Communications in Mathematical Physics

, Volume 322, Issue 2, pp 277–302 | Cite as

Stability of Frustration-Free Hamiltonians



We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call Local Topological Quantum Order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by Bravyi et al. on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace. We conclude with a list of open problems relevant to spectral gaps and topological quantum order.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aharonov, D., Arad, I., Landau, Z., Vazirani, U.: Quantum Hamiltonian complexity and the detectability lemma. [quant-ph], 2011
  2. 2.
    Audenaert K.M.R.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A: Math. Theor. 40, 8127 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Commun. Math. Phys. 309, 835 (2012)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Borgs C., Kotecký R., Ueltschi D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409 (1996)ADSMATHCrossRefGoogle Scholar
  5. 5.
    Bravyi S., Haah J.: On the energy landscape of 3D spin Hamiltonians with topological order. Phys. Rev. Lett. 107, 150504 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Bravyi S., Hastings M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Bravyi S., Hastings M.B., Verstraete F.: Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order. Phys. Rev. Lett. 97, 050401 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    Chen X., Gu Z.-C., Wen X.-G.: Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B 83, 035107 (2011)ADSCrossRefGoogle Scholar
  10. 10.
    Datta N., Fernández R., Fröhlich J.: Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states. J. Stat. Phys. 84, 455 (1996)ADSMATHCrossRefGoogle Scholar
  11. 11.
    Datta N., Fernández R., Fröhlich J., Rey-Bellet L.: Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. Helv. Phys. Acta 69, 752 (1996)MathSciNetMATHGoogle Scholar
  12. 12.
    Fannes M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31, 291 (1973)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Fannes M., Nachtergaele B., Werner R.: Finitely Correlated States on Quantum Spin Chains. Commun. Math. Phys. 144, 443 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83, 042330 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Hastings M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    Hastings, M.B.: An area law for one dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007)Google Scholar
  17. 17.
    Hastings, M.B.: Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance. [math-ph], 2010
  18. 18.
    Hastings M.B.: Topological Order at Non-Zero Temperature. Phys. Rev. Lett. 107, 210501 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    Hastings M.B., Koma T.: Spectral Gap and Exponential Decay of Correlations. Commun. Math. Phys. 265, 781 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Hastings, M.B., Michalakis, S.: Quantization of Hall conductance for interacting electrons without averaging assumptions. [math-ph], 2009
  21. 21.
    Hastings M., Wen X.: Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    Ingham A.E.: A note on Fourier Transforms. J. London Math. Soc. 9, 29 (1934)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kato T.: Continuity of the Map \({S \to |S|}\) for Linear Operators. Proc. Japan Acad. 49, 3 (1973)Google Scholar
  24. 24.
    Kay A.: Capabilities of a Perturbed Toric Code as a Quantum Memory. Phys. Rev. Lett. 107, 270502 (2011)CrossRefGoogle Scholar
  25. 25.
    Kennedy T., Tasaki H.: Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains. Commun. Math. Phys. 147, 431–484 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Kitaev A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Klich I.: On the stability of topological phases on a lattice. Ann. Phys. 325, 2120 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Lieb E.H., Robinson D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251 (1972)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Nachtergaele B.: The spectral gap for some quantum spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565 (1996)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Nachtergaele B., Ogata Y., Sims R.: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124, 1 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems. Commun. Math. Phys. 286, 1073 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Nachtergaele B., Sims R.: Lieb-Robinson Bounds and the Exponential Clustering Theorem. Commun. Math. Phys. 265, 119 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Nachtergaele, B., Sims, R.: Locality Estimates for Quantum Spin Systems. In: Sidoravicius, V. (ed.) New Trends in Mathematical Physics. Selected contributions of the XVth International Congress on Mathematical Physics, Berlin-Heidelberg-Newyork: Springer Verlag, 2009, pp. 591–614Google Scholar
  34. 34.
    Nachtergaele, B., Sims, R.: Lieb-Robinson Bounds in Quantum Many-Body Physics. In: Sims, R., Ueltschi, D. (eds), Entropy and the Quantum. Contemporary Mathematics, 529, Providence RI: Amer. Math. Soc., 2010, pp. 141–176Google Scholar
  35. 35.
    Nussinov Z., Ortiz G.: Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems. Phys. Rev. B 77, 064302 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    Osborne T.J.: Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Osborne, T.J.: Private communication Google Scholar
  38. 38.
    Pérez-García D., Verstraete F., Cirac I., Wolf M.: PEPS as unique ground states of local Hamiltonians. Quant. Inf. Comp. 8, 0650 (2008)Google Scholar
  39. 39.
    Prémont-Schwarz I., Hamma A., Klich I., Markopoulou-Kalamara F.: Lieb-Robinson bounds for commutator-bounded operators. Phys. Rev. A. 81, 040102(R) (2010)ADSGoogle Scholar
  40. 40.
    Prémont-Schwarz I., Hnybida J.: Lieb-Robinson bounds on the speed of information propagation. Phys. Rev. A. 81, 062107 (2010)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Sachdev, S.: Quantum phase transitions. Cambridge: Cambridge University Press, 2000Google Scholar
  42. 42.
    Schuch N., Cirac I., Pérez-García D.: PEPS as ground states: Degeneracy and topology. Ann. Phys. 325, 2153 (2010)ADSMATHCrossRefGoogle Scholar
  43. 43.
    Schuch N., Pérez-García D., Cirac I.: Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B 84, 165139 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    Spitzer W.L., Starr S.: Improved Bounds on the Spectral Gap Above Frustration-Free Ground States of Quantum Spin Chains. Lett. Math. Phys. 63, 165 (2003)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Stark C., Imamoglu A., Renner R.: Localization of Toric Code Defects. Phys. Rev. Lett. 107, 030504 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    Yarotsky D.: Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems. Commun. Math. Phys. 261, 799 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  47. 47.
    Wootton J.R., Pachos J.K.: Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories. Phys. Rev. Lett. 107, 030503 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    Yoshida B.: Feasibility of self-correcting quantum memory and thermal stability of topological order. Ann. Phys. 326, 2566 (2011)ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Quantum Information and Matter, CaltechPasadenaUSA
  2. 2.Department of PhysicsNicolaus Copernicus UniversityTorunPoland
  3. 3.Department of PhysicsOregon State UniversityCorvallisUSA

Personalised recommendations