WIAS Preprint No. 2713, (2020)

Site-monotonicity properties for reflection positive measures with applications to quantum spin systems


  • Lees, Benjamin
  • Taggi, Lorenzo

2010 Mathematics Subject Classification

  • 82B20 82B26 82B41 05A05


  • Quantum spin systems, random loop models, infrared bound, phase transitions




We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model, lattice permutations, thus extending the previous results of Lees and Taggi (2019).

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