Hierarchy of energy scales and field-tunable order by disorder in dipolar-octupolar pyrochlores

Dipolar-octupolar pyrochlore magnets in a strong external magnet field applied in the [110] direction are known to form a “chain” state, with subextensive degeneracy. Magnetic moments are correlated along one-dimensional chains carrying effective Ising degrees of freedom which are noninteracting on the mean-field level. Here we investigate this phenomenon in detail, including the effects of quantum fluctuations. We identify two distinct types of chain phases, both featuring distinct subextensive, classical ground-state degeneracy. Focusing on one of the two kinds, we discuss lifting of the classical degeneracy by quantum fluctuations. We map out the ground-state phase diagram as a function of the exchange couplings, using linear spin wave theory and real-space perturbation theory. We find a hierarchy of energy scales in the ground-state selection, with the effective dimensionality of the system varying in an intricate way as the hierarchy is descended. We derive an effective two-dimensional anisotropic triangular lattice Ising model with only three free parameters which accounts for the observed behavior. Connecting our results to experiment, they are consistent with the observation of a disordered chain state in ${\mathrm{Nd}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. We also show that the presence of two distinct types of chain phases has consequences for the field-induced breakdown of the apparent $U\left(1\right)$ octupolar quantum liquid phase recently observed in ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$.

Figure 1

(a) Pyrochlore lattice separated into chains parallel ($\alpha$ chains) and perpendicular ($\beta$ chains) to the external field $\mathit{H}\parallel (1,1,0)$. (b) The two classical ground-state orientations for a single tetrahedron with the local easy axis along the local [111] direction. While the magnetic moments on the $\alpha$ chains are pinned by the external field $\mathit{h}$, those on the $\beta$ chains are perpendicular to the field and thus each retain one independent Ising-like degree of freedom. On one of the tetrahedra we also indicate the four fcc sublattices $0,1,2,3$. The local magnetic moments on each sublattice point into a direction $\pm {\widehat{\mathit{z}}}_i$, where the ${\widehat{\mathit{z}}}_i$ are defined to point into the tetrahedron.

Figure 2

Zero-temperature mean-field phase diagram of dipolar-octupolar pyrochlores with an applied field along the [110] direction as a function of exchange couplings and field strength, for $J_z>0$. The first column is for fixed octupolar coupling $J_y=0$ and the second and third coupling are for fixed magnetic fields $h=J_z/2$ and $h=4J_z$, respectively. The top and bottom rows have a fixed mixing angle $\theta =\pi /4$ and $\theta =0.33\pi$ respectively [cf. Eq. (5) for the meaning of the parameters]. The phase diagram does not change qualitatively for any value of $\theta$ except for the singular limits $\theta \to 0$, $\theta \to \pi /2$, and $\theta \to \pi$. As $h\to 0$ (not shown), while the low-field all-in-all-out (${\mathrm{AIAO}}_{\lambda }$) phases are stable, the chain phases (${\mathrm{Chain}}_{\lambda }^{(*)}$) are not. In the dipolar case ($\lambda =x,z$), at exactly zero field the chain phases become the respective spin-ice phase (${\mathrm{SI}}_{\lambda }$), with extensive instead of subextensive degeneracy. For the octupolar chain phase, as the field is lowered starting in the ${\mathrm{Chain}}_Y$ phase, there is actually a second-order transition to an intermediate phase ${\mathrm{Chain}}_Y^{*}$ (see also Fig. 9), in which each $\alpha$ chain carries an additional Ising degree of freedom. The ${\mathrm{Chain}}_Y^{*}$ phase is connected to the octupolar ice phase ${\mathrm{SI}}_Y$ at zero field. At large fields, the orientation of magnetic moments in the ${\mathrm{AIAO}}_{\lambda }$ phases get significantly distorted with respect to their zero field orientations. However, even as $h\to \infty$ they retain their twofold degeneracy and nonzero order parameter (${\sum }_i{S}_i^{\lambda }\ge SN/2$).

Figure 3

Transition form the ${\mathrm{Chain}}_Y$ to the ${\mathrm{Chain}}_Y^{*}$ phase as function of $J_y$ for fixed field $h=J_z$ and $\theta =\pi /4$. The plot shows the staggered octupolar moment on the $\alpha$ chains [Eq. (10)], which is nonzero only in the ${\mathrm{Chain}}_Y^{*}$ phase.

