Long-range order in the XXZ model

Hidetoshi Nishimori, Kenn Kubo, Yukiyasu Ozeki, Yasuhiro Tomita, Tatsuya Kishi

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3 Citations (Scopus)


The existence of long-range order is proved under certain conditions for the antiferromagnetic quantum spin system with anisotropic interactions (XXZ model) on the simple cubic or the square lattice. In three dimensions (the simple cubic lattice), finite long-range order exists at sufficiently low temperatures for any anisotropy Δ(≥0) if S≥1, and for 0≤Δ<0.29 (XY-like) or Δ>1.19 (Ising-like) if S=1/2. In two dimensions (the square lattice), ground-state long-range order exists under the following conditions: for any anisotropy (Δ≥0) if S≥3/2; 0≤Δ<0.032 (XY-like) or 0.67<Δ<1.34 (almost isotropic) or Δ>1.80 (Ising-like) if S=1;Δ>1.93 (Ising-like) if S=1/2. We conjecture that the two-dimensional spin-1/2 XY model (Δ=0) has finite ground-state long-range order. Numerical evidence supporting this conjecture is given.

Original languageEnglish
Pages (from-to)259-277
Number of pages19
JournalJournal of Statistical Physics
Issue number1-2
Publication statusPublished - 1989 Apr
Externally publishedYes


  • Long-range order
  • anisotropy
  • quantum effects

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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