Quantum Effects in a Weakly Frustrated $S=1/2$ Two-Dimensional Heisenberg Antiferromagnet in an Applied Magnetic Field

We have studied the two-dimensional $S=1/2$ square-lattice antiferromagnet $\mathrm{Cu}(\mathrm{pz}{)}_2({\mathrm{ClO}}_4{)}_2$ (where pz denotes pyrazine), using neutron inelastic scattering and series expansion calculations. We show that the presence of antiferromagnetic next-nearest-neighbor interactions enhances quantum fluctuations associated with resonating valence bonds. Intermediate magnetic fields lead to a selective tuning of resonating valence bonds and a spectacular inversion of the zone-boundary dispersion, providing novel insight into 2D antiferromagnetism in the quantum limit.

Figure 1

Zone-boundary spin dispersion at zero field and $H\sim J$. Color plot of the normalized scattering intensity $I(\mathbf{Q},\omega )$ per ${\mathrm{Cu}}^{2+}$ at $T=80\mathrm{mK}$, showing the dispersion from $(\pi ,0)$ to $(\pi /2,\pi /2)$ at zero field and ${\mu }_{0}H=14.9\mathrm{T}$ under otherwise identical conditions. Both panels present smoothed data obtained by performing six constant $\mathbf{Q}$ scans from $(\pi /2,\pi /2)$ to $(\pi ,\pi )$ with a 0.05 meV energy step.

Figure 2

Series expansion calculations of the zone-boundary dispersion. Theoretical magnon dispersion for different values of the next-nearest-neighbor exchange interaction $J_2$ in zero magnetic field (left panel) and in a ${\mu }_{0}H=14.9\mathrm{T}$ magnetic field (right panel). The value of the nearest-neighbor interaction $J$ has been normalized to fit the experimental data. The normalization constants are shown in the legend.

Figure 3

Field dependence of zone-boundary excitations. Energy scans at $(\pi ,0)$ and $(\pi /2,\pi /2)$ are shown in (a) and (b), respectively. The data were measured at $T=80\mathrm{mK}$. The black solid line in (b) represents linear spin-wave excitations convoluted with the resolution function. Apart from the continuum region, all of the experimentally measured peaks are resolution limited. The field-dependent onset of magnetic scattering at $(\pi ,0)$ and $(\pi /2,\pi /2)$ as a function of energy is shown in (c) by red triangles and black circles, respectively. The dashed red and black lines represent the estimate from the series expansion calculations with $J_1=1.54\mathrm{meV}$ and $J_2=0.02J_1$, respectively.

Figure 4

Spin dispersion for $H\sim J$. (a) The spin-wave dispersion in $\mathrm{Cu}(\mathrm{pz}{)}_2({\mathrm{ClO}}_4{)}_{)}$ measured at ${\mu }_{0}H=12\mathrm{T}$ and $T=80\mathrm{mK}$. The red line represents the spin-wave dispersion with $J_1=1.54\mathrm{meV}$ and $J_2=0.02J_1$. (b),(c) Energy scans at $(0.6\pi ,0.6\pi )$ and $(0.7\pi ,0.7\pi )$ provide evidence of a mode with a field-induced gap. (d) Constant-energy scattering at 1 meV energy transfer providing evidence for the Goldstone mode. The solid line in (b)–(d) corresponds to a convolution of a Gaussian with the resolution function, demonstrating that the excitations are resolution limited.

Figure 5

Field dependence of the zone-center excitation. Energy scans performed at $(\pi ,\pi )$ at ${\mu }_{0}H=2\mathrm{T}$, ${\mu }_{0}H=6\mathrm{T}$, and ${\mu }_{0}H=10\mathrm{T}$ (a) and the gap energy at the AFM point plotted as the function of the field (b). The data were measured at $T=80\mathrm{mK}$. The solid lines in (a) are the fits of a Gaussian function convoluted with the resolution function. The filled circles in (b) represent the experimental data. The solid line in (b) is the linear fit of the gap energy. Black circles in inset (c) show the measured intensities as a function of the magnetic field, and the curve is the scattering intensity calculated using linear spin-wave theory.