Figure 4

Effective triangular lattice Ising model. (a) The smallest linked clusters that yield a correction to the classical energy and distinguish between classical ground states. The part of the energy correction that distinguishes between classical ground states is proportional to the product of the Ising variables on the chains they connect. (b) The resulting effective anisotropic triangular lattice Ising model. (c) The phase diagram of the anisotropic triangular lattice Ising model as a function of the two nearest-neighbors couplings for the case of negligible next-nearest-neighbor coupling [38]. (d) Lifting of the subextensive degeneracy in the zigzag phase by next-nearest-neighbor coupling $J_{\mathrm{ch}}^{″}$.

Figure 5

Hierarchy of energy scales in ground-state selection in the ${\mathrm{Chain}}_Z$ phase. The classical ground-state energy is separated from the excited states on a scale set by $J_z$. On an energy scale of $\mathrm{\Delta }$, which for most values of exchange couplings lies in a range of ${10}^{-6}\text{–}{10}^{-4}{J}_z$ when $h\sim J_z$, quantum fluctuations split the classical ground-state manifold into the ferromagnetic state and the band of zigzag state. The choice of a single zigzag state happens again on a much lower energy scale $B$, which for most values of parameters lies in a range of ${10}^{-9}\text{–}{10}^{-7}{J}_z$. The numerical values for the scales are obtained from linear spin wave theory (Figs. 6 and 7).

Figure 6

Order by disorder (OBD) phase diagram of dipolar-octupolar pyrochlores for mixing angle $\theta =\pi /4$, together with the two relevant energy scales of the ground-state selection, as a function of exchange parameters. We indicate the ground state found by linear spin wave theory (LSWT) by color. Then, Degen denotes that all ground states are degenerate, i.e., no OBD is observed. ZZ denotes that a zigzag state is selected, but that the zigzag states are degenerate within the numerical precision (cf. the right column in the respective regions). FM denotes that a ferromagnetic chain configuration is selected. The striped (Str) and kinked (Kink) configuration are both zigzag configurations [Fig. 4]. We also indicate the transition between the FM and ZZ phases as computed from real space perturbation theory (RSPT) by a black line. We plot the phase diagram for two field values $h=S{J}_z/2$ (a) and $h=4S{J}_z$ (b). As anticipated from real space perturbation theory, the extent of the zigzag phase is significantly reduced as the external field is increased. This makes it possible to tune between different ground states in some regions of the phase diagram. In all cases, it is evident that the energy scales of ground-state selection between the ferromagnetic and the zigzag states are much larger than those of ground-state selection within the set of zigzag states.

Figure 7

Order by disorder (OBD) phase diagram of dipolar-octupolar pyrochlores for mixing angle $\theta =0.33\pi$, together with the two relevant energy scales of the ground-state selection, as a function of exchange parameters. We indicate the ground state found by linear spin wave theory (LSWT) by color. Then, Degen denotes that all ground states are degenerate, i.e., no OBD is observed. ZZ denotes that a zigzag state is selected, but that the zigzag states are degenerate within the numerical precision (cf. the right column in the respective regions). FM denotes that a ferromagnetic chain configuration is selected. The striped (Str) and kinked (Kink) configuration are both zigzag configurations [Fig. 4]. We also indicate the transition between the FM and ZZ phases as computed from real space perturbation theory (RSPT) by a black line. We plot the phase diagram for two field values $h=S{J}_z/2$ (a) and $h=4S{J}_z$ (b). As anticipated from real space perturbation theory, the extent of the zigzag phase is significantly reduced as the external field is increased. This makes it possible to tune between different ground states in some regions of the phase diagram. In all cases, it is evident that the energy scales of ground-state selection between the ferromagnetic and the zigzag states are much larger than those of ground-state selection within the set of zigzag states.

Figure 8

Difference of the zero point energy of the ferromagnetic chain configuration and the zigzag configuration from LSWT as a function of field for fixed exchange parameters. Top: For $J_x=0.9J_z$, $J_y=0.45J_z$, it is possible to tune the ground state of the system by means of the external field. Bottom: For $J_x=J_y=0.8J_z$, the ground state is always a zigzag configuration, however the OBD energy scale is much larger. The splitting of the zigzag states in both cases is imperceptible on the scale of the plot.

Figure 9

Transition from the ${\mathrm{Chain}}_Y^{*}$ to the ${\mathrm{Chain}}_Y$ phase as a function of the external field, for exchange parameters used to model ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ in Ref. [27]. The plot shows the staggered octupolar moment on the $\alpha$ chains [Eq. (10)], which is nonzero only in the ${\mathrm{Chain}}_Y^{*}$ phase. Classically, the octupolar ice phase ${\mathrm{SI}}_Y$ exists only for strictly zero field $h=0$, however in the quantum case there would be a finite window at low field with a quantum spin liquid ground state.

Figure 10

(a) The two distinct smallest clusters connecting two chains, marked in red and blue. Their lowest order contribution is of order 4 and they cancel out exactly. (b) A configuration of perturbation on a hexagon that does not contribute to the effective coupling. (c) The four inequivalent configurations that do contribute to $J_{\mathrm{ch}}$. (d) A possible sequence of intermediate states for the first configuration in (c). In (b) and (c) we label the sites with reference to the sublattice ${\mathcal{L}}_{\nu }$ they belong to, consistent with the hexagon indicated in green in Fig. 4

Figure 11

The four inequivalent configurations of perturbations on a hexagon contributing to $J_{\mathrm{ch}}^{\prime }$. We label the sites with reference to the sublattice ${\mathcal{L}}_{\nu }$ they belong to, consistent with the hexagon indicated in pink in Fig. 4.

Figure 12

Diagrams arising in RSPT in the ${\mathrm{Chain}}_Y$ phase. (a) Example of a configuration that does not contribute, with one possible direction of spin-flip hopping indicated. Whenever there is a double spin-flip acting on the $\beta$ chain (site 1, 2), the two perturbations on edge (1,2) and $(2,3^{\prime })$ will have to act in the same way (i.e., both with $S^{+}$ or both with $S^{-}$) on site 1 and 2, respectively, leading to a constant contribution. (b) The four perturbations contributing to $J_{\mathrm{ch}}^y$. Note that the product of matrix elements does not depend on the order of application of the perturbations, so that the $\mathrm{\Gamma }$ factors can be computed as before. (c) A distribution of perturbations on the hexagon whose product of matrix elements yields a contribution to $J_{\mathrm{ch}}^{y\prime }$. However, the diagram is not valid since there is no possible order in which, starting from the ground state, it yields the ground state also as a final state. For spin-1/2, there are actually no allowed diagrams using only $V_{ij}^{\left(1\right)}$, $V_{ij}^{\left(2\right)}$, and contributing to $J_{\mathrm{ch}}^{\prime }$, even at higher orders in RSPT.

Figure 13

Benchmarking the effective model using linear spin wave theory (LSWT) using the 14 ground states of the effective model and 1000 random chain configurations. (a) Maximum residue $R$ between a zero point energy (per site) predicted by the effective model and computed by LSWT. We give its absolute value as well as divided by the total bandwidth of zero point energies $B$, which is a scale given by the effective nearest-neighbor chain couplings. (b) Comparison between the nearest-neighbor effective chain couplings as calculated from sixth real-space perturbation theory ($J_{\mathrm{ch}}^{(\prime )\left(6\right)}$) and as obtained from LSWT using Eq. (19) ($J_{\mathrm{ch}}^{(\prime )\mathrm{SW}}$).

Figure 14

A possible sequence of intermediate states at fourth-order degenerate perturbation theory. The full sequence applied here is $V_i{V}_j{V}_j{V}_i$. Tetrahedra on which the pseudospin two-in-two-out rule is violated are indicated in red, with energy penalties $\mathrm{\Delta }{E}_{\mathrm{tet}}$ indicated next to them. Note that all valid orders of application of the $V_i$, $V_j$ will have the same set of intermediate energies since the ground state is not allowed as an intermediate state.

Figure 15

Order by disorder as a function of temperature for exchange parameters as estimated for ${\mathrm{Nd}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ [see Eq. (20)]. The plot shows the gap $\mathrm{\Delta }$ between the ferromagnetic chain configuration and the zigzag band. For low temperatures, it approaches the gap in zero point energies, while for high temperatures, the gap vanishes.