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Communications in Mathematical Physics

, Volume 352, Issue 3, pp 1205–1263 | Cite as

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization III

  • Yoshiko OgataEmail author
Article

Abstract

In this paper, we consider the classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in [O1]. We consider the C 1-classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that (1) are given by translation invariant finite range interactions, (2) are gapped in the bulk, (3) are frustration-free, (4) have uniformly bounded ground state degeneracy on finite intervals, and (5) have a unique bulk ground state. We show that for the bulk-classification, any such Hamiltonians are equivalent.

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References

  1. BDN.
    Bachmann S., Dybalski W., Naaijkens P.: Lieb–Robinson bounds, Arveson spectrum and Haag–Ruelle scattering theory for gapped quantum spin systems. Ann. Henri Poincaré 17, 1737 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. BMNS.
    Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309(3), 835–871 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. BN12a.
    Bachmann, S., Nachtergaele, B.: Product vacua with boundary states and the classification of gapped phases. Commun. Math. Phys. 329(2), 509–544 (2014)Google Scholar
  4. BN12b.
    Bachmann S., Nachtergaele B.: Product vacua with boundary states. Phys. Rev. B 86(3), 035149 (2012)ADSCrossRefGoogle Scholar
  5. BR96.
    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1996)zbMATHGoogle Scholar
  6. BO.
    Bachmann S., Ogata Y.: C 1-Classification of gapped parent Hamiltonians of quantum spin chains. Commun. Math. Phys. 338, 1011–1042 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. CGW1.
    Chen X., Gu Z.-C., Wen X.-G.: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B 82(15), 155138 (2010)ADSCrossRefGoogle Scholar
  8. CGW2.
    Chen X., Gu Z.-C., Wen X.-G.: Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B 83(3), 035107 (2011)ADSCrossRefGoogle Scholar
  9. FNW.
    Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144(3), 443–490 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. FNW2.
    Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated pure states. J. Funct. Anal. 144, 443–490 (1992)zbMATHGoogle Scholar
  11. HW.
    Hastings M., Wen X.-G.: Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72(4), 045141 (2005)ADSCrossRefGoogle Scholar
  12. KN.
    Koma T., Nachtergaele B.: The spectral gap of the ferromagnetic XXZ chain. Lett. Math. Phys. 40(1), 1–16 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. M1.
    Matsui, T.: A characterization of pure finitely correlated states. Infinite Dimens. Anal. Quantum Probab. Relat. Topics 01, 647–661 (1998)Google Scholar
  14. N.
    Nachtergaele B.: The spectral gap for some spin chains with discrete symmetry breaking. Commun. Math. Phys. 175(3), 565–606 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. NS.
    Nachtergaele, B., Sims, R.: Locality estimates for quantum spin systems. New Trends in Mathematical Physics. Selected contributions of the XVth International Congress on Mathematical Physics, Springer Verlag, pp. 591–614 (2009)Google Scholar
  16. O1.
    Ogata Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I. Commun. Math. Phys. 348(3), 847–895 (2016)ADSCrossRefzbMATHGoogle Scholar
  17. O2.
    Ogata, Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification II. Commun. Math. Phys. 348(3), 897–957 (2006)Google Scholar
  18. SPWC.
    Sanz M., Pérez-García D., Wolf M.M., Cirac J.I.: A quantum version of Wielandt’s inequality. IEEE Trans. Inform. Theory 56(9), 4668–4673 (2010)MathSciNetCrossRefGoogle Scholar
  19. SPC.
    Schuch N., Pérez-García D., Cirac J.I.: Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B 84(16), 165139 (2011)ADSCrossRefGoogle Scholar
  20. SS.
    Spitzer W., Starr S.: Improved bounds on the spectral gap above frustration-free ground states of quantum spin chains. Lett. Math. Phys. 63(2), 165–177 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. SW.
    Szehr O., Wolf M.: Connected components of irreducible maps and 1D quantum phases. J. Math. Phys. 57, 081901 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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