Recent experiments on Ce$_2$Zr$_2$O$_7$ suggest that this material may host a novel form of quantum spin ice, a three-dimensional quantum spin liquid with an emergent photon. The Ce$^{3+}$ local moments on the pyrochlore lattice are described by pseudospin 1/2 degrees of freedom, whose components transform as dipolar and octupolar moments under symmetry operations. In principle, there exist four possible quantum spin ice regimes, depending on whether the Ising component is in the dipolar/octupolar channel, and two possible flux configurations of the emergent gauge field. In this work, using exact diagonalization and molecular dynamics, we investigate the equal-time and dynamical spin structure factors in all four quantum spin ice regimes using quantum and classical computations. Contrasting the distinct signatures of quantum and classical results for the four possible quantum spin ice regimes and elucidating the role of quantum fluctuations, we show that the quantum structure factor computed for the $π$-flux octupolar quantum spin ice regime is most compatible with the neutron scattering results on Ce$_2$Zr$_2$O$_7$.
We derive analytic forms for spin waves in pyrochlore magnets with dipolar-octupolar interactions, such as ${\rm Nd}_2{\rm Zr}_2{\rm O}_7$. We obtain full knowledge of the diagonalized magnonic Hamiltonian within the linear spin wave approximation. We also consider the effect of a "breathing mode" as a perturbation of this system. The breathing mode lifts the degeneracy of the upper band of the spin wave dispersion along the direction $X\to W$ in $k$-space.
We consider a class of d- and f-electron systems in which dipolar-octupolar Kramers doublets arise on the sites of the pyrochlore lattice. For such doublets, two components of the pseudospin transform like a magnetic dipole, while the other transforms like a component of the magnetic octupole tensor. Based on a symmetry analysis, we construct and study models of dipolar-octupolar doublets in itinerant and localized limits. In both limits, the resulting models are of surprisingly simple form. In the itinerant limit, we find topological insulating behavior. In the localized limit, the most gener…
The dipolar-octupolar pyrochlore oxides R$_2$M$_2$O$_7$ (R=Ce, Sm, Nd) represent an important opportunity in the search for three dimensional Quantum Spin Liquid (QSL) ground states. Their low energy physics is governed by an alluringly simple XYZ Hamiltonian, enabling theoretical description with only a small number of free parameters. Meanwhile, recent experiments on Ce pyrochlores strongly suggest QSL physics. Motivated by this, we present here a complete analysis of the ground state phase diagram of dipolar-octupolar pyrochlores. Combining cluster mean field theory, variational arguments a…
Recently there have been a number of experiments on Ce$_2$Zr$_2$O$_7$ and Ce$_2$Sn$_2$O$_7$, suggesting that these materials host a three-dimensional quantum spin liquid with emergent photons and fractionalized spinon excitations. However, the interpretation of the data to determine the precise nature of the quantum spin liquids is still under debate. The Kramers doublet in Ce$^{3+}$ local moment offers unusual pseudo-spin degrees of freedom as the $x$ and $z$ components transform as a dipole and $y$ component as an octupole. This leads to a variety of possible quantum spin liquid (or quantum …
While it is established that the pinch point scattering pattern in spin ice arises from an emergent coulomb phase associated with magnetic moment that is divergence-free, more complex Hamiltonians can introduce a divergence-full part. If these two parts remain decoupled, they give rise to the co-existence of distinct features. Here we show that the moment in ${\rm Nd_2Hf_2O_7}$ forms a static long-range ordered ground state, a flat, gapped pinch point excitation and dispersive excitations. These results confirm recent theories which predict that the dispersive modes, which arise from the diver…
Over the past two decades, the magnetic ground states of all rare earth titanate pyrochlores have been extensively studied, with the exception of Sm$_2$Ti$_2$O$_7$. This is, in large part, due to the very high absorption cross-section of naturally-occurring samarium, which renders neutron scattering infeasible. To combat this, we have grown a large, isotopically-enriched single crystal of Sm$_2$Ti$_2$O$_7$. Using inelastic neutron scattering, we determine that the crystal field ground state for Sm$^{3+}$ is a dipolar-octupolar doublet with Ising anisotropy. Neutron diffraction experiments reve…
Recent experiments on the dipolar-octupolar pyrochlore compound Ce$_2$Zr$_2$O$_7$ indicate that it may realize a three-dimensional quantum spin liquid (QSL). In particular, the analyses of available data suggest that the system is in a region of parameter space, where the so-called $π$-flux $U(1)$ octupolar quantum spin ice (QSI) with an emergent photon could be realized. On the other hand, because the system is far from the perturbative classical spin ice regime, it is unclear whether the quantum ground state is primarily a coherent superposition of 2-in-2-out configurations as in the canonic…
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. Focussing on one of the two kinds, we discuss lifting of the c…
Quantum spin liquid (QSL) is a disordered state of quantum-mechanically entangled spins commonly arising from frustrated magnetic dipolar interactions. However, QSL in some pyrochlore magnets can also come from frustrated magnetic octupolar interactions. Although the key signature for both dipolar and octupolar interaction-driven QSL is the presence of a spin excitation continuum (spinons) arising from the spin quantum number fractionalization, an external magnetic field-induced ferromagnetic order will transform the spinons into conventional spin waves in a dipolar QSL. By contrast, in an oct…
