed higher spin VBS states, J. Phys. A: Math. Theor. 45 (2012), 175303, 14 pages, arXiv:1201.5927.
  33. Santos R.A., Paraan F.N.C., Korepin V.E., Klümper A., Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states, Europhys. Lett. 98 (2012), 37005, 6 pages, arXiv:1112.0517.
  34. Stéphan J.M., Misguich G., Alet F., Geometric entanglement and Affleck-Ludwig boundary entropies in critical XXZ and Ising chains, Phys. Rev. B 82 (2010), 180406(R), 4 pages, arXiv:1007.4161.
  35. Totsuka K., Suzuki M., Hidden symmetry breaking in a generalized valence-bond solid model, J. Phys. A: Math. Gen. 27 (1994), 6443-6456.
  36. Totsuka K., Suzuki M., Matrix formalism for the VBS-type models and hidden order, J. Phys. Condens. Matter 7 (1995), 1639-1662.
  37. Verstraete F., Martín-Delgado M.A., Cirac J.I., Diverging entanglement length in gapped quantum spin systems, Phys. Rev. Lett. 92 (2004), 087201, 4 pages, quant-ph/0311087.
  38. Wei T.C., Das D., Mukhopadyay S., Vishveshwara S., Goldbart P.M., Global entanglement and quantum criticality in spin chains, Phys. Rev. A 71 (2005), 060305(R), 4 pages, quant-ph/0405162.
  39. Wei T.C., Vishveshwara S., Goldbart P.M., Global geometric entanglement in transverse-field XY spin chains: finite and infinite systems, Quantum Inf. Comput. 11 (2011), 326-354, arXiv:1012.4114.
  40. Xu Y., Katsura H., Hirano T., Korepin V.E., Entanglement and density matrix of a block of spins in AKLT model, J. Stat. Phys. 133 (2008), 347-377, arXiv:0802.3221.
  41. Zhang J., Wei T.C., Laflamme R., Experimental quantum simulation of entanglement in many-body systems, Phys. Rev. Lett. 107 (2011), 010501, 4 pages, arXiv:1104.0275.

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