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A Relation Between Disorder Chaos and Incongruent States in Spin Glasses on \({\mathbb{Z}^d}\)

  • L.-P. ArguinEmail author
  • C. M. Newman
  • D. L. Stein
Article
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Abstract

We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards–Anderson model on \({\mathbb{Z}^d}\). The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.

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Notes

Acknowledgements

The authors thank the referee for many important suggestions that led to simplifications of some of the proofs. The authors are also grateful to Nick Read for insightful remarks and for an important correction to Proposition 2.9, as well as Aernout van Enter and Jon Machta for useful comments on the manuscript. The research of LPA is supported in part by U.S. NSF Grant DMS-1513441 and by U.S. NSF CAREER DMS-1653602. The research ofCMNwas supported in part by U.S. NSF Grants DMS-1207678 and DMS-1507019. The research of DLS was supported in part by U.S. NSF Grant DMS-1207678. DLS thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCity University of New York, Baruch College and Graduate CenterNew YorkUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiShanghaiChina
  4. 4.Department of Physics and Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  5. 5.NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU ShanghaiShanghaiChina
  6. 6.Santa Fe InstituteSanta FeUSA

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