Magnetic Field Response of Dipolar-Octupolar Quantum Spin Ice

TL;DR

This work analyzes dipolar-octupolar quantum spin ice in Ce-based pyrochlores under magnetic fields using gauge mean-field theory (GMFT) and projective symmetry group (PSG) analysis. Spins are mapped to a $U(1)$ gauge theory with spinon matter, allowing exploration beyond perturbative limits and yielding flux backgrounds determined by symmetry. Phase diagrams for field directions [110], [111], and [001] reveal both conventional 0-flux and $oldsymbol{\pi}$-flux QSI phases, plus a novel staggered flux state unique to the [110] field, with first-order field-induced transitions and field-direction–dependent signatures. The predicted static and dynamical spin structure factors provide concrete neutron-scattering fingerprints to guide experiments on Ce$_2$(Zr,Sn,Hf)$_2$O$_7$ and to test the DO-QSI scenario.

Abstract

Dipolar-octupolar (DO) pyrochlore systems Ce$_2$(Zr,Sn,Hf)$_2$O$_7$ have garnered much attention as recent investigations suggest that they may stabilize a novel quantum spin ice (QSI), a quantum spin liquid (QSL) with an emergent $U(1)$ gauge field. In particular, the experimentally estimated microscopic exchange parameters place Ce$_2$Zr$_2$O$_7$ in the $π$-flux QSI regime, and recent neutron scattering experiments have corroborated some key theoretical predictions. On the other hand, to make a definitive conclusion, more multifaceted experimental signatures are desirable. In this regard, recent neutron scattering investigation of the magnetic field dependence of the spin correlations in Ce$_2$Zr$_2$O$_7$ may provide valuable information. However, there have not been any comprehensive theoretical studies for comparison. In this work, we provide such information using gauge mean-field theory (GMFT), allowing for theoretical investigation beyond the perturbative regime. In particular, we construct the phase diagrams for the [110], [111], and [001] field directions. Furthermore, we demonstrate the distinctive evolution of the equal-time and dynamical spin structure factors as a function of the magnetic field for each field direction. These predictions will help future experiments confirm the true nature of the DO-QSI.

Figures (18)

• Figure 1: (a) The pyrochlore lattice. The down-pointing (up-pointing) tetrahedrons are colored blue (green). The parent diamond lattice sites located at the center of each tetrahedron are illustrated by spheres. (b) A single diamond unit cell with the four pyrochlore sublattices labeled. Black arrows denote the local pseudospin axis $\hat{\mathbf{z}}_\mu$ on each sublattice. The parent diamond lattice sites are labeled with $\mathbf{r}_A$ and $\mathbf{r}_B$. (c) Inequivalent hexagonal plaquettes. The plaquette $F_{320}$, which involves pyrochlore sublattices 0, 2, and 3, is highlighted in red. Polarized pseudospin $S^z$ configuration in the presence of only the Zeeman term (i.e., $J_{yy}/h\to 0$ and $J_{\pm}/h\to 0$) for a field along the (d) [001], (e) [110], and (f) [111] directions. The blue sites in (e) denote the $\beta$ chains that are decoupled from the field. For the [111] field in (f), the red sites form a Kagome plane, whereas the green sites make up triangular planes. Notice that the light and dark green sites are on different planes along the $[111]$ axis.
• Figure 2: Hexagonal plaquette fluxes for all allowed phases. (a) 0-flux and (b) $\pi$-flux phase with all plaquettes being threaded by 0 and $\pi$ flux, respectively. (c) $(0,\pi,\pi,0)$ phase where the plaquettes touching the edge highlighted in green are $\pi$-flux and the rests, 0-flux. (d) $(\pi,0,0,\pi)$ phase where the plaquettes touching the edge highlighted in green are 0-flux and the rests, $\pi$-flux.
• Figure 3: (1) Phase diagram as a function of transverse coupling $J_\pm$ and magnetic field strength $h$ for fields along the (1a) [110], (1b) [111], and (1c) [001] directions. Blue denotes the $\pi$-flux phase, red the $0$-flux phase, and purple the ($0,\pi,\pi,0$) phase. The black arrows at $J_\pm=-0.03$ and $J_\pm=0.03$ and the yellow arrows at $J_\pm=-0.3$ indicate the regions in parameter space where the SSSF and DSSF are calculated. (2) Static spin structure factors in the global frame of $\pi$-flux QSI at $J_\pm=-0.3$ for a [110] field (2a),(2d), a [111] field (2b), (2e), and a [001] field (2c), (2f). (3) Dynamical spin structure factors at finite magnetic fields for a [110] field (3a), a [111] field (3b), and a [001] field (3c).
• Figure 4: Normalized magnetization $g_{zz}\mu_B$ per pyrochlore site along the applied magnetic field, $|\mathbf{m}|=\sum_\mu \hat{\mathbf{n}}\cdot \hat{\mathbf{z}}_\mu\langle S^z_\mu \rangle/4$, with $\mathbf{n}$ parallel to (a) [110], (b) [111], and (c) [001].
• Figure 5: Static spin structure factors in the global frame $\mathcal{S}^{zz}(\mathbf{q})$ with a field in the $[110]$ direction along the $(h,-h,l)$ plane for a transverse coupling of (a), (c) $J_\pm/J_{yy} = 0.03$ and (b), (d) $J_\pm/J_{yy} = -0.03$ at a field strength of (a)-(b) $h/J_{yy}=0$ and (c)-(d) $h/J_{yy}=0.4$. (d) is in the $(0,\pi,\pi,0)$ phase. The First Brillouin zone is highlighted in white, and high symmetry points are labeled accordingly.