The large-$N$ expansion of the quasi-two-dimensional quantum nonlinear $\sigma$ model is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at $N=\infty$, the renormalized coordination number introduced by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)] is a universal number in the limit of $J^{\prime }∕J\to 0$. Moreover, similar scaling relations proposed by Hastings and Mudry [Phys. Rev. Lett. 96, 027215 (2006)] are derived at $N=\infty$ for the three-dimensional static spin susceptibility at the wave vector $(\pi ,\pi ,0)$, as well as for the instantaneous structure factor at the same wave vector. We then use $1∕N$ corrections to study the relation between interplane coupling, correlation length, and critical temperature, and show that the universal scaling relations lead to logarithmic corrections to previous mean-field results.

• Received 1 June 2006

DOI:https://doi.org/10.1103/PhysRevB.74.184407