Skip to main content
SpringerLink
Log in

Quantum phase transitions in a frustration-free spin chain based on modified Motzkin walks

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here, we further modify the Motzkin walks using the elements of a symmetric inverse semigroup as basis states on each step of the walk. This change alters the number of paths allowed in the Motzkin walks and by introducing an appropriate term in the Hamiltonian with a tunable parameter we show that we can jump from a state that violates the area law logarithmically to a state that obeys the area law providing an example of quantum phase transition in a one-dimensional system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005)

    Article  ADS  Google Scholar 

  2. R. Orus, Ann. Phys. 349, 117 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  3. S.R. White, Phys. Rev. Lett. 69, 2863 (1992)

    Article  ADS  Google Scholar 

  4. M.B. Hastings, JSTAT 73, P08024 (2007)

    Google Scholar 

  5. I. Arad, A. Kitaev, Z. Landau, U. Vazirani, https://doi.org/arXiv:1301.1162 [quant-ph] (2013)

  6. T. Nishioka, S. Ryu, T. Takayanagi, J. Phys. A42, 504008 (2009)

    Google Scholar 

  7. J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)

    Article  ADS  Google Scholar 

  8. N. Laflorencie, Phys. Rep. 643, 1 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  9. D.N. Page, Phys. Rev. Lett. 71, 1291 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  10. S.K. Foong, S. Kanno, Phys. Rev. Lett. 72, 1148 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  11. S. Sen, Phys. Rev. Lett. 77, 1 (1996)

    Article  ADS  Google Scholar 

  12. S. Irani, J. Math. Phys. 51, 022101 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  13. S. Yan, D.A. Huse, S.R. White, Science 332, 1173 (2011)

    Article  ADS  Google Scholar 

  14. D. Gottesman, M.B. Hastings, New J. Phys. 12, 025002 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  15. G. Vitagliano, A. Riera, J.I. Latorre, https://doi.org/arXiv:1003.1292 [quant-ph] (2010)

  16. G. Ramírez, J.R. Laguna, G. Sierra, J. Stat. Mech. 2014, P10004 (2014)

    Article  Google Scholar 

  17. O. Salberger, V. Korepin, https://doi.org/arXiv:1605.03842 [quant-ph] (2016)

  18. S. Bravyi, L. Caha, R. Movassagh, D. Nagaj, P.W. Shor, Phys. Rev. Lett. 109, 207202 (2012)

    Article  ADS  Google Scholar 

  19. L. Huijse, B. Swingle, Phys. Rev. B 87, 035108 (2013)

    Article  ADS  Google Scholar 

  20. R. Movassagh, https://doi.org/arXiv:1606.09313 [quant-ph] (2016)

  21. A. Osterloh, L. Amico, G. Falci, R. Fazio, Nature 416, 608 (2002)

    Article  ADS  Google Scholar 

  22. P. Calabrese, J. Cardy, J. Phys. A 42, 504005 (2009)

    Article  MathSciNet  Google Scholar 

  23. C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 443 (1994)

    Article  ADS  Google Scholar 

  24. C. Callan, F. Wilczek, Phys. Lett. B 333, 55 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  25. M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006)

    Article  ADS  Google Scholar 

  26. D. Gioev, I. Klich, Phys. Rev. Lett. 96, 100503 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  27. R. Movassagh, P.W. Shor, Proc. Natl. Acad. Sci. 113, 201605716 (2016)

    Article  Google Scholar 

  28. O. Salberger, T. Udagawa, Z. Zhang, H. Katsura, I. Klich, V. Korepin, J. Stat. Mech. 1706, 063103 (2017)

    Article  Google Scholar 

  29. Z. Zhang, I. Klich, https://doi.org/arXiv:1702.03581 [cond-mat.stat-mech] (2017)

  30. Z. Zhang, A. Ahmadain, I. Klich, https://doi.org/arXiv:1606.07795 [quant-ph] (2016)

  31. J. Kellendonk, M.V. Lawson, J. Algebra 224, 140 (2000)

    Article  MathSciNet  Google Scholar 

  32. M. Senechal, Quasicrystals and Geometry (Cambridge University Press, 1995)

  33. R. Exel, D. Goncalves, C. Starling, https://doi.org/arXiv:1106.4535 [math.OA] (2011)

  34. P. Padmanabhan, S.-J. Rey, D. Teixeira, D. Trancanelli, JHEP 1705, 136 (2017)

    Article  ADS  Google Scholar 

  35. V.V. Wagner, J. Lond. Math. Soc. 29, 411 (1954)

    Google Scholar 

  36. M.V. Lawson, Inverse Semigroups – The Theory of Partial Symmetries (World Scientific, 1998)

Download references

Author information

Authors and Affiliations

  1. Fields, Gravity & Strings, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34037, Republic of Korea

    Fumihiko Sugino & Pramod Padmanabhan

Authors
  1. Fumihiko Sugino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pramod Padmanabhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Fumihiko Sugino or Pramod Padmanabhan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sugino, F., Padmanabhan, P. Quantum phase transitions in a frustration-free spin chain based on modified Motzkin walks. Eur. Phys. J. Spec. Top. 227, 269–284 (2018). https://doi.org/10.1140/epjst/e2018-00080-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2018-00080-2

Search

Navigation