Experimental evidence for field-induced emergent clock anisotropies in the XY pyrochlore ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The XY pyrochlore antiferromagnet ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ exhibits a rare case of $Z_6$ discrete symmetry breaking in its ${\psi }_2$ magnetic ground state. Despite being well-studied theoretically, systems with high discrete symmetry breakings are uncommon in nature. Thus, ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ provides an experimental playground for the study of broken $Z_n$ symmetry, for $n>2$. A recent theoretical work examined the effect of a magnetic field on a pyrochlore lattice with broken $Z_6$ symmetry and applied it to ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. This study predicted multiple domain transitions depending on the crystallographic orientation of the magnetic field, inducing rich and controllable magnetothermodynamic behavior. In this work, we present neutron scattering measurements on ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ with a magnetic field applied along the [001] and [111] directions and provide experimental observation of these exotic domain transitions. In a [001] field, we observe a ${\psi }_2$ to ${\psi }_3$ transition at a critical field of $0.18\pm 0.05$ T. We are thus able to extend the concept of the spin-flop transition, which has long been observed in Ising systems, to higher discrete $Z_n$ symmetries. In a [111] field, we observe a series of domain-based phase transitions for fields of $0.15\pm 0.03$ T and $0.40\pm 0.03$ T. We show that these field-induced transitions are consistent with the emergence of twofold, threefold, and possibly sixfold Zeeman terms. Considering all the possible ${\psi }_2$ and ${\psi }_3$ domains, these Zeeman terms can be mapped onto an analog clock—exemplifying a literal clock anisotropy. Lastly, our quantitative analysis of the [001] domain transition in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is consistent with order-by-disorder as the dominant ground state selection mechanism.

Figure 1

(a) The $k=0,{\mathrm{\Gamma }}_5$ magnetic structure on the pyrochlore $\left(Fd\overline{3}m\right)$ lattice, which is constructed by the ${\psi }_2$ (red) and ${\psi }_3$ (blue) basis vectors. Within the local XY plane, six specific spin orientations are allowed by (b) ${\psi }_2$ and six interleaving angles are allowed by (c) ${\psi }_3$. The ensemble of the full ${\psi }_2$ and ${\psi }_3$ states mimic a literal clock and can be used to represent the different anisotropic Zeeman terms via the selection of different hours (angles) on the clock.

Figure 2

The elastic scattering of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ above and below the Neel ordering transition for crystals aligned in the (a,b) (H,H,L) plane, (c,d) (H,K,0) plane, and (e,f) (K+H,K-H, -2K) plane. At high temperature, the intensity of the (220) Bragg peak is purely structural, while at low temperature a magnetic Bragg peak also forms on the (220) position. Each of these data sets has been integrated in energy from $-0.1$ to 0.1 meV, which is approximately the resolution of elastic scattering in this experiment. Within the magnetically ordered state, this integration picks up a component of the magnetic inelastic scattering, giving the appearance of a significantly broadened peak. The dashed white lines indicate the areas of integration described in the text.

Figure 3

Representative selection of elastic cuts over the (220) Bragg peak in varying magnetic field strength, for fields applied along (a) the [1-10] direction, (c) the [001] direction, and (e) the [111] direction. The solid lines in these panels are the fits to the Bragg peak, which were used to extract the integrated intensity. The resultant field dependence of the magnetic elastic intensity at (220) with the field applied along the (b) [1-10], (d) [001], and (f) [111] direction, revealing multiple low field domain transitions in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The red curves in these panels correspond to the theoretically predicted domain transitions [17]. Note that the (2-20) Bragg position is symmetrically equivalent to (220), as it is referred to in the main text.

Figure 4

Magnetic field dependence of the inelastic neutron scattering spectrum of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ centered on the (220) Bragg position. These energy spectra are plotted along (a) the [H,H,0] direction for a [1-10] field, (b) the [2,K,0] direction for a [001] field, and (c) the [1+K′,-2K′,K′-1] direction for a [111] field. For the [1-10] field orientation, there is an immediate diminution of the quasi Goldstone-mode excitations at 0.5 T. For the [001] field orientation, the quasi-Goldstone mode excitations at (220) are intensified by the application of a field up to 1 T. In a [111] field, the quasi-Goldstone mode excitations have their intensity continuously decreased for fields ranging from 0.15 to 1 T. Above 1.5 T, the response of the spin wave spectra is due to the transition towards the field polarized state for all field orientations.

Figure 5

The inelastic intensity centered on (220) as a function of energy for varying (a) [1-10] magnetic fields, (b) [001] magnetic fields, and (c) [111] magnetic fields. In a [1-10] field, the low energy scattering is completely suppressed upon the application of a 0.5 T field. In a [001] field, the inelastic intensity at low energies strongly increases for fields as small as 0.1 T. Further increasing the field shifts the spectral weight to higher energies. In a [111] field, the inelastic spectra is unaffected by fields up to 0.15 T. At larger [111] fields, the intensity at low energies continuously decreases. The integrations performed in $Q$ are given in the experimental methods section.

Figure 6

The scattered intensity at the (220) Bragg position in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ as a function of the angle, $\theta$, in the XY local plane. The red dots represent the six ${\psi }_2$ domains and the blue dots represent the six ${\psi }_3$ domains. In (a) the two ${\psi }_2$ domains circled in black are the ones selected when a magnetic field of order 0.1 T is applied along the [1-10] direction. In (b) the four domains circled in black are the four ${\psi }_2$ states immediately selected in a [001] field. These states then cant towards the two ${\psi }_3$ states indicated by the black squares, which are selected by a $0.18\pm 0.05$ T [001] magnetic field. In (c) the three ${\psi }_2$ domains circled in black are the ones selected for magnetic fields up to 0.15 T applied along the [111] direction. At higher fields, these domains are predicted to split by an angle, $\theta$, as indicated by the arrows. Our measurement in a [111] field measured the (2-20) Bragg peak, which is symmetrically equivalent to (220).

Figure 7

(a) A typical fit of the scattering at the (220) Bragg peak position using both a Gaussian function and a Lorentzian function, to capture the elastic and inelastic contributions to the scattering, respectively. The field dependence of the Lorentzian contribution of the scattering for a (b) [1-10] magnetic field, (c) [001] magnetic field, and (d) [111] magnetic field. The field dependence of the Lorentzian contribution tracks the intensity of the low energy quasi-Goldstone-mode excitations.