Diluted quantum antiferromagnets: Spin excitations and long-range order

We have studied the static and dynamic magnetic properties of two-dimensional (2D) and quasi-two-dimensional, spin-S, quantum Heisenberg antiferromagnets diluted with spinless vacancies. Using spin-wave theory and the T-matrix approximation we have calculated the staggered magnetization M(x,T), the neutron scattering dynamical structure factor S(k,ω), the 2D magnetic correlation length ξ(x,T) and, for the quasi- (2D) case, the Néel temperature T_N(x). We find that in two dimensions a hydrodynamic description of excitations in terms of spin waves breaks down at wavelengths larger than l/a∼e^π/4x, x being the impurity concentration and a the lattice spacing. We find signatures of localization associated with the scale l, and interpret this scale as the localization length of magnons. The spectral function for momenta a^-1≫k≫l^-1 consists of two distinct parts: (i) a damped quasiparticle peak at an energy c₀k≳ω≫ω₀, with abnormal damping Γ_k∼xc₀k, where ω₀∼c₀l^-1, c₀ is the bare spin-wave velocity; and (ii) a non-Lorentian localization peak at ω∼ω₀. For k≲l^-1 these two structures merge, and the spectrum becomes incoherent. The density of states acquires a constant term, and exhibits an anomalous peak at ω∼ω₀ associated with low-energy localized excitations. These anomalies lead to a substantial enhancement of the magnetic specific heat C_M at low temperatures. Although the dynamical properties are significantly modified, we show that D=2 is not the lower critical dimension for this problem. We find that at small x the average staggered magnetization at the magnetic site is M(x,0)≃S-Δ-Bx, where Δ is the zero-point spin deviation and B≃0.21 is independent of the value of S; the Néel temperature T_N(x)≃(1-A_sx) T_N(0), where A_s=π-2/π+B/(S-Δ) is weakly S dependent. Our results are in quantitative agreement with recent Monte Carlo simulations and experimental data for S=1/2, 1, and 5/2. In our approach long-range order persists up to a high concentration of impurities x_c which is above the classical percolation threshold x_p≈0.41. This result suggests that long-range order is stable at small x, and can be lost only around x≃x_p where approximations of our approach become invalid.