Skip to main content
SpringerLink
Log in

Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

A Correction to this article was published on 27 June 2018

This article has been updated

Abstract

We establish bounds on the decay of time-dependent multipoint correlation functionals of one-dimensional quasi-free fermions in terms of the decay properties of their two-point function. At a technical level, this is done with the help of bounds on certain bordered determinants and pfaffians. These bounds, which we prove, go beyond the well-known Hadamard estimates. Our main application of these results is a proof of strong (exponential) dynamical localization of spin-correlation functions in disordered XY-spin chains.

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

Change history

  • 27 June 2018

    On the occasion of the extension of part of our results to complex

References

  1. Abdul-Rahman H., Stolz G.: A uniform area law for the entanglement of eigenstates in the disordered XY chain. J. Math. Phys. 56, 121901 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Aizenman M., Graf J.M.: Localization bounds for an electron gas. J. Phys. A: Math. Gen. 31, 6783–6806 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Aizenman, M., Warzel, S.: Random Operators: Disorder Effects on Spectra and Dynamics. GSM 168. AMS (2015)

  4. Araki H.: Gibbs states of a one dimensional quantum lattice. Commun. Math. Phys. 14, 120–157 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Araki H.: On quasifree states of CAR and Bogoliubov automorphisms. Publ. RIMS Kyoto 6, 385–442 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Brandt U., Jacoby K.: Exact results for the dynamics of one-dimensional spin chains. Z. Physik B 25, 181–187 (1976)

    Article  ADS  Google Scholar 

  8. Bravyi S.: Lagrangian representation for fermionic linear optics. Quantum Inf. Comput. 5, 216–238 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Bravyi S., König R.: Disorder-assisted error correction in Majorana chains. Commun. Math. Phys. 316, 641–692 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chapman J., Stolz G.: Localization for random block operators related to the XY spin chain. Ann Henri Poincaré 16, 405–435 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Probability and Its Applications. Birkhäuser, Basel (1990)

  12. Elgart A., Shamis M., Sodin S.: Localisation for non-monotone Schrödinger operators. J. Eur. Math. Soc. 16, 909–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Friesdorf M., Werner A.H., Brown W., Scholz V.B., Eisert J.: Many-body localisation implies that eigenvectors are matrix-product states. Phys. Rev. Lett. 114, 170505 (2015)

    Article  ADS  Google Scholar 

  14. Hamza E., Sims R., Stolz G.: Dynamical localization in disordered quantum spin systems. Commun. Math. Phys. 315, 215–239 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Jordan P., Wigner E.: Über das Paulische Äquivalenzverbot. Z. Phys. 47, 631–651 (1928)

    Article  ADS  MATH  Google Scholar 

  16. Klein A., Perez J.F.: Localization in the ground-state of the one dimensional XY model with a random transverse field. Commun. Math. Phys. 128, 99–108 (1990)

    Article  ADS  MATH  Google Scholar 

  17. Lieb E., Schultz Th., Mattis M.D.: Two soluble models of an antiferromagnetic spin chain. Ann. Phys. 16, 407–466 (1961)

    Article  ADS  MATH  Google Scholar 

  18. McCoy B.M., Barouch E., Abraham B.: Statistical mechanics of the XY model. IV. Time-dependent spin-correlation functions. Phys. Rev. A 4, 2331–2341 (1971)

    Article  ADS  Google Scholar 

  19. Muir, T.: A Treatise on the Theory of Determinants. New York, Dover (1960). Revised and Enlarged by William H. Metzler

  20. Nachtergaele, B., Sims, R.: Locality Estimates for Quantum Spin Systems. New Trends in Mathematical Physics. In: Selected Contributions of the XVth International Congress on Mathematical Physics, pp. 591–614. Springer, Berlin (2009)

  21. Niemeijer Th.: Some exact calculations on a chain of spins \({\frac{1}{2}}\). Physica 36, 377–419 (1967)

    Article  ADS  Google Scholar 

  22. Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Grundlehren der Mathematischen Wissenschaften, vol. 297. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  23. Ruelle D.: Statistical mechanics of a one dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Seiringer R., Warzel S.: Decay of correlations and absence of superfluidity in the disordered Tonks–Girardeau gas. New J. Phys. 18, 035002 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  25. Stolze J., Nöppert A., Müller G.: Gaussian, exponential, and power-law decay of time-dependent correlation functions in quantum spin chains. Phys. Rev. B. 52, 4319–4326 (1995)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Robert Sims

  2. Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747, Garching, Germany

    Simone Warzel

Authors
  1. Robert Sims
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Warzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Sims.

Additional information

Communicated by M. Salmhofer

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sims, R., Warzel, S. Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices. Commun. Math. Phys. 347, 903–931 (2016). https://doi.org/10.1007/s00220-016-2612-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2612-0

Search

Navigation