Spin dynamics in the presence of competing ferromagnetic and antiferromagnetic correlations in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

In this work, we show that the zero-field excitation spectra in the quantum spin ice candidate pyrochlore compound ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is a continuum characterized by a very broad and almost flat dynamical response, which extends up to $1\text{–}1.5$ meV, coexisting or not with a quasielastic response depending on the wave vector. The spectra do not evolve between 50 mK and 2 K, indicating that the spin dynamics is only little affected by the temperature in both the short-range correlated and ordered regimes. Although classical spin dynamics simulations qualitatively capture some of the experimental observations, we show that they fail to reproduce this broad continuum. In particular, the simulations predict an energy scale twice smaller than the experimental observations. This analysis is based on a careful determination of the exchange couplings, able to reproduce both the zero-field diffuse scattering and the spin wave spectrum rising in the field polarized state. According to this analysis, ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ lies at the border between a ferro- and an antiferromagnetic phase. These results suggest that the unconventional ground state of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is governed by strong quantum fluctuations arising from the competition between those phases. The observed spectra may correspond to a continuum of deconfined spinons as expected in quantum spin liquids.

Figure 1

Specific heat $C$ (red squares) and magnetization $M/H$ (blue dots) vs temperature $T$. The specific heat is the long time specific heat (see text) and the magnetization was measured in a field of 3.83 mT, with a step of 10 mK every 1350 s.

Figure 2

Specific heat $C$ vs temperature $T$ for several magnetic fields. Open symbols correspond to the long-time specific heat ($t\approx 600$ s) and full symbols to the short-time specific heat ($t\approx 9$ s).

Figure 3

(a) Elastic intensity map in the $\left(hh\ell \right)$ scattering plane at $T=0.05$ K. White lines correspond to the Brillouin zones boundaries. Black dashed lines denote the directions probed in the inelastic neutron scattering experiments (see Sec. 2b2). (b) Elastic intensity (full blue dots) along the $(hh2-h)$ direction obtained by integrating (a) over $\delta Q=0.4$ rlu in the perpendicular direction. The lines with open symbols show the calculated elastic scattering functions obtained from classical spin dynamics simulations for different parameter sets $J_2=-0.26,-0.29,-0.31$ meV (open gray), $-0.326$ meV (open blue circles), and $T=0.4$ K (see Sec. 3e). (c) Spin-flip elastic intensities (full symbols) along the $(hh2-h)$ direction at $T=0.9$ K with polarization applied along $y$ (blue empty dots) and $z$ (red squares). The blue and red lines with open symbols correspond to calculated elastic scattering functions $S_y(\mathbf{Q},\omega =0)$ and $S_z(\mathbf{Q},\omega =0)$ obtained from classical spin dynamics simulations for $J_2=-0.32$ meV (see Sec. 3e).

Figure 4

Excitation spectra along high symmetry directions $\left(hh2\right),\left(00\ell \right)$, and $(hh2-h)$ at $T=0.05$ K. The blue area stands for the background. Dashed red and blue plain lines are the result of the fit considering a dual response consisting in quasielastic and inelastic contributions. Each contribution is represented in red while the total is represented in blue.

Figure 5

Excitations spectra at several wave vectors close to [(a), (b), and (d)] or away from [(c) and (e)] the rods of diffuse scattering, and temperature from $T=50$ to 850 mK.

Figure 6

(a) Spin polarized neutron intensity $S(\mathbf{Q},\omega )$ taken at $\mathbf{Q}=(1,1,2)$ and $T=4.5$ K. Red squares, blue circles, and green triangles, respectively, denote the spin-flip contribution in the $z,y$, and $x$ polarization channels. The inset shows the calculated spectra, see Sec. 3e at $T=0.9$ K. (b) Spin flip intensity with polarization along $x$ at 4.5 (red squares) and 2 K (blue circles). (c) shows the imaginary part of the magnetic susceptibility ${\chi }^{\prime \prime }(\mathbf{Q},\omega )$ at $\mathbf{Q}=(1,1,2)$ for 4.5 (red squares), 2 (blue circles), and 0.05 K (green triangles).

