Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz

  • Glen Evenbly
  • Guifre Vidal
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 176)


The goal of this chapter is to provide an introduction to the multi-scale entanglement renormalization ansatz (MERA) and its application to the study of quantum critical systems. The MERA, in its scale-invariant form, is seen to offer direct numerical access to the scale-invariant operators of a critical theory. As a result, given a critical Hamiltonian on the lattice, the scale-invariant MERA can be used to characterize the underlying conformal field theory. The performance of the MERA is benchmarked for several critical quantum spin chains, namely Ising, Potts, XX and (modified) Heisenberg models, and an insightful comparison with results obtained using a matrix product state is made. The extraction of accurate conformal data, such as scaling dimensions and operator product expansion coefficients of both local and non-local primary fields, is also illustrated.


Ground State Energy Conformal Field Theory Reduce Density Matrix Density Matrix Renormalization Group Tensor Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    G. Evenbly, G. Vidal, Tensor Network States and Geometry. J. Stat. Phys. 145, 891–898 (2011)Google Scholar
  2. 2.
    M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144, 443 (1992)Google Scholar
  3. 3.
    S. Ostlund, S. Rommer, Phys. Rev. Lett. 75, 3537 (1995)Google Scholar
  4. 4.
    S. Rommer, S. Ostlund, Phys. Rev. B 55, 2164 (1997)Google Scholar
  5. 5.
    F. Verstraete, J.I. Cirac, arXiv:cond-mat/0407066v1 (2004)Google Scholar
  6. 6.
    V. Murg, F. Verstraete, J.I. Cirac, Phys. Rev. A 75, 033605 (2007)Google Scholar
  7. 7.
    J. Jordan, R. Orus, G. Vidal, F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 101, 250602 (2008)Google Scholar
  8. 8.
    G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)Google Scholar
  9. 9.
    G. Vidal, Phys. Rev. Lett. 101, 110501 (2008)Google Scholar
  10. 10.
    G. Evenbly, G. Vidal, Phys. Rev. B 81, 235102 (2010)Google Scholar
  11. 11.
    L. Cincio, J. Dziarmaga, M.M. Rams, Phys. Rev. Lett. 100, 240603 (2008)Google Scholar
  12. 12.
    G. Evenbly, G. Vidal, New J. Phys. 12, 025007 (2010)Google Scholar
  13. 13.
    M. Aguado, G. Vidal, Phys. Rev. Lett. 100, 070404 (2008)Google Scholar
  14. 14.
    V. Giovannetti, S. Montangero, R. Fazio, Phys. Rev. Lett. 101, 180503 (2008)Google Scholar
  15. 15.
    G. Evenbly, G. Vidal, Phys. Rev. B 79, 144108 (2009)Google Scholar
  16. 16.
    R.N.C. Pfeifer, G. Evenbly, G. Vidal, Phys. Rev. A 79, 040301(R) (2009)Google Scholar
  17. 17.
    G. Evenbly, G. Vidal, Phys. Rev. Lett. 102, 180406 (2009)Google Scholar
  18. 18.
    R. Koenig, B.W. Reichardt, G. Vidal, Phys. Rev. B 79, 195123 (2009)Google Scholar
  19. 19.
    S. Montangero, M. Rizzi, V. Giovannetti, R. Fazio, Phys. Rev. B 80, 113103 (2009)Google Scholar
  20. 20.
    G. Vidal, in Understanding Quantum Phase Transitions, ed. by L.D. Carr (Taylor & Francis, Boca Raton, 2010)Google Scholar
  21. 21.
    G. Evenbly, G. Vidal, Phys. Rev. Lett. 104, 187203 (2010)Google Scholar
  22. 22.
    G. Evenbly, P. Corboz, G. Vidal, Phys. Rev. B 82, 132411 (2010)Google Scholar
