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

Abstract

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 \textit{Lees and Taggi (2019)}.

References in corpus (4)

• Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N
• Loop correlations in random wire models
• Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction
• Existence of Néel order in the S=1 bilinear-biquadratic Heisenberg model via random loops