Communications in Mathematical Physics

, Volume 318, Issue 3, pp 809–830 | Cite as

Thermodyamic Bounds on Drude Weights in Terms of Almost-conserved Quantities

  • Enej Ilievski
  • Tomaž Prosen


We consider one-dimensional translationally invariant quantum spin (or fermionic) lattices and prove a Mazur-type inequality bounding the time-averaged thermodynamic limit of a finite-temperature expectation of a spatio-temporal autocorrelation function of a local observable in terms of quasi-local conservation laws with open boundary conditions. Namely, the commutator between the Hamiltonian and the conservation law of a finite chain may result in boundary terms only. No reference to techniques used in Suzuki’s proof of Mazur bound is made (which strictly applies only to finite-size systems with exact conservation laws), but Lieb-Robinson bounds and exponential clustering theorems of quasi-local C* quantum spin algebras are invoked instead. Our result has an important application in the transport theory of quantum spin chains, in particular it provides rigorous non-trivial examples of positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ spin 1/2 chain (Prosen, in Phys Rev Lett 106:217206, 2011).


Spin Chain Thermodynamic Limit Quantum Spin Gibbs State Linear Response Theory 
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  1. 1.
    Affleck, I.: Private communication, 2011Google Scholar
  2. 2.
    Araki H.: Gibbs States of a One Dimensional Quantum Lattice. Commun. Math. Phys. 14, 120–157 (1969)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1 & 2, Second Edition. Berlin: Springer-Verlag, 1997Google Scholar
  4. 4.
    Bravyi S., Hastings M.B., Verstraete F.: Lieb-Robinson bounds and the generation of correlations and the topological quantum order. Phys. Rev. Lett. 97, 050401 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    Grabowski M.P., Mathieu P.: Structure of the conservation laws in integrable spin chains with short range interactions. Ann. Phys. (N.Y.) 243, 299–371 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Grabowski M.P., Mathieu P.: The Structure of Conserved Charges in Open Spin Chains. J. Phys. A: Math. & Gen. 29, 7635–7650 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Ilievski, E., Prosen, T.: High-temperature expansion of spin Drude weight of anisotropic Heisenberg spin chain. Preprint 2011Google Scholar
  8. 8.
    Jakšić V., Ogata Y., Pillet C.-A.: The Green-Kubo Formula and the Onsager Reciprocity Relations in Quantum Statistical Mechanics. Commun. Math. Phys. 265, 721–738 (2006)ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Heidrich-Meisner F., Honecker A., Cabra D.C., Brenig W.: Zero-frequency transport properties of one-dimensional spin-1/2 systems. Phys. Rev. B 68, 134436 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    Khinchin A.I.: Korrelationstheorie der stationaren stochastischen Processe. Math. Annalen 190, 604–615 (1934)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lieb E., Robinson D.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Matsui T.: On the algebra of fluctuations in quantum spin chains. Ann. Henri Poincaré 4, 63–83 (2002)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Mazur P.: Non-ergodicity of phase functions in certain systems. Physica 43, 533–545 (1969)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Nachtergaele B., Ogata Y., Sims R.: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124, 1–13 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Prosen T.: Open XXZ spin chain: An exact nonequilibrium steady state and strict bound on ballistic transport. Phys. Rev. Lett. 106, 217206 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Rigol M., Shastry B.S.: Drude weight in systems with open boundary conditions. Phys. Rev. B 77, 161101(R) (2008)ADSCrossRefGoogle Scholar
  17. 17.
    Suzuki M.: Ergodicity, constants of motion, and bounds for susceptibilities. Physica 51, 277–291 (1971)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Zotos X., Naef F., Prelovšek P.: Transport and conservation laws. Phys. Rev. B 55, 11029–11032 (1997)ADSCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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