Hydrodynamic relation in a two-dimensional Heisenberg antiferromagnet in a field

The spin stiffness ${\rho }_s$ of a two-dimensional (2D) Heisenberg antiferromagnet depends nonanalytically on external magnetic field. We demonstrate that the hydrodynamic relation between ${\rho }_s$, the uniform susceptibility $\chi$, and the spin-wave velocity $c$ is not violated by such a behavior because similar nonanalytic terms from all three quantities mutually cancel out. In this work, explicit expressions for the field-dependent spin stiffness and for the magnon velocity of the 2D square-lattice antiferromagnet are obtained by direct calculation to order $1/S$ and in the whole range of magnetic fields.

Figure 1

(Color online) Left: angle of twist $\phi$ in the $x_0\text{−}{y}_0$ plane of spins in one sublattice with respect to the other. Right: field-induced canting by the angle $\theta$.

Figure 2

(Color online) $1/S$ quantum correction to the spin stiffness $-\delta {\rho }_s/JS$ as a function of the field $h$. Inset: ${\rho }_s/J{S}^2$ vs $h$ for $S=1/2$; dashed line is ${\rho }_s^{\text{cl}}/J{S}^2$ vs $h$.

Figure 3

(Color online) $1/S$ quantum correction to the susceptibility $JS\delta \chi$ as a function of the field $h$. Inset: $J\chi$ vs $h$ for $S=1/2$; dashed line is $J{\chi }_{\text{cl}}$.

Figure 4

(Color online) $1/S$ quantum correction to the spin-wave velocity $\Delta c/J$ as a function of the field $h$. Inset: $c/2JS$ vs $h$ for $S=1/2$; dashed line is $c_0/2JS$ vs $h$.

Figure 5

(Color online) Cancellation of the $1/S$ quantum corrections in the hydrodynamic relation (28) as a function of the field $h$ (solid line). Dashed lines show contributions of individual terms in Eq. (28).