Journal of Low Temperature Physics

, Volume 192, Issue 1–2, pp 75–87 | Cite as

First-Principle Construction of U(1) Symmetric Matrix Product States

  • Mykhailo V. Rakov


The algorithm to calculate the sets of symmetry sectors for virtual indices of U(1) symmetric matrix product states (MPS) is described. The principal differences between open (OBC) and periodic (PBC) boundary conditions are stressed, and the extension of PBC MPS algorithm to projected entangled pair states is outlined.


Matrix product states U(1) symmetry Open boundary conditions 



Useful discussions with Ian P. McCulloch are highly appreciated. The author also thanks Physikalisch-Technische Bundesanstalt (Braunschweig, Germany) for provision of Mathematica software during guest visits.


  1. 1.
    H.J. Mikeska, A.K. Kolezhuk, in Quantum Magnetism, vol. 645, Lecture Notes in Physics, ed. by U. Schollwöck, J. Richter, D.J.J. Farnell, R.F. Bishop (Springer, Berlin Heidelberg, 2004), pp. 1–83CrossRefGoogle Scholar
  2. 2.
    W. Brenig, Phys. Rep. 251, 153 (1995)ADSCrossRefGoogle Scholar
  3. 3.
    X. Chen, Z.C. Gu, X.G. Wen, Phys. Rev. B 82, 155138 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)ADSCrossRefGoogle Scholar
  5. 5.
    A. Kitaev, Ann. Phys. 321, 2 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    X.G. Wen, Int. J. Mod. Phys. B 4, 239 (1990)ADSCrossRefGoogle Scholar
  7. 7.
    L.M. Duan, E. Demler, M.D. Lukin, Phys. Rev. Lett. 91, 090402 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    A. Micheli, G.K. Brennen, P. Zoller, Nat. Phys. 2, 341 (2006)CrossRefGoogle Scholar
  9. 9.
    A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, New York, 1994)CrossRefGoogle Scholar
  10. 10.
    D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    H.P. Büchler, M. Hermele, S.D. Huber, M.P.A. Fisher, P. Zoller, Phys. Rev. Lett. 95, 040402 (2005)CrossRefGoogle Scholar
  12. 12.
    M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Nature 415, 39 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    F.H.L. Essler, H. Frahm, F. Gohmann, A. Klümper, V.E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)ADSCrossRefGoogle Scholar
  15. 15.
    R. Orús, Ann. Phys. 349, 117 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    U. Schollwöck, Ann. Phys. 326, 96 (2011)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    S. Singh, R.N.C. Pfeifer, G. Vidal, Phys. Rev. B 83, 115125 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    I.P. McCulloch, J. Stat. Mech. (2007).
  19. 19.
    F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)ADSCrossRefGoogle Scholar
  20. 20.
    M.V. Rakov, M. Weyrauch, B. Braiorr-Orrs, Phys. Rev. B 93, 054417 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    M.V. Rakov, unpublishedGoogle Scholar
  22. 22.
    B. Bauer, P. Corboz, R. Orús, M. Troyer, Phys. Rev. B 83, 125106 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    J.W. Mei, J.Y. Chen, H. He, X.G. Wen, Phys. Rev. B 95, 235107 (2017)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of PhysicsKyiv National UniversityKyivUkraine

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