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Large Deviations in Quantum Lattice Systems: One-Phase Region

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Abstract

We give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.

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References

  • H. Araki (1969) ArticleTitleGibbs states of a one dimensional quantum lattice Commun. Math. Phys. 14 120–157 Occurrence Handle1969CMaPh..14..120A Occurrence Handle273958 Occurrence Handle10.1007/BF01645134 Occurrence Handle0199.28001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • H. Araki (1974) ArticleTitleOn the equivalence of the KMS condition and the variational principal for quantum lattice systems Commun. Math. Phys. 38 1–10 Occurrence Handle1974CMaPh..38....1A Occurrence Handle475531 Occurrence Handle10.1007/BF01651545 Occurrence Handle0289.46048

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • H. Araki and H. Moriya, Equilibrium statistical mechanics of fermion lattice systems, Preprint, 2003.

  • M. van den Berg, J. T. Lewis and J. V. Pulé Large deviations and the boson gas, in Stochastic Mechanics and Stochastic Processes, Lecture Notes in Mathematics 1325 (Springer, Berlin, 1988) pp. 24–39.

  • M. van (1988) ArticleTitleden Berg J. T. Lewis and J. V. Pulé, The large deviation principle and some models of an interacting boson gas, Commun. Math. Phys. 118 61–85 Occurrence Handle1988CMaPh.118...61V

    ADS  Google Scholar 

  • I. Bjelakovic, T. Kröger, R. Siegmund-Schultze and A. Szkola, The Shannon–McMillan theorem for ergodic quantum lattice systems, Preprint, 2003.

  • T. I. Bjelakovic R Kröger (2003) Siegmund-Schultze and A. a quantum version of Breiman’s Theorem, Preprint Szkola, Chained typical subspaces

    Google Scholar 

  • O. Bratteli D. W. Robinson (1981) Operator Algebras and Quantum Statistical Mechanics. Vols. 1 and 2. Texts and Monographs in Physics Springer Berlin Berlin

    Google Scholar 

  • F. Comets (1986) ArticleTitleGrandes déviations pour des champs de Gibbs sur ℤd C. R. Acad. Sci. Paris Sér. I Math. 303 511–513 Occurrence Handle865873 Occurrence Handle0606.60035

    MathSciNet  MATH  Google Scholar 

  • N. Datta R. Fernández J. Fröhlich L. Rey-Bellet (1996) ArticleTitleLow-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy Helv. Phys. Acta 69 752–820 Occurrence Handle1428028 Occurrence Handle0867.47049

    MathSciNet  MATH  Google Scholar 

  • A. Dembo O. Zeitouni (1998) Large Deviations Techniques and Applications, Applications of Mathematics 38 EditionNumber2 Springer New York Occurrence Handle10.1007/978-1-4612-5320-4 Occurrence Handle0896.60013

    Book  MATH  Google Scholar 

  • J.-D. Deuschel D. W. Stroock H. Zessin (1991) ArticleTitleMicrocanonical distributions for lattice gases Commun. Math. Phys. 139 83–101 Occurrence Handle1991CMaPh.139...83D Occurrence Handle1116411 Occurrence Handle10.1007/BF02102730 Occurrence Handle0727.60025

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Entropy R. S. Ellis (1985) Large Deviations and Statistical Mechanics. Grundlehren der mathematischen Wissenschaften 271. Springer Berlin Heidelberg New York

    Google Scholar 

  • H. Föllmer S. Orey (1988) ArticleTitleLarge deviations for the empirical field of a Gibbs measure Ann. Probab. 16 961–977 Occurrence Handle942749 Occurrence Handle10.1214/aop/1176991671 Occurrence Handle0648.60028

    Article  MathSciNet  MATH  Google Scholar 

  • G. Gallavotti J. L. Lebowitz V. Mastropietro (2002) ArticleTitleLarge deviations in rarefied quantum gases J. Statist. Phys. 108 831–861 Occurrence Handle1933435 Occurrence Handle10.1023/A:1019766826534 Occurrence Handle1025.82004

    Article  MathSciNet  MATH  Google Scholar 

  • H.-O. Georgii (1993) ArticleTitleLarge deviations and maximum entropy principle for interacting random fields on \bZd Ann. Probab. 21 1845–1875 Occurrence Handle1245292 Occurrence Handle10.1214/aop/1176989002 Occurrence Handle0790.60031

    Article  MathSciNet  MATH  Google Scholar 

  • D. Goderis A. Verbeure P. Vets (1989) ArticleTitleNoncommutative central limits Prob. Th. Rel. Fields 82 527–544 Occurrence Handle1002899 Occurrence Handle10.1007/BF00341282 Occurrence Handle0695.46023

    Article  MathSciNet  MATH  Google Scholar 

  • D. Goderis A. Verbeure P. Vets (1990) ArticleTitleDynamics of fluctuations for quantum lattice systems Comm. Math. Phys. 128 533–549 Occurrence Handle1990CMaPh.128..533G Occurrence Handle1045883 Occurrence Handle10.1007/BF02096872 Occurrence Handle0702.46050

