Equivalence between the nonlinearσmodel and the spin-(1/2 antiferromagnetic Heisenberg model: Spin correlations inLa2CuO4

We study the continuum limit of the quantum nonlinear σ model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (${100}^2$×8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point $g_c$ that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model’s renormalization group β function close to the critical point. Using the β function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve ξ/$a_{\sigma }$=f(T/$T_{\sigma }$). Even though the lattice spacing vanishes, a finite unit of length $a_{\sigma }$ and a temperature scale $T_{\sigma }$ is generated via dimensional transmutation. Assuming that the nonlinear σ model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units $a_{\sigma }$ and $T_{\sigma }$ to the lattice spacing $a_H$ and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales $a_{\sigma }$,$T_{\sigma }$ and $a_H$, J is obtained, and f(T/$T_{\sigma }$) can be accurately approximated by an exponential of $T_{\sigma }$/T below $g_c$. We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.