Scaling relations in quasi-two-dimensional Heisenberg antiferromagnet

The large-N expansion of the quasi-two-dimensional quantum nonlinear $\sigma$ model (QNLSM) is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at $N=\mathrm{\infty }$, the renormalized coordination number introduced by Yasuda $\mathit{\text{et}}\mathit{\text{al.}}$, Phys. Rev. Lett. $\mathbf{\text{94}}$, 217201 (2005), is a universal number in the limit of $J^{\mathrm{\prime }}\mathrm{/}J\to 0$. Moreover, similar scaling relations proposed by Hastings and Mudry, Phys. Rev. Lett. $\mathbf{\text{96}}$, 027215 (2006), are derived at $N=\mathrm{\infty }$ 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.