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.
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…
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…
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…
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…
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…
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 ex…
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…
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…
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…
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…
The spin dynamics in the geometrically frustrated pyrochlore antiferromagnet $\rm Gd_2Sn_2O_7$ is studied by means of the electron spin resonance. In the ordered phase ($T_N = 1$ K), we have detected three gapped resonance modes. Their values agree well with the developed spin-wave theory which takes into account the Heisenberg nearest-neighbor exchange, the single-ion anisotropy and the long-range dipolar interactions. The theory also predicts a fourth lowest-frequency gap, which lies beyond the experimental range of frequencies, but determines the exponential decrease of the specific heat at…
We present a mean-field theory for magnetic field driven transitions in dipolar coupled gadolinium titanate Gd$_2$Ti$_2$O$_7$ pyrochlore system. Low temperature neutron scattering yields a phase that can be regarded as a 8 sublattice antiferromagnet, in which long-ranged ordered moments and fluctuating moments coexist. Our theory gives parameter regions where such a phase is realized, and predicts several other phases, with transitions amongst them driven by magnetic field as well as temperature. We find several instances of {\em local} disorder parameters describing the transitions.
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…
Influence of controlled disorder on the dipolar spin ice state of Ho-based pyrochlores
Oak Ridge National Laboratory（橡树岭国家实验室）；University of California Santa Barbara（加州大学圣塔芭芭拉分校）；Perry College of Mathematics, Computing, and Sciences, University of West Georgia（佐治亚西部大学珀里数学、计算与科学学院）；Japan Atomic Energy Agency（日本原子能机构）；Brigham Young University（杨百翰大学）
The low energy spin excitation spectrum of the breathing pyrochlore Ba3Yb2Zn5O11 has been investigated with inelastic neutron scattering. Several nearly resolution limited modes with no observable dispersion are observed at 250 mK while, at elevated temperatures, transitions between excited levels become visible. To gain deeper insight, a theoretical model of isolated Yb3+ tetrahedra parametrized by four anisotropic exchange constants is constructed. The model reproduces the inelastic neutron scattering data, specific heat, and magnetic susceptibility with high fidelity. The fitted exchange pa…
We present a comprehensive experimental and theoretical study of the pyrochlore Tb$_2$Ge$_2$O$_7$, an exemplary realization of a material whose properties are dominated by competition between magnetic dipolar and electric quadrupolar correlations. The dipolar and quadrupolar correlations evolve over three distinct regimes that we characterize via heat capacity, elastic and inelastic neutron scattering. In the first regime, above $T^*=1.1$ K, significant quadrupolar correlations lead to an intense inelastic mode that cannot be accounted for within a scenario with solely magnetic dipole-dipole c…
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。
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）
We study Ising pyrochlores by means of Monte Carlo simulations. We cover a set of exchange constants ranging from the frustrated ferromagnetic case (spin-ice) to the fully-ordered "all-in--all-out" antiferromagnet in the dipolar model, reinterpreting the results --as in an ionic system-- in terms of a temperature vs. magnetic charge density phase diagram. In spite of its spin nature and the presence of both double and single non-conserved magnetic charges, the dipolar model gives place to a phase diagram which is quite comparable with those previously obtained for on-lattice systems of electri…
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…
Despite a well-ordered pyrochlore crystal structure and strong magnetic interactions between the Dy$^{3+}$ or Ho$^{3+}$ ions, no long range magnetic order has been detected in the pyrochlore titanates Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$. To explore the actual magnetic phase formed by cooling these materials, we measure their magnetization dynamics using toroidal, boundary-free magnetization transport techniques. We find that the dynamical magnetic susceptibility of both compounds has the same distinctive phenomenology, that is indistinguishable in form from that of the dielectric permittiv…
From a microscopic model for the pyrochlore antiferromagnet Tb2Sn2O7, including the crystal field Hamiltonian and interactions between the angular momenta, we compute an effective pseudospin-1/2 Hamiltonian Heff$ that incorporates perturbatively in the effective interactions the effect of excited crystal field levels. We obtain the semiclassical ground states of Heff and find a region of parameter space with a two-in/two-out spin ice configuration on each tetrahedron with ordering wavevector q=0 and with spins canted away from the local Ising axes as found in Tb2Sn2O7. This ground state can al…
We study spin correlations for the highly frustrated classical pyrochlore lattice antiferromagnets with O(N) symmetry in the limit T->0. We conjecture that a local constraint obeyed by the extensively degenerate ground states dictates a dipolar form for the asymptotic spin correlations, at all N $\ne$ 2 for which the system is paramagnetic down to T=0. We verify this conjecture in the cases N=1 and N=3 by simulations and to all orders in the 1/N expansion about the solvable N=infinity limit. Remarkably, the N=infinity formulae are an excellent fit, at all distances, to the correlators at N=3 a…
We investigate spin correlations in the dipolar Heisenberg antiferromagnet Gd2Sn2O7 using polarised neutron-scattering measurements in the correlated paramagnetic regime. Using Monte Carlo methods, we show that our data are sensitive to weak further-neighbour exchange interactions of magnitude ~0.5% of the nearest-neighbour interaction, and are compatible with either antiferromagnetic next-nearest neighbour interactions, or ferromagnetic third-neighbour interactions that connect spins across hexagonal loops. Calculations of the magnetic scattering intensity reveal rods of diffuse scattering al…
We report experimental signatures of topological transitions among the Weyl semimetal states of pyrochlore Pr2Ir2O7, where the Kondo coupling between the Ir topological electrons and the spin-ice like orders of Pr moments plays a decisive role. The magnetic-field dependence of resistivity and the Hall conductivity exhibits a plateau and a sharp jump associated with a magnetic-field hysteresis, similar to a liquid-gas-like transition in dipolar spin ice system. Furthermore, the Kondo coupling is controlled by the hydrostatic pressure, revealing that the field-induced displacement of Weyl points…
The search for quantum spin liquids (QSL) -- topological magnets with fractionalized excitations -- has been a central theme in condensed matter and materials physics. While theories are no longer in short supply, tracking down materials has turned out to be remarkably tricky, in large part because of the difficulty to diagnose experimentally a state with only topological, rather than conventional, forms of order. Pyrochlore systems have proven particularly promising, hosting a classical Coulomb phase in the spin ices Dy/Ho$_2$Ti$_2$O$_7$, with subsequent proposals of candidate QSLs in other p…
The Raman spectroscopy and AC and DC magnetization of Dy$_{2-x}$Eu$_x$Ti$_2$O$_7$ have been investigated. In Raman Spectroscopy, the systematic shift in all phonon modes with Eu content in Dy$_{2-x}$Eu$_x$Ti$_2$O$_7$ confirms that Dy$^{3+}$ ion is substituted by Eu3+ ions. High concentration of Eu induces the dipolar exchange interactions and crystal-field interactions in Dy$_{2-x}$Eu$_x$Ti$_2$O$_7$. Rich Eu content samples (x=1.8 and 1.9) show the existence of wasp-waisted hysteresis loop and that can be attributed to the coexistence of dipolar field and anisotropy exchange interaction. AC su…
We report the low-temperature magnetothermal properties and spin dynamics of a mixed pyrochlore, Dy2GaSbO7 through the coordination of measurements of dc magnetization, ac susceptibility, and heat capacity, and CF computation. In Dy2GaSbO7, the spins freeze at temperature Tice = 3.1 K, corresponding to dynamical spin-freezing into a disordered state in compliance with the spin-ice rules, but with enhanced zero-point entropy, unlike cannonical dipolar