Quantum Heisenberg antiferromagnets in a uniform magnetic field: Nonanalytic magnetic field dependence of the magnon spectrum

We reexamine the 1/S correction to the self-energy of the gapless magnon of a D-dimensional quantum Heisenberg antiferromagnet in a uniform magnetic field h using a hybrid approach between 1/S expansion and nonlinear sigma model, where the Holstein-Primakoff bosons are expressed in terms of Hermitian field operators representing the uniform and the staggered components of the spin operators [N. Hasselmann and P. Kopietz, Europhys. Lett. 74, 1067 (2006)]. By integrating over the field associated with the uniform spin fluctuations, we obtain the effective action for the staggered spin fluctuations on the lattice, which contains fluctuations on all length scales and does not have the cutoff ambiguities of the nonlinear sigma model. We show that in dimensions D <= 3, the magnetic-field dependence of the spin-wave velocity (c) over tilde_(h) is nonanalytic in h(2), with (c) over tilde_(h)-(c) over tilde_(0)proportional to h(2) ln vertical bar h vertical bar in D=3, and (c) over tilde_(h)-(c) over tilde (0)proportional to vertical bar h vertical bar in D=2. The frequency-dependent magnon self-energy is found to exhibit an even more singular magnetic-field dependence, implying a strong momentum dependence of the quasiparticle residue of the gapless magnon. We also discuss the problem of spontaneous magnon decay and show that in D>1 dimensions, the damping of magnons with momentum k is proportional to vertical bar k vertical bar(2D-1) if spontaneous magnon decay is kinematically allowed.