Skip to main content
SpringerLink
Log in

Disordered Fermions on Lattices and Their Spectral Properties

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤd, including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type \(\mathop {\rm {III}}\) von Neumann algebra (with the type \(\mathop {\rm {III_{0}}}\) component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type \(\mathop {\rm {III_{\lambda }}}\) for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.

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

References

  1. Accardi, L., Kozyrev, S.: Lectures on quantum interacting particle systems. In: Accardi, L., Fagnola F. (eds.) Quantum Interacting Particle Systems. QP–PQ, vol. 14, pp. 1–195. World Scientific, Singapore (2002)

    Chapter  Google Scholar 

  2. Accardi, L., Fidaleo, F., Mukhamedov, F.: Quantum Markov states and chains on the CAR algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, 165–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Araki, H.: Remarks on spectra of modular operators of von Neumann algebras. Commun. Math. Phys. 28, 267–278 (1972)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Araki, H.: Operator algebras and statistical mechanics. In: Dell’Antonio G., Doplicher S., Jona-Lasinio G. (eds.) Mathematical Problems in Theoretical Physics, Proc. Internat. Conf., Rome, 1977. Lecture Notes in Physics, vol. 90, pp. 94–105. Springer, Berlin (1978)

    Google Scholar 

  6. Araki, H.: Standard potentials for non-even dynamics. Commun. Math. Phys. 262, 161–176 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Araki, H., Moriya, H.: Equilibrium statistical mechanics of fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Araki, H., Moriya, H.: Conditional expectations relative to a product state and the corresponding standard potentials. Commun. Math. Phys. 246, 113–132 (2004)

    Article  ADS  MATH  Google Scholar 

  9. Barreto, S.D.: A quantum spin system with random interactions I. Proc. Indian Acad. Sci. 110, 347–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barreto, S.D., Fidaleo, F.: On the structure of KMS states of disordered systems. Commun. Math. Phys. 250, 1–21 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Barreto, S.D., Fidaleo, F.: Some topics in quantum disordered systems. Atti. Sem. Mat. Fis. Univ. Modena e Reggio Emillia 53, 215–234 (2005)

    MathSciNet  Google Scholar 

  12. Bellissard, J., van Elst, A., Schulz-Baldes, H.: Noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Binder, K., Young, A.P.: Spin glass: experimental facts, theoretical concepts and open questions. Rev. Mod. Phys. 58, 801–976 (1986)

    Article  ADS  Google Scholar 

  14. Borchers, H.-J.: Characterization of inner ∗-automorphisms of W ∗-algebras. Publ. RIMS Kyoto Univ. 10, 11–49 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (1981)

    Google Scholar 

  16. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin (1981)

    MATH  Google Scholar 

  17. Connes, A.: Une classification des facteurs de type III. Ann. Scient. Éc. Norm. Sup. 6, 133–252 (1973)

    MathSciNet  MATH  Google Scholar 

  18. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987)

    MATH  Google Scholar 

  19. Dixmier, J.: C ∗-Algebras. North-Holland, Amsterdam (1977)

    Google Scholar 

  20. Driessler, W.: On the type of local algebras in quantum field theory. Commun. Math. Phys. 53, 295–297 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Edwards, R.G., Anderson, P.W.: Theory of spin glasses. J. Phys. F 5, 965–974 (1975)

    Article  ADS  Google Scholar 

  22. van Enter, A.C.D., van Hemmen, J.L.: The thermodynamic limit for long range random systems. J. Stat. Phys. 32, 141–152 (1983)

    Article  ADS  Google Scholar 

  23. van Enter, A.C.D., van Hemmen, J.L.: Statistical mechanical formalism for spin-glasses. Phys. Rev. A 29, 355–365 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  24. van Enter, A.C.D., Maes, C., Schonmann, R.H., Shlosman, S.: The Griffiths singularity random field. Amer. Math. Soc. Trans. 198, 51–58 (2000)

    Google Scholar 

  25. Fidaleo, F.: KMS states and the chemical potential for disordered systems. Commun. Math. Phys. 262, 373–391 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Fidaleo, F.: Fermi Markov states. J. Operator Theory (to appear)

  27. Fidaleo, F., Mukhamedov, F.: Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras. Probab. Math. Stat. 24, 401–418 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233, 1–12 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. van Hemmen, J.L., Palmer, L.G.: The replica method and a solvable spin glass model. J. Phys. A: Math. Gen. 12, 563–580 (1979)

    Article  ADS  Google Scholar 

  30. van Hemmen, J.L., Palmer, L.G.: The thermodynamic limit and the replica method for short-range random systems. J. Phys. A: Math. Gen. 15, 3881–3890 (1982)