  23. 23.
    L. Cincio, J. Dziarmaga, A.M. Oles, Phys. Rev. B 82, 104416 (2010)Google Scholar
  24. 24.
    S.R. White, Phys. Rev. Lett. 69, 2863 (1992)Google Scholar
  25. 25.
    S.R. White, Phys. Rev. B 48, 10345 (1993)Google Scholar
  26. 26.
    U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005)Google Scholar
  27. 27.
    U. Schollwoeck, Ann. Phys. 326, 96 (2011)Google Scholar
  28. 28.
    G.K.-L. Chan, J.J. Dorando, D. Ghosh, J. Hachmann, E. Neuscamman, H. Wang, T. Yanai, An introduction to the density matrix renormalization group ansatz in quantum chemistry. in Frontiers in Quantum Systems in Chemistry and Physics, ed. by S. Wilson, P.J. Grout, J. Maruani, G. DelgadoBarrio, P. Piecuch, vol. 18 of Prog. Theor. Chem. Phys. pp. 49–65, 2008Google Scholar
  29. 29.
    G. Vidal, Phys. Rev. Lett. 91, 147902 (2003)Google Scholar
  30. 30.
    G. Vidal, Phys. Rev. Lett. 93, 040502 (2004)Google Scholar
  31. 31.
    A.J. Daley, C. Kollath, U. Schollweock, G. Vidal, J. Stat. Mech. Theor. Exp. P04005 (2004)Google Scholar
  32. 32.
    S.R. White, A.E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004)Google Scholar
  33. 33.
    L. Tagliacozzo, T.R. de Oliveira, S. Iblisdir, J.I. Latorre, Phys. Rev. B 78, 024410 (2008)Google Scholar
  34. 34.
    F. Pollmann, S. Mukerjee, A. Turner, J.E. Moore, Phys. Rev. Lett. 102, 255701 (2009)Google Scholar
  35. 35.
    J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996)Google Scholar
  36. 36.
    P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory (Springer, New York, 1997)Google Scholar
  37. 37.
    M. Henkel, Conformal Invariance and Critical Phenomena (Springer, New York, 1999)Google Scholar
  38. 38.
    See appendix of the preprint version of this Chapter, arXiv:1109.5334v1 (2011)Google Scholar
  39. 39.
    S. Singh, R.N.C. Pfeifer, G. Vidal, Phys. Rev. A 82, 050301 (2010)Google Scholar
  40. 40.
    S. Singh, R.N.C. Pfeifer, G. Vidal, Phys. Rev. B 83, 115125 (2011)Google Scholar
  41. 41.
    G. Vidal et al., Phys. Rev. Lett. 90, 227902 (2003)Google Scholar
  42. 42.
    P. Calabrese, J. Cardy, J. Stat. Mech. P06002 (2004)Google Scholar
  43. 43.
    P. Pfeuty, Ann. Phys. 57, 79–90 (1970)Google Scholar
  44. 44.
    T.W. Burkhardt, I. Guim, J. Phys. A Math. Gen. 18, L33–L37 (1985)Google Scholar
  45. 45.
    J. Sólyom, P. Pfeuty, Phys. Rev. B 24, 218 (1981)Google Scholar
  46. 46.
    E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961)Google Scholar
  47. 47.
    S. Eggert, Phys. Rev. B 54, 9612 (1996)Google Scholar
  48. 48.
    I.P. McCulloch, arXiv:0804.2509v1 (2008)Google Scholar
  49. 49.
    J.I. Latorre, E. Rico, G. Vidal, Quant. Inf. Comput. 4, 48–92 (2004)Google Scholar
  50. 50.
    G. Evenbly, R.N.C. Pfeifer, V. Pico, S. Iblisdir, L. Tagliacozzo, I.P. McCulloch, G. Vidal, Phys. Rev. B 82, 161107(R) (2010)Google Scholar
  51. 51.
    P. Silvi, V. Giovannetti, P. Calabrese, G.E. Santoro, R. Fazio, J. Stat. Mech. L03001 (2010)Google Scholar
  52. 52.
    G. Evenbly, G. Vidal, in preparationGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Quantum Information and Matter, California Institute of TechnologyPasadenaUSA
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations