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Site-monotonicity properties for reflection positive measures with applications to quantum spin systems. (English) Zbl 1467.82019

Summary: 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, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and \(XY\) model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates – improving previous positivity results which hold for the Cesàro sum. We also 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 and lattice permutations, thus extending the previous results of the authors [Commun. Math. Phys. 376, No. 1, 487–520 (2020; Zbl 1445.60075)].

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanics of magnetic materials

Citations:

Zbl 1445.60075
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References:

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