Figure 7

Sketches of the spin configurations on one tetrahedron in the (a) ${\psi }_2$, (b) ${\psi }_3$, and (c) FM phases. (d) Phase diagram calculated as a function of temperature and of exchange coupling varying along the optimal one-dimensional parameter space determined from the spin-wave fitting procedure. The black line roughly indicates the range of parameters for which the transition toward the ferromagnetic order is first order.

Figure 8

(a) Experimental spectra obtained in reference [8] at $H=2$ T and $T=0.05$ mK. [(b)–(d)] Examples of simulations, performed in the same temperature and field conditions, giving a good agreement with the data: (b) $(J_1,J_2,J_3,J_4)=(-0.09,-0.19,-0.25,0.005)$, (c) $(J_1,J_2,J_3,J_4)=(0.04,-0.29,-0.26,0.024)$, and (d) $(J_1,J_2,J_3,J_4)=(-0.02,-0.34,-0.29,0.036)$ meV.

Figure 9

Exchange interaction sets determined based on the field-induced spin waves at $H=2$ and 5 T. The green, red, and blue shaded areas in (b) correspond to the range of parameters giving a good agreement with the field induced spin-wave spectra. Exchange interactions reported in Refs. [8, 23] are indicated. Those reported in Refs [9, 16] are out of the parameter range of the figure, and are instead indicated in Fig. 15 of Appendix pp2. The fit goodness defined by ${\chi }^2$ is given along the 1D parameter line in (a) (see text).

Figure 10

Elastic scattering function $S(\mathit{Q},\omega =0)$ in the $(h,h,l)$ plane obtained from spin dynamics simulations for $J_2=-0.26,-0.29,-0.31,-0.319,-0.326,-0.345$ meV at $T=0.4$ K.

Figure 11

Comparison between the measured [8] (a) and the calculated (b) spin-wave spectra at 5 and 2 T along $(h,h,0)$ as well as the measured [16] (c) and the calculated (d) energy integrated diffuse scattering in the $\left(hh\ell \right)$ plane. The diffraction measurements are carried out with the spin of the neutrons polarized along the $z$ direction. In (c) and (d), SF and NSF are the spin-flip and non-spin-flip contributions, respectively, while the “total” is the sum of both contributions. All calculations are performed for the optimal set of parameters ${J_i}^0=(-0.03\left(2\right),-0.32\left(1\right),-0.28\left(3\right),0.02\left(2\right))$ meV.

Figure 12

$S(\mathbf{Q},\omega )$ along high symmetry directions: (a) neutron data (this work) taken at $T=0.05$ K; (b) calculated spectra obtained from classical spin dynamics simulations in the short-range correlated regime ($T=0.4$ K); and (c) RPA spectra obtained in the FM phase.

Figure 13

Calculated $S(\mathbf{Q},\omega )$ obtained from classical spin dynamics simulations at different temperatures $T=0.3,0.4,0.5,0.6$, and $0.9$ K, and for relevant wave vectors $\mathbf{Q}=(1,1,1),(2,2,0),(1/2,1/2,3/2)$, and $(0,0,2)$. Insets show the neutron data at the same wave vectors.

Figure 14

(Top) Example of raw data fitted according to the series of spectral bands $R_i\left(\omega \right)$. (Bottom) Relative weights $A_i/{\sum }_j{A}_j$ of the different bands for the data recorded at the base temperature along $\left(hh2\right),\left(00\ell \right),\left(11\ell \right),(hh2-h),$ and $(hh3-h)$. $A_i$ is defined via the spectral function $(1+n\left(\omega \right)]{\sum }_{i=0,\cdots ,6}{R}_i\left(\omega \right)-R_i(-\omega )$, where $R_i\left(\omega \right)$ is constant, equal to $A_i$ in a given energy range centered at $E_{i=0,1,2,3,4,5,6}=0.19,0.36,0.53,0.70,0.87,1.04$, and $1.21$ meV, with a fixed width of $2{\mathrm{\Delta }}_r=170 \mu \mathrm{eV}$.

Figure 15

Correspondence between the different exchange coupling sets. The green, red, and blue shaded areas correspond to the range of parameters giving a good agreement with the field induced spin-wave spectra (see Sec. 3c).