Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Communications in Mathematical Physics

Volumn 309, Issue 3, 2012, Pages 835-871

  Bachmann, Sven ^a   Michalakis, Spyridon ^b   Nachtergaele, Bruno ^a   Sims, Robert ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^c University of Arizona   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84856121008     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-011-1380-0     Document Type: Article

References (52)

1. Albanese C.: Unitary dressing transformations and exponential decay below threshold for quantum spin systems. I, II. Commun. Math. Phys. 134, 1-27 (1990).
2. Albanese C.: Unitary dressing transformations and exponential decay below threshold for quantum spin systems. III, IV. Commun. Math. Phys. 134, 237-272 (1990).
3. Amour L., Levy-Bruhl P., Nourrigat J.: Dynamics and Lieb-Robinson estimates for lattices of interacting anharmonic oscillators. Colloq. Math. 118, 609-648 (2010).
4. Baillie R., Borwein D., Borwein J. M.: Surprising Sinc Sums and Integrals. Am. Math. Mon. 115, 888-901 (2008).
5. Birman M. S., Solomyak M.: Double Operator Integrals in a Hilbert Space. Integr. Eq. Oper. Th. 47, 131-168 (2003).
6. Borgs C., Kotecký R., Ueltschi D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409-446 (1996).
7. Bravyi S., Hastings M., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010).
8. Bravyi, S., Hastings, M.: A short proof of stability of topological order under local perturbations. http://arxiv. org/abs/1001. 4363v1 [math-ph], 2010.
9. 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).
10. 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, 155138 (2010).
11. Chen X., Gu Z.-C., Wen X.-G.: Classification of Gapped Symmetric Phases in 1D Spin Systems. Phys. Rev. B 83, 035107 (2011).
12. Cramer, M., Serafini, A., Eisert, J.: Locality of dynamics in general harmonic quantum systems. In: Quantum information and many body quantum systems, Ericsson, M., Montangero, S. (eds.), CRM Series, no. 8. Pisa: Edizioni della Normale, 2008, pp. 51-72.
13. Dagotto E., Rice T. M.: Surprises on the Way from One- to Two-Dimensional Quantum Magnets: The Ladder Materials. Science 271, 618-623 (1996).
14. 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, 455-534 (1996).
15. 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, 752-820 (1996).
16. Dziubańsky J., Hernández E.: Band-limited wavelets with subexponential decay. Canad. Math. Bull. 41, 398-403 (1998).
17. Fannes M., Nachtergaele B., Werner R.: Finitely Correlated States on Quantum Spin Chains. Commun. Math. Phys. 144, 443-490 (1992).
18. Hamza E., Michalakis S., Nachtergaele B., Sims R.: Approximating the ground state of gapped quantum spin systems. J. Math. Phys. 50, 095213 (2009).
19. Hastings M.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004).
20. Hastings M.: An area law for one dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007).
21. 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.
22. Hastings M., Koma T.: Spectral Gap and Exponential Decay of Correlations. Commun. Math. Phys. 265, 781-804 (2006).
23. Hastings, M., Michalakis, S.: Quantization of Hall conductance for interacting electrons without averaging assumptions. http://arxiv/org/abs/0911. 4706v1 [math-ph], 2009.
24. Hastings M., Wen X.: Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B. 72, 045141 (2005).
25. Ingham A. E.: A note on Fourier Transforms. J. London Math. Soc. 9, 29-32 (1934).
26. Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1980).
27. Kennedy T.: Long range order in the anisotropic quantum ferromagnetic Heisenberg model. Commun. Math. Phys. 100, 447-462 (1985).
28. Kennedy T., Tasaki H.: Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains. Commun. Math. Phys. 147, 431-484 (1992).
29. 2 symmetry breaking in Haldane gap antiferromagnets. Phys. Rev. B 45, 304-307 (1992).
30. Kitaev A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2-30 (2003).
31. Laughlin R. B.: The Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Phys. Rev. Lett. 50, 1395 (1983).
32. Lieb E. H., Robinson D. W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251-257 (1972).
33. Lieb E. H., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407-466 (1961).
34. Matsui T.: Uniqueness of the translationally invariant ground state in quantum spin systems. Commun. Math. Phys. 126, 453-467 (1990).
35. Nachtergaele B.: The spectral gap for some quantum spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565-606 (1996).
36. Nachtergaele B., Ogata Y., Sims R.: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124, 1-13 (2006).
37. Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems. Commun. Math. Phys. 286, 1073-1098 (2009).
38. Nachtergaele B., Schlein B., Sims R., Starr S., Zagrebnov V.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22, 207-231 (2010).
39. Nachtergaele B., Sims R.: Lieb-Robinson Bounds and the Exponential Clustering Theorem. Commun. Math. Phys. 265, 119-130 (2006).
40. Nachtergaele B., Sims R.: A multi-dimensional Lieb-Schultz-Mattis theorem. Commun. Math. Phys. 276, 437-472 (2007).
41. thInternational Congress on Mathematical Physics, Berlin-Heidelberg-New York: Springer Verlag, 2009, pp. 591-614.
42. Nachtergaele, B., Sims, R.: Lieb-Robinson Bounds in Quantum Many-Body Physics. In: Sims, R., Ueltschi, D. (eds), Entropy and the Quantum, Contemporary Mathematics, Volume 529, Providence, RI: Amer. Math. Soc., 2010, pp. 141-176.
43. Nachtergaele, B., Scholz, V. B., Werner, R. F.: Local approximation of observables and commutator bounds. http://arxiv/org/abs/1103. 5663v1 [math-ph], 2011.
44. Osborne T. J.: Simulating adiabatic evolution of gapped spin systems. J. Phys. A. 75, 032321 (2007).
45. Prémont-Schwarz I., Hamma A., Klich I., Markopoulou-Kalamara F.: Lieb-Robinson bounds for commutator-bounded operators. Phys. Rev. A. 81, 040102 (2010).
46. Prémont-Schwarz I., Hnybida J.: Lieb-Robinson bounds with dependence on interaction strengths. Phys. Rev. A. 81, 062107 (2010).
47. Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness. Methods of Modern Mathematical Physics, Vol. 2, London: Academic Press 1975.
48. Sachdev, S.: Quantum phase transitions. Cambridge University Press 2000.
49. Schuch N., Pérez-García D., Cirac I.: Classifying quantum phases using MPS and PEPS. Phys. Rev. B. 84, 165139 (2011).
50. Schuch N., Harrison S. K., Osborne T. J., Eisert J.: Information propagation for interacting particle systems. Phys. Rev. A. 84, 032309 (2011).
51. Tsui D. C., Stormer H. L., Gossard A. C.: Two-Dimensional Magnetotransport in the Extreme Quantum Limit. Phys. Rev. Lett. 48, 1559-1562 (1982).
52. Yarotsky D.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Commun. Math. Phys. 261, 799-819 (2006).