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 $D\le 3$, the magnetic-field dependence of the spin-wave velocity ${\tilde{c}}_{-}\left(h\right)$ is nonanalytic in $h^2$, with ${\tilde{c}}_{-}\left(h\right)-{\tilde{c}}_{-}\left(0\right)\propto h^2\text{ln}\left|h\right|$ in $D=3$, and ${\tilde{c}}_{-}\left(h\right)-{\tilde{c}}_{-}\left(0\right)\propto \left|h\right|$ 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 $\mathit{k}$ is proportional to ${\left|\mathit{k}\right|}^{2D-1}$ if spontaneous magnon decay is kinematically allowed.

Figure 1

(Color online) Spin configuration $⟨{\mathit{S}}_i⟩=S{\hat{\mathit{m}}}_i$ in the classical ground state of a two-sublattice antiferromagnet subject to a uniform magnetic field $\mathit{h}=h{\mathit{e}}_x$ in the $x$ direction. The hypercubic lattice can be divided into two sublattices, labeled A and B, such that the nearest neighbors of a given site all belong to the other sublattice. The solid square denotes a site of the A sublattice and a solid circle denotes a site of the B sublattice. Here ${\vartheta }_0$ is the classical canting angle between the direction of the staggered magnetization ${\mathbf{\Omega }}_i$ and the local spin direction ${\hat{\mathit{m}}}_i$.

Figure 2

(Color online) Graphical representation of the interaction vertices $V_{+}\left(K_1,K_2,K_3\right)$ and $V_{-}\left(K_1,K_2,K_3\right)$ defined in Eqs. (3.28, 3.29). Solid lines represent the gapless field $X_{-}$, while dashed lines correspond to the gapped field $X_{+}$. The shape of the symbols reflects the symmetry of the vertices with respect to the permutation of the labels.

Figure 3

(Color online) Feynman diagrams of the self-energy corrections to second order in the three-legged vertices [see Eqs. (4.3, 4.4)]. The slashed tadpole diagrams give frequency-independent contributions of order $1/S$, which are analytic functions of the magnetic field. Since in this work we are only interested in the frequency-dependent part of the self-energy, we shall omit the tadpole diagrams.

Figure 4

Evolution of the spin-wave velocity of the gapless magnon as a function of the external magnetic field for $S=1/2$. The full line shows the spin-wave velocity obtained numerically from Eq. (B1) normalized by the zero-field value ${\tilde{c}}_0\approx 1.16c_0$ for $S=1/2$ (see Ref. 4). The dashed line shows the prediction of Eq. (5.16a). Good agreement is obtained in the limit of vanishing fields, which confirms that the leading field dependence is described by Eq. (5.16a).