Journal of Statistical Physics

, Volume 170, Issue 4, pp 684–699 | Cite as

Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model



It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.


Transverse field Ising model Quantum spin systems Gaussian random field The FKG inequality The Ghirlanda–Guerra identities The Lie–Trotter–Suzuki formula Interpolation 



It is pleasure to thank R. M. Woloshyn for reading the manuscript and helpful suggestions. I am grateful to M. Aoyagi for discussions in early stage of this work. I would like to thank the anonymous referees for essential comments.


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Authors and Affiliations

  1. 1.Department of Physics, GS & CSTNihon UniversityTokyoJapan

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