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • R. B. Israel (1979) Convexity in the Theory of Lattice Gases. Princeton series in physics Princeton University Press Princeton

    Google Scholar 

  • O. E. Lanford III (1973) Entropy and equilibrium states in classical statistical mechanics, in Statistical Mechanics and Mathematical Problems. Lecture Notes in Physics 20 Springer Berlin Heidelberg New York 1–113

    Google Scholar 

  • J. L. Lebowitz M. Lenci H. Spohn (2000) ArticleTitleLarge deviations for ideal quantum systems J. Math. Phys. 41 1224–1243 Occurrence Handle2000JMP....41.1224L Occurrence Handle1757957 Occurrence Handle10.1063/1.533185 Occurrence Handle1034.82034

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. T. Lewis C.-E. Pfister W. G. Sullivan (1994) ArticleTitleThe equivalence of ensembles for lattice systems: some examples and a counterexample J. Stat. Phys. 77 397–419 Occurrence Handle1994JSP....77..397L Occurrence Handle1300540 Occurrence Handle10.1007/BF02186849 Occurrence Handle0837.60096

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. T. Lewis, C.-E. Pfister and W. G. Sullivan, Entropy, concentration of probability and conditional limit theorems, Markov Process. Related Fields 1:319–386 (1995).

    MathSciNet  MATH  Google Scholar 

  • T. Matsui (2002) ArticleTitleBosonic central limit theorem for the one-dimensional XY model Rev. Math. Phys. 14 675–700 Occurrence Handle1932661 Occurrence Handle10.1142/S0129055X02001272 Occurrence Handle1027.82006

    Article  MathSciNet  MATH  Google Scholar 

  • T. Matsui, On the algebra of fluctuation in quantum spin chains, Preprint (2002).

  • K. Netocny and F. Redig, Large deviations for quantum spin systems, Preprint (2004).

  • P. Ohya D. Petz (1993) Quantum Entropy and its Use Texts and Monographs in Physics Springer Berlin Occurrence Handle10.1007/978-3-642-57997-4 Occurrence Handle0891.94008

    Book  MATH  Google Scholar 

  • S. Olla (1988) ArticleTitleLarge deviations for Gibbs random fields Probab. Th. Rel. Fields 77 343–357 Occurrence Handle931502 Occurrence Handle10.1007/BF00319293 Occurrence Handle0621.60031

    Article  MathSciNet  MATH  Google Scholar 

  • D. Petz G. A. Raggio A. Verbeure (1989) ArticleTitleAsymptotics of Varadhan-type and the Gibbs variational principle Commun. Math. Phys. 121 271–282 Occurrence Handle1989CMaPh.121..271P Occurrence Handle985399 Occurrence Handle10.1007/BF01217806 Occurrence Handle0682.46054

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • D. Petz (1990) First steps towards a Donsker and Varadhan theory in operator algebras, in Quantum Probability and Applications, V, Lecture Notes in Mathematics 1442 Springer Berlin 311–319

    Google Scholar 

  • C.-E. Pfister, Thermodynamical aspects of classical lattice systems, in In and Out of Equilibrium (Mambucaba, 2000), Progr. Probab., 51 (Birkhäuser Boston, Boston, MA, 2002), pp. 393–472.

  • G. A. Raggio R. F. Werner (1989) ArticleTitleQuantum statistical mechanics of general mean field systems Helv. Phys. Acta 62 980–1003 Occurrence Handle1034151 Occurrence Handle0938.82501

    MathSciNet  MATH  Google Scholar 

  • G. A. Raggio R. F. Werner (1991) ArticleTitleThe Gibbs variational principle for inhomogeneous mean-field systems Helv. Phys. Acta 64 633–667 Occurrence Handle1127778 Occurrence Handle0938.82500

    MathSciNet  MATH  Google Scholar 

  • D. Ruelle (1965) ArticleTitleCorrelation functionals J. Math.Phys. 6 201–220 Occurrence Handle1965JMP.....6..201R Occurrence Handle201174 Occurrence Handle10.1063/1.1704272 Occurrence Handle0127.21501

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • D. Ruelle (1999) Statistical Mechanics: Rigorous Results World Scientific River Edge, NJ Occurrence Handle10.1142/4090 Occurrence Handle1016.82500

    Book  MATH  Google Scholar 

  • B. Simon (1993) The Statistical Mechanics of Lattice Gases Vol. I. Princeton series in physics Princeton University Press Princeton

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, 07030

    Marco Lenci

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003

    Luc Rey-Bellet

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  1. Marco Lenci
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  2. Luc Rey-Bellet
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Correspondence to Marco Lenci.

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Lenci, M., Rey-Bellet, L. Large Deviations in Quantum Lattice Systems: One-Phase Region. J Stat Phys 190 (Suppl 1), S2 (2023). https://doi.org/10.1007/s10955-004-3015-8

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