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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Statistical Physics

J. Math. Phys. 53, 023301 (2012); http://dx.doi.org/10.1063/1.3679069 (20 pages)

Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems

Michael Aizenman1, Rafael L. Greenblatt2, and Joel L. Lebowitz3

1Departments of Physics and Mathematics, Princeton University, Princeton, New Jersey 08544-8019, USA
2Dipartamento di Matematica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
3Departments of Mathematics and Physics, Rutgers University, Piscataway, New Jersey 08854-8019, USA

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(Received 19 April 2011; accepted 4 January 2012; published online 1 February 2012)

We prove that for quantum lattice systems in d ⩽ 2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T = 0. For systems with continuous symmetry the statement extends up to d ⩽ 4 dimensions. This establishes for quantum systems the existence of the Imry–Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. EXAMPLES OF THE ROUNDING EFFECT
    1. Transverse field Ising model
    2. Isotropic Heisenberg model
  3. A GENERAL FORMULATION
    1. The system and its Hamiltonian
    2. The quenched free energy and its infinite volume limit
    3. Quenched disorder with continuous symmetry
  4. STATEMENT OF THE MAIN RESULTS
    1. Two perspectives on 1st order phase transitions
    2. Thermodynamic formulation
    3. Statistical mechanical implications (no long range order)
  5. THE FREE-ENERGY-DIFFERENCE FUNCTIONAL
  6. UPPER BOUNDS ON THE FREE ENERGY DIFFERENCE
    1. A general surface bound
    2. An improved bound for systems with continuous symmetry
  7. STOCHASTIC LOWER BOUNDS ON THE LOCAL FREE ENERGY DIFFERENCE
  8. CONCLUSION OF THE PROOFS OF THE MAIN RESULTS
    1. Existence of an infinite-system limit for free energy fluctuations
      1. Finite temperature
      2. Absolutely continuous random fields
    2. Concluding the proofs of Theorems 4.1 and 4.2

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KEYWORDS and PACS

Keywords

lattice theory, phase transformations, quantum theory

PACS

  • 03.65.Ta

    Foundations of quantum mechanics; measurement theory

  • 05.50.+q

    Lattice theory and statistics (Ising, Potts, etc.)

  • 05.70.Fh

    Phase transitions: general studies

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3679069

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979).

    J. Z. Imbrie, Phys. Rev. Lett. 53, 1747 (1984).

    J. Bricmont and A. Kupiainen, Phys. Rev. Lett. 59, 1829 (1987).

    P. Goswami, D. Schwab, and S. Chakravarty, Phys. Rev. Lett. 100, 015703 (2008).

    R. L. Greenblatt, M. Aizenman, and J. L. Lebowitz, Phys. Rev. Lett. 103, 197201 (2009).

    D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992).

    R. Fisch, Phys. Rev. B 76, 214435 (2007).



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