Experimental identification of quantum spin ice (QSI), a U(1) quantum spin liquid on the pyrochlore lattice hosting emergent photons, is a major challenge in frustrated magnets. In this work, we propose ultrasound measurements as a novel tool for probing the emergent photons of various QSI phases. Our analysis includes QSI phases in non-Kramers doublet compounds such as $\rm{Pr}_2 \rm{Zr}_2 \rm{O}_7$ as well as dipolar-octupolar Kramers doublet compounds such as $\rm{Ce}_2 \rm{Zr}_2 \rm{O}_7$. The latter may host emergent photons associated with an octupolar component which renders them diffic…
Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling d…
Motivated by experimental and theoretical interest in realizing multipolar orders in $d$-orbital materials, we discuss the quantum magnetism of $J\!=\!2$ ions which can be realized in spin-orbit coupled oxides with $5d^2$ transition metal ions. Based on the crystal field environment, we argue for a splitting of the $J\!=\!2$ multiplet, leading to a low lying non-Kramers doublet which hosts quadrupolar and octupolar moments. We discuss a microscopic mechanism whereby the combined perturbative effects of orbital repulsion and antiferromagnetic Heisenberg spin interactions leads to ferro-octupola…
We present a muon spin relaxation study on the Ising pyrochlore Nd$_2$Zr$_2$O$_7$ which develops an "all-in-all-out" magnetic order below 0.4~K. At 20~mK far below the ordering transition temperature, the zero-field muon spin relaxation spectra show no static features and can be well described by a dynamical Gaussian-broadened Gaussian Kubo-Toyabe function indicating strong fluctuations of the ordered state. The spectra of the paramagnetic state (below 4.2~K) reveal anomalously slow paramagnetic spin dynamics and show only small difference with the spectra of the ordered state. We find that th…
Quantum magnets with spin $J=2$, which arise in spin-orbit coupled Mott insulators, can potentially display multipolar orders. We carry out an exact diagonalization study of a simple octahedral crystal field Hamiltonian for two electrons, incorporating spin-orbit coupling (SOC) and interactions, finding that either explicitly including the $e_g$ orbitals, or going beyond the rotationally invariant Coulomb interaction within the $t_{2g}$ sector, causes a degeneracy breaking of the $J\!=\!2$ level degeneracy. This can lead to a low-lying non-Kramers doublet carrying quadrupolar and octupolar mom…
Symmetry plays a fundamental role in our understanding of both conventional symmetry breaking phases and the more exotic quantum and topological phases of matter. We explore the experimental signatures of symmetry enriched U(1) quantum spin liquids (QSLs) on the pyrochlore lattice. We point out that the Ce local moment of the newly discovered pyrochlore QSL candidate Ce$_2$Sn$_2$O$_7$, is a dipole-octupole doublet. The generic model for these unusual doublets supports two distinct symmetry enriched U(1) QSL ground states in the corresponding quantum spin ice regimes. These two U(1) QSLs are du…
Classical symmetry enriched topological orders and distinct monopole charges for dipole-octupole spin ices
International Center for Quantum Materials, Peking University（北京大学量子材料中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；The University of Hong Kong, Shenzhen Institute of Research and Innovation（香港大学深圳研究院）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
Single Crystal Diffuse Neutron Scattering Study of the Dipole-Octupole Quantum Spin Ice Candidate Ce$_2$Zr$_2$O$_7$: No Apparent Octupolar Correlations Above $T = 0.05$ K
本研究通过单晶漫散射中子衍射测量与数值连接团簇计算，发现在 $T = 0.05$ K 时 Ce$_2$Zr$_2$O$_7$ 仅表现出偶极自旋冰关联，未观察到八极关联散射的证据。
Observation of Dipolar Spin-ice--like Correlations in the Quantum Spin Ice Candidate Ce$_2$Sn$_2$O$_7$
McMaster University（麦克马斯特大学）；Clemson University（克莱姆森大学）；Harvard University（哈佛大学）；Max Planck Institute for Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Oak Ridge National Laboratory（橡树岭国家实验室）；Université de Montréal（蒙特利尔大学）
We point out the experimental relevance and the detection scheme of symmetry enriched U(1) quantum spin liquids (QSLs) outside the perturbative spin-ice regime. Recent experiments on Ce-based pyrochlore QSL materials suggest that the candidate QSL may not be proximate to the well-known spin ice regime, and thus differs fundamentally from other pyrochlore QSL materials. We consider the possibility of the $π$-flux U(1) QSL favored by frustrated transverse exchange interactions rather than the usual quantum spin ice. It was previously suggested that both dipolar U(1) QSL and octupolar U(1) QSL ca…
Discovery of novel spin-orbital entangled quantum ground states paves an important avenue for controllable quantum materials via unique couplings to the lattice and other external perturbations. In this work, motivated by recent experiments on cubic heavy fermion materials with multipolar local moments, we theoretically investigate strongly-interacting spin-orbital entangled quantum ground states in multipolar quantum impurity systems. Here itinerant electrons are interacting with the local moments carrying quadrupolar and octupolar moments, in contrast to the conventional Kondo problem with d…
Nd2Hf2O7, belonging to the family of geometrically frustrated cubic rare earth pyrochlore oxides, was recently identified to order antiferromagnetically below T_N = 0.55 K with an all-in/all-out arrangement of Nd3+ moments, however with a much reduced ordered state moment. Herein we investigate the spin dynamics and crystal field states of Nd2Hf2O7 using muon spin relaxation (muSR) and inelastic neutron scattering (INS) measurements. Our muSR study confirms the long range magnetic ordering and shows evidence for coexisting persistent dynamic spin fluctuations deep inside the ordered state down…
In spin ice research, small variations in structure or interactions drive a multitude of different behaviors, yet the collection of known materials relies heavily on the `227' pyrochlore structure. Here, we present thermodynamic, structural and inelastic neutron scattering data on a new spin-ice material, MgEr$_2$Se$_4$, which contributes to the relatively under-explored family of rare-earth spinel chalcogenides. X-ray and neutron diffraction confirm a normal spinel structure, and places Er$^{3+}$ moments on an ideal pyrochlore sublattice. Measurement of crystal electric field excitations with…