    Article  ADS  Google Scholar 

  31. Herman, R.H., Longo, R.: A note on the Γ-spectrum of an automorphism group. Duke Math. J. 47, 27–32 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis I. Springer, Berlin (1979)

    Google Scholar 

  33. Kastler, D.: Equilibrium states of matter and operator algebras. In: Symposia Mathematica, vol. XX, pp. 49–107. Academic Press, San Diego (1976)

  34. Kelley, J.L.: General Topology. Springer, Berlin (1975)

    MATH  Google Scholar 

  35. Kishimoto, A.: Equilibrium states of a semi-quantum lattice system. Rep. Math. Phys. 12, 341–374 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Külske, C.: (Non-)Gibbsianness and phase transitions in random lattice spin models. Markov Process. Relat. Fields 5, 357–383 (1999)

    MATH  Google Scholar 

  37. Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différence finies aléatoires. Commun. Math. Phys. 78, 201–246 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  38. Longo, R.: Algebraic and modular structure of von Neumann algebras of physics. Proc. Symp. Pure Math. 38, 551–566 (1982)

    MathSciNet  Google Scholar 

  39. Longo, R.: An analogue of the Kac-Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Matsui, T.: Ground states of fermions on lattices. Commun. Math. Phys. 182, 723–751 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Matsui, T.: Quantum statistical mechanics and Feller semigroup. Quantum Probab. Commun. 10, 101–123 (1998)

    MathSciNet  ADS  Google Scholar 

  42. Matsui, T.: On spectral gap, U(1) symmetry and split property in qauntum spin chains. arXiv:0808.1537v1 [math-ph]

  43. Mezard, M., Parisi, G., Virasoro, M.A.: Spin-Glass Theory and Beyond. World Scientific, Singapore (1986)

    Google Scholar 

  44. Newman, M.N.: Topics in Disordered Systems. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  45. Newman, M.N., Stein, D.L.: Ordering and broken symmetry in short-ranged spin glasses. J. Phys.: Condens. Matter 15, R1319 (1998)

    Article  ADS  Google Scholar 

  46. Nielsen, O.A.: Direct Integral Theory. Marcel Dekker, New York (1980)

    MATH  Google Scholar 

  47. Pedersen, G.K.: C ∗-Algebras and Their Automorphism Groups. Academic Press, London (1979)

    MATH  Google Scholar 

  48. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792–1796 (1975)

    Article  ADS  Google Scholar 

  49. Størmer, E.: Spectra of states, and asymptotically Abelian C ∗-algebras. Commun. Math. Phys. 28, 279–294 (1972)

    Article  ADS  Google Scholar 

  50. Strǎtilǎ, S.: Modular Theory in Operator Algebras. Abacus Press, Tunbridge Wells (1981)

    Google Scholar 

  51. Strǎtilǎ, S., Zsidó, L.: Lectures on von Neumann Algebras. Abacus Press, Tunbridge Wells (1979)

    Google Scholar 

  52. Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics and All That. Princeton University Press, Princeton (2000)

    MATH  Google Scholar 

  53. Sunder, V.S.: An Invitation to von Neumann Algebras. Springer, Berlin (1986)

    Google Scholar 

  54. Sutherland, C.E.: Crossed products, direct integrals and Connes’ classification of type III-factors. Math. Scand. 40, 209–214 (1977)

    MathSciNet  MATH  Google Scholar 

  55. Takesaki, M.: Theory of Operator Algebras III. Springer, Berlin (1979)

    Book  Google Scholar 

  56. Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  57. Talagrand, M.: The generalized Parisi formula. C. R. Acad. Sci. Paris, Ser. I 337, 111–114 (2003)

    MathSciNet  MATH  Google Scholar 

  58. Tanaka, T.: Moment problem in replica method. Interdisciplin Inf. Sci. 13, 17–23 (2007)

    Article  MATH  Google Scholar 

  59. Tasaki, H.: The Hubbard model—an introduction and selected rigorous results. J. Phys.: Condens. Matter 10, 4353 (1998)

    Article  ADS  Google Scholar 

  60. Wreszinski, W.F., Salinas, S.A.: Disorder and Competition in Soluble Lattice Models. Series on Advances in Statistical Mechanics, vol. 9. World Scientific, Singapore (1993)

    Google Scholar 

  61. Young, A.P. (ed.): Spin Glasses and Random Fields. World Scientific, Singapore (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Padre Conceicao College of Engineering, Verna Goa, 403 722, India

    Stephen Dias Barreto

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, 00133, Italy

    Francesco Fidaleo

Authors
  1. Stephen Dias Barreto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Fidaleo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Fidaleo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barreto, S.D., Fidaleo, F. Disordered Fermions on Lattices and Their Spectral Properties. J Stat Phys 143, 657–684 (2011). https://doi.org/10.1007/s10955-011-0197-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0197-8

Keywords

Search

Navigation