On the behaviour of a two-dimensional Heisenberg antiferromagnet at very low temperatures

We present an analytical formula for the ratio of the physical spin correlation length of a two-dimensional Heisenberg antiferromagnet on a square lattice, and the one which is actually computed in numerical simulations. This latter correlation length is deduced from the second moment of the structure factor at the antiferromagnetic momentum Q. We show that the ratio is very close to one, in agreement with a previously obtained numerical result based on the 1/N expansion. The two-dimensional Heisenberg antiferromagnet on a square lattice is one of the most extensively studied systems in condensed-matter physics. The interest in this model is twofold. On one hand, the Heisenberg antiferromagnet models a large number of real materials including parent compounds of high-Tc superconductors. On the other hand, its low-energy physics is adequately described by a field-theoretical σ -model thus allowing one to find similarities between condensed-matter physics and field theory. The low-temperature behaviour of the Heisenberg antiferromagnet is understood in great detail [1–3]. For short-range interaction, the ground state is ordered unless one fine tunes the couplings between nearest and further neighbours. The ordered ground state is characterized by a sublattice order parameter, N0, spin stiffness, ρs , and transverse susceptibility, χ⊥ = c−2ρs , where c is the spin-wave velocity. At any finite temperature, however, the system is disordered due to thermal fluctuations. The disordering means that the equal-time spin–spin correlation function decays exponentially with the distance, as e−r/ξ . The length scale ξ is the physical spin correlation length. Various approaches to 2D antiferromagnets all predict [1, 2, 4] that in the renormalized-classical region (T ρs), which we consider here, ξ is exponentially large in T at low T and behaves as ξ ∼ exp (2πρs/T ). Equal-time spin correlations at large distances can also be described by a static structure factor S(k) for k near the antiferromagnetic momentum Q = (π, π). At finite temperatures, S(Q) scales as ξ 2 and is therefore also exponential in T . The exponential temperature dependences of ξ and ofS(Q)have been verified in numerical simulations [5, 6], and by analysing the neutron scattering and NMR data for La2CuO4 and Sr2CuO2Cl2 [2]. The accuracy of numerical simulations is however so high that one can not only check the temperature dependences but also compare the absolute value of the spin correlation length with the exact expression for ξ obtained some time ago by Hasenfratz and Niedermayer (see below). Recently, two groups [5, 6] performed such a detailed comparison and found a good agreement with the Hasenfratz and Niedermayer formula at very low T . This comparison, however, requires care, as in numerical simulations one in fact measures not the physical spin correlation length ξ , but another length scale which differs from ξ by a 0953-8984/99/160169+06$19.50 © 1999 IOP Publishing Ltd L169 L170 Letter to the Editor constant factor which is not necessarily close to one. The point is that in numerical simulations one measures the spin structure factor S(k) in the momentum space. Meanwhile, the physical spin correlation length is associated with the real-space behaviour of the structure factor: at large distances S(r) ∝ e−r/ξ . To extract this ξ from S(k), one has to move to the imaginary k-axis. Then ξ−1 is the scale at which S(k) has a pole: S−1(k = iξ−1) = 0 [2]. In numerical simulations, however, the structure factor is evaluated only for real values of the momentum k. By agreement, the correlation length is identified as a second moment of S(k) for k = Q, i.e., as ξ̃ = (−S−1(Q) dS(k)/dk|k→Q) [5, 6]. For the Lorentzian form of S(k), S(k) ∝ ((Q− k)2 +m0), both ξ and ξ̃ are equal to the mean-field spin excitation gapm−1 0 and are therefore identical. However, the 1/N calculations for the O(N) σ -model rigorously demonstrated that S(k) has a Lorentzian form only in the limit N → ∞, while for arbitrary N , and, in particular, for physical N = 3, the momentum dependence of S(k) is different from a simple Lorentzian [2]. In this situation, ξ̃ and the physical spin correlation length ξ differ by some constant factor. To proceed further, we quote the exact theoretical result [2–4] ξ−1/m = ( 8 e )1/(N−2) 1 0(1 + 1/(N − 2)) (1)