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Stability of Frustration-Free Hamiltonians

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Abstract

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.

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References

  1. Aharonov, D., Arad, I., Landau, Z., Vazirani, U.: Quantum Hamiltonian complexity and the detectability lemma. http://arxiv.org/abs/1011.3445v5 [quant-ph], 2011

  2. Audenaert K.M.R.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A: Math. Theor. 40, 8127 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Commun. Math. Phys. 309, 835 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Borgs C., Kotecký R., Ueltschi D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409 (1996)

    Article  ADS  MATH  Google Scholar 

  5. Bravyi S., Haah J.: On the energy landscape of 3D spin Hamiltonians with topological order. Phys. Rev. Lett. 107, 150504 (2011)

    Article  ADS  Google Scholar 

  6. Bravyi S., Hastings M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Fannes M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31, 291 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Fannes M., Nachtergaele B., Werner R.: Finitely Correlated States on Quantum Spin Chains. Commun. Math. Phys. 144, 443 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83, 042330 (2011)

    Article  ADS  Google Scholar 

  15. Hastings M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)

    Article  ADS  Google Scholar 

  16. Hastings, M.B.: An area law for one dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007)

  17. Hastings, M.B.: Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance. http://arxiv.org/abs/1001.5280v2 [math-ph], 2010

  18. Hastings M.B.: Topological Order at Non-Zero Temperature. Phys. Rev. Lett. 107, 210501 (2011)

    Article  ADS  Google Scholar 

  19. Hastings M.B., Koma T.: Spectral Gap and Exponential Decay of Correlations. Commun. Math. Phys. 265, 781 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Hastings, M.B., Michalakis, S.: Quantization of Hall conductance for interacting electrons without averaging assumptions. http://arxiv.org/abs/0911.4706v1 [math-ph], 2009

  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)

    Article  ADS  Google Scholar 

  22. Ingham A.E.: A note on Fourier Transforms. J. London Math. Soc. 9, 29 (1934)

    Article  MathSciNet  Google Scholar 

  23. Kato T.: Continuity of the Map \({S \to |S|}\) for Linear Operators. Proc. Japan Acad. 49, 3 (1973)

    Google Scholar 

  24. Kay A.: Capabilities of a Perturbed Toric Code as a Quantum Memory. Phys. Rev. Lett. 107, 270502 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Kitaev A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Klich I.: On the stability of topological phases on a lattice. Ann. Phys. 325, 2120 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Lieb E.H., Robinson D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  29. Nachtergaele B.: The spectral gap for some quantum spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Nachtergaele B., Ogata Y., Sims R.: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124, 1 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems. Commun. Math. Phys. 286, 1073 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Nachtergaele B., Sims R.: Lieb-Robinson Bounds and the Exponential Clustering Theorem. Commun. Math. Phys. 265, 119 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. 591–614

  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. 141–176

  35. Nussinov Z., Ortiz G.: Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems. Phys. Rev. B 77, 064302 (2008)

    Article  ADS  Google Scholar 

  36. Osborne T.J.: Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  37. Osborne, T.J.: Private communication

  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. Prémont-Schwarz I., Hamma A., Klich I., Markopoulou-Kalamara F.: Lieb-Robinson bounds for commutator-bounded operators. Phys. Rev. A. 81, 040102(R) (2010)

    ADS  Google Scholar 

  40. Prémont-Schwarz I., Hnybida J.: Lieb-Robinson bounds on the speed of information propagation. Phys. Rev. A. 81, 062107 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  41. Sachdev, S.: Quantum phase transitions. Cambridge: Cambridge University Press, 2000

  42. Schuch N., Cirac I., Pérez-García D.: PEPS as ground states: Degeneracy and topology. Ann. Phys. 325, 2153 (2010)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  45. Stark C., Imamoglu A., Renner R.: Localization of Toric Code Defects. Phys. Rev. Lett. 107, 030504 (2011)

    Article  ADS  Google Scholar 

  46. Yarotsky D.: Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems. Commun. Math. Phys. 261, 799 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Wootton J.R., Pachos J.K.: Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories. Phys. Rev. Lett. 107, 030503 (2011)

    Article  ADS  Google Scholar 

  48. Yoshida B.: Feasibility of self-correcting quantum memory and thermal stability of topological order. Ann. Phys. 326, 2566 (2011)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Quantum Information and Matter, Caltech, Pasadena, CA, 91125, USA

    Spyridon Michalakis

  2. Department of Physics, Nicolaus Copernicus University, Torun, Poland

    Justyna P. Zwolak

  3. Department of Physics, Oregon State University, Corvallis, OR, 97331, USA

    Justyna P. Zwolak

Authors
  1. Spyridon Michalakis
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  2. Justyna P. Zwolak
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Correspondence to Spyridon Michalakis.

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Communicated by M. B. Ruskai

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Michalakis, S., Zwolak, J.P. Stability of Frustration-Free Hamiltonians. Commun. Math. Phys. 322, 277–302 (2013). https://doi.org/10.1007/s00220-013-1762-6

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  • DOI: https://doi.org/10.1007/s00220-013-1762-6

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