Towards a global phase diagram of Ce-based dipolar-octupolar pyrochlore magnets under magnetic fields

Abstract

Recent experiments have established a strong case for Ce$_2$(Zr, Sn, Hf)$_2$O$_7$ to host $π$-flux quantum spin ice (QSI). However, an irrefutable conclusion still requires strong, multifaceted evidence. In dipolar-octupolar (DO) compounds, external magnetic fields only strongly couple with the dipolar component $τ_z$ along its local z-axis in contrast to octupolar components $τ^{x,y}$. This gives rise to the unique ways magnetic fields interact with the system and, in turn, provides us with a variety of tuning knobs to generate comprehensive experimental results. In this work, we focus on magnetic fields along the (110), (111), and (001) directions and present a plethora of remarkable experimental signatures to probe the underlying physics of $π$-flux QSI using gauge mean field theory (GMFT) and Monte Carlo simulations. In particular, we present unique signatures in magnetic field-dependent phase diagrams, equal-time and dynamical structure factors, and magnetostriction.

Figures (4)

• Figure 1: Phase diagrams containing dipolar parameter set $(J_{xx},J_{yy},J_{zz},J_{xz})=(0.063, 0.062, 0.011,0)$meV (a-c) and the octupolar parameter set $(J_{xx},J_{yy},J_{zz},J_{xz})=(0.062, 0.063, 0.011,0)$meV (d-f) for Ce$_2$Zr$_2$O$_7$ denoted by the black arrow under field directions (111) (a,d), (001) (b,e), and (110) (c,f) by looking at the parameter spaces with $J_{\pm\pm}=0.2J_\parallel$ (a-c), $J_{\pm\pm}=-0.2J_\parallel$ (d-f), respectively. Blue denotes $\pi$-flux phase; red denotes 0-flux phase; and purple denotes the staggered $(0,\pi,\pi,0)$ phase. The green star in (c) denotes the parameter set for which the SSSF is computed. The insets: (a) an example hexagonal plaquette for which the flux thread through; (b) shows a pyrochlore parent unit cell denoting the pyrochlore coordinates $\mathbf{R}_\mu$ and the sublattice-indexed coordinate systems $\mathbf{r}_\alpha$ (see Sec. \ref{['I-sec:Appendix_DO_compound']} in Ref. supplementary for definitions); (c) an example of the coupled $\alpha$ chain in red and the decoupled $\beta$ chain in blue under a (110) field.
• Figure 2: Phase Diagrams containing octupolar Ce$_2$Hf$_2$O$_7$ parameter $(J_{xx},J_{yy},J_{zz},J_{xz})=(0.020, 0.047, 0.013,-0.008)$meV denoted by the black arrows by picking the parameter space where $J_{\pm\pm}=-0.09J_{yy}$, $\theta=-0.58$ under magnetic field direction (111) (a), (001) (b), and (110) (c). Blue denotes $\pi$-flux phase; red denotes 0-flux phase; and green denotes the staggered $(\pi,0,0,\pi)$ phase. The yellow star denotes the parameter set in which we compute the SSSF with.
• Figure 3: Static spin structure factor (1.a-1.b, 2.a-2.b) and dynamical spin structure factor (1.c, 2.c) with the proposed parameters of octupolar Ce$_2$Hf$_2$O$_7$ and dipolar Ce$_2$Zr$_2$O$_7$ under a (110) magnetic field with field strength $h=0.15J_{\parallel}$ (1.a-1.c) and $h=0.08J_{\parallel}$ (2.a-2.c). The inset in (1.b) shows $\mathcal{S}_{\text{NSF}}$ of an energetically less favourable 0-flux QSI under the octupolar Ce$_2$Hf$_2$O$_7$ parameters. Here, the DSSF is calculated along a path going through high symmetry points where $X=(1,0,0)$, $W=(1,-1/2,0)$, $K=(3/4,-3/4,0)$, $L=(1/2,1/2,1/2)$, $U=(1/4, 1/4, 1)$, $W'=(0,1/2,1)$ The side panels in (1.c) and (2.c) show the momentum integrated DSSF $\int d\mathbf{q}\mathcal{S}(\mathbf{q},\omega)$.
• Figure 4: Magnetostriction signatures (a,b) $L^{(001)}_{[111]}$, $L^{(001)}_{[110]}$, $L^{(001)}_{[001]}$ and $S_{PM}^{x}= S_0^x-S_1^x-S_2^x+S_3^x$ (c,d) under the application of a (001) magnetic field for (a,c) octupolar parameter set and (b,d) dipolar parameter set of Ce$_2$Hf$_2$O$_7$.