Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state

Abstract

We present the general theory of clean, two-dimensional, quantum Heisenberg antiferromagnets which are close to the zero-temperature quantum transition between ground states with and without long-range N\'eel order. While some of our discussion is more general, the bulk of our theory will be restricted to antiferromagnets in which the N\'eel order is described by a three-vector order parameter. For N\'eel-ordered states, ``nearly critical'' means that the ground-state spin stiffness, ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$, satisfies ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$\ensuremath{\ll}J, where J is the nearest-neighbor exchange constant, while ``nearly critical'' quantum-disordered ground states have an energy gap, \ensuremath{\Delta}, towards excitations with spin 1, which satisfies \ensuremath{\Delta}\ensuremath{\ll}J. The allowed temperatures, T, are also smaller than J, but no restrictions are placed on the values of ${\mathit{k}}_{\mathit{B}}$T/${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ or ${\mathit{k}}_{\mathit{B}}$T/\ensuremath{\Delta}. Under these circumstances, we show that the wave vector and/or frequency-dependent uniform and staggered spin susceptibilities, and the specific heat, are completely universal functions of just three thermodynamic parameters. On the ordered side, these three parameters are ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$, the T=0 spin-wave velocity c, and the ground-state staggered moment ${\mathit{N}}_{0}$; previous works have noted the universal dependence of the susceptibilities on these three parameters only in the more restricted regime of ${\mathit{k}}_{\mathit{B}}$T\ensuremath{\ll}${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$. On the disordered side the three thermodynamic parameters are \ensuremath{\Delta}, c, and the spin-1 quasiparticle residue scrA. Explicit results for the universal scaling functions are obtained by a 1/N expansion on the O(N) quantum nonlinear \ensuremath{\sigma} model, and by Monte Carlo simulations. These calculations lead to a variety of testable predictions for neutron scattering, NMR, and magnetization measurements. Our results are in good agreement with a number of numerical simulations and experiments on undoped and lightly doped ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$${\mathrm{Sr}}_{\mathrm{\ensuremath{\delta}}}$${\mathrm{CuO}}_{4}$.