Magnon interactions in the frustrated pyrochlore ferromagnet ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The frustrated rare-earth pyrochlore ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is remarkable among magnetic materials: despite a ferromagnetically ordered ground state it exhibits a broad, nearly gapless, continuum of excitations. This broad continuum connects smoothly to the sharp one-magnon excitations expected, and indeed observed, at high magnetic fields, raising the question: how does this picture of sharp magnons break down as the field is lowered? In this paper, we consider the effects of magnon interactions in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, showing that their inclusion greatly extends the reach of spin-wave theory. First, we show that magnon interactions shift the phase boundary between the (splayed) ferromagnet (SFM) and the antiferromagnetic ${\mathrm{\Gamma }}_5$ phase so that ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ lies very close to it. Next, we show how the high-field limit connects to lower fields; this includes corrections to the critical fields for the [111] and $\left[1\overline{1}0\right]$ directions, bringing them closer to the observed experimental values, as well as accounting for the departures from linear spin-wave theory that appear in [001] applied fields below $3\mathrm{T}$ [Thompson et al., Phys. Rev. Lett. 119, 057203 (2017)]. Turning to low fields, though the extent of the experimentally observed broadening is not quite reproduced, we find a rough correspondence between nonlinear spin-wave theory and inelastic neutron scattering data on both a single-crystal sample as well as on a powder sample [Peçanha-Antonio et al., Phys. Rev. B 96, 214415 (2017)]. We conclude with an outlook on implications for future experimental and theoretical work on ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and related materials, highlighting the importance of proximity to the splayed ferromagnet-${\mathrm{\Gamma }}_5$ phase boundary and its potential role in intrinsic or extrinsic explanations of the low-field physics of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 1

Pyrochlore lattice of ${\mathrm{Yb}}^{3+}$ ions in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The three symmetry-related nearest-neighbor bond types ($x$, $y$, and $z$) are labeled, along with the three magnetic field directions we consider in this work. Also shown are the directions of the magnetic moments, ${\mathit{\mu }}_r$, of the ${\mathrm{Yb}}^{3+}$ ions in the splayed ferromagnet (SFM) state expected (classically) at zero field for the parameters of Thompson et al. [42].

Figure 2

The three diagram classes contributing at leading order to the magnon self-energy, ${\mathit{\Sigma }}^R(\mathit{k},\omega )$. Propagators are free Green's functions (denoted as —) including both normal and anomalous parts. Each diagram is connected to two external legs (denoted as $—$). Single four-magnon interaction vertices (denoted as $\blacksquare$) and pairs of three-magnon interaction vertices (denoted as $•$) are included [see Eqs. (11c), (11d), and (11e)].

Figure 3

Phase diagram in a region of nearest-neighbor exchange parameters near the boundary of the splayed ferromagnet (SFM) and ${\mathrm{\Gamma }}_5$ phases, as a function of the strength of (dual) Dzyaloshinskii-Moriya coupling $\left|D\right|/J$ and the (dual) symmetric anisotropic couplings $(K+\mathrm{\Gamma })/J$. The remaining parameter is chosen so that the exchanges of Thompson et al. [42] live in this plane (as shown), with $(K-\mathrm{\Gamma })/J=-0.096$. Classically, the ${\mathrm{\Gamma }}_5$ manifold is the ground state for $(K+\mathrm{\Gamma })/J\lesssim 0$ and the SFM states for $(K+\mathrm{\Gamma })/J\gtrsim 0$. Magnon interactions lead to an instability of the SFM at larger $(K+\mathrm{\Gamma })/J$, with spin-wave theory breaking down until the classical boundary at $(K+\mathrm{\Gamma })/J\sim 0$. For the parameters of Eq. (17), this is a shift of $+0.03\mathrm{meV}$. The exchange parameters of Thompson et al. [42] lie very close to the renormalized, nonlinear spin-wave theory (NLSWT) phase boundary.

Figure 4

Spin-wave gap ($\mathrm{\Delta }$) at wave vector [000] computed in linear spin-wave theory (LSWT) and nonlinear spin-wave theory (NLSWT) as a function of applied [001] magnetic field for the parameters of Thompson et al. [42]. The splayed ferromagnet becomes unstable at a small (but finite) field $\sim 2.8\mathrm{mT}$, due to its close proximity to the phase boundary to the ${\mathrm{\Gamma }}_5$ phase (see Fig. 3). The sharp softening of this mode occurs only in a small field range below $\left|\mathit{B}\right|\lesssim 0.2\mathrm{T}$. The region where spin-wave theory has broken down is indicated.

Figure 5

Spin-wave gap ($\mathrm{\Delta }$) at wave vector [000] computed in linear spin-wave theory (LSWT) and nonlinear spin-wave theory (NLSWT) as a function of applied (a) $\left[1\overline{1}0\right]$ and (b) [111] magnetic fields for the parameters of Thompson et al. [42]. For $\left[1\overline{1}0\right]$ fields in NLSWT there is an instability at $\left|\mathit{B}\right|\sim 0.61\mathrm{T}$ out of the splayed ferromagnet phase, not far from the experimental value, $\left|\mathit{B}\right|\sim 0.5\pm 0.125\mathrm{T}$, of Ross et al. [15]. For [111] fields, NLSWT becomes unstable at $\left|\mathit{B}\right|\sim 0.65\mathrm{T}$, close to the experimentally determined field Scheie et al. [48] and above the classical critical field of $0.46\mathrm{T}$. The region where spin-wave theory has broken down is indicated.

Figure 6

Dynamical structure factor in nonlinear spin-wave theory, along a path in momentum space for a [001] magnetic field of strength (a) $\left|\mathit{B}\right|=5\mathrm{T}$, (b) $3\mathrm{T}$, (c) $2\mathrm{T}$, (d) $1\mathrm{T}$, (e) $0.5\mathrm{T}$, and (f) $0.0^{+}\mathrm{T}$ for the parameters of Thompson et al. [42]. Intensity shows $\mathcal{S}(\mathit{q},\omega )$ [Eq. (13)]; the dots mark the one-magnon energies in linear spin-wave theory, and the upward pointing (downward pointing) symbols indicate the lower (upper) boundary of the two-magnon continuum. Overall intensity is arbitrary but consistent across panels. The instability at zero field can be seen in the soft mode in panel (f), near [020].

Figure 7

Dynamical structure factor in nonlinear spin-wave theory along a path in momentum space for a $\left[1\overline{1}0\right]$ magnetic field of strength (a) $\left|\mathit{B}\right|=5\mathrm{T}$ and (b) $2\mathrm{T}$, for the parameters of Thompson et al. [42]. Intensity shows $\mathcal{S}(\mathit{q},\omega )$ [Eq. (13)]; the dots mark the one-magnon energies in linear spin-wave theory, and the upward pointing (downward pointing) symbols indicate the lower (upper) boundary of the two-magnon continuum. Overall intensity is arbitrary but consistent across panels.

Figure 8

Dynamical structure factor in nonlinear spin-wave theory along a path in momentum space for [111] magnetic fields of strength (a) $\left|\mathit{B}\right|=3\mathrm{T}$ and (b) $1.5\mathrm{T}$ for the parameters of Thompson et al. [42]. Intensity shows $\mathcal{S}(\mathit{q},\omega )$ [Eq. (13)]; the dots mark the one-magnon energies in linear spin-wave theory, and the upward pointing (downward pointing) symbols indicate the lower (upper) boundary of the two-magnon continuum. Overall intensity is arbitrary but consistent across panels.

Figure 9

Comparison of inelastic neutron scattering intensity, $I(\mathit{q},\omega )$ [Eq. (14)], in linear spin-wave theory (LSWT), nonlinear spin-wave theory (NLSWT), and the experimental data of Thompson et al. [42] along $\left[h00\right]$ for four different fields along [001], (a) $\left|\mathit{B}\right|=5\mathrm{T}$, (b) $1.5\mathrm{T}$, (c) $0.21\mathrm{T}$, and (d) $0.0^{+}\mathrm{T}$. Intensities in all plots are averaged over wave vectors along transverse directions over the range $l,k=[-0.2,0.2]$. Overall scale is chosen to match experimental data and is consistent across Figs. 9, 10, and 11. The dashed lines indicate the boundaries of the experimental data, and features outside these boundaries in the experimental panels correspond to the NLSWT calculation.

Figure 10

Comparison of inelastic neutron scattering intensity, $I(\mathit{q},\omega )$ [Eq. (14)], in linear spin-wave theory (LSWT), nonlinear spin-wave theory (NLSWT), and the experimental data of Thompson et al. [42] along $\left[\overline{\zeta }\zeta 0\right]$ for four different fields along [001], (a) $\left|\mathit{B}\right|=5\mathrm{T}$, (b) $1.5\mathrm{T}$, (c) $0.21\mathrm{T}$, and (d) $0.0^{+}\mathrm{T}$. Intensities in all plots are averaged over wave vectors along transverse directions over the range $l,k=[-0.2,0.2]$. Overall scale is chosen to match experimental data and is consistent across Figs. 9, 10, and 11. The dashed lines indicate the boundaries of the experimental data, and features outside these boundaries in the experimental panels correspond to the NLSWT calculation.

Figure 11

Comparison of inelastic neutron scattering intensity, $I(\mathit{q},\omega )$ [Eq. (14)], in linear spin-wave theory (LSWT), nonlinear spin-wave theory (NLSWT), and the experimental data of Ref. [42] along $[-1+\xi ,1+\xi ,0]$ for four different fields along [001], (a) $\left|\mathit{B}\right|=5\mathrm{T}$, (b) $1.5\mathrm{T}$, (c) $0.21\mathrm{T}$, and (d) $0.0^{+}\mathrm{T}$. Intensities in all plots are averaged over wave vectors along transverse directions over the range $l,k=[-0.2,0.2]$. Overall scale is chosen to match experimental data and is consistent across Figs. 9, 10, and 11. The dashed lines indicate the boundaries of the experimental data, and features outside these boundaries in the experimental panels correspond to the NLSWT calculation.

Figure 12

Comparison of nonlinear spin-wave theory (NLSWT) calculation of powder-averaged inelastic neutron scattering intensity [see Eq. (14)] and zero-field experimental data of Peçanha-Antonio et al. [40]. A small field has been applied in the theoretical calculation to resolve the zero-field instability for the parameters of Ref. [42] (see Sec. 3b), as well as some broadening to mimic the finite experimental energy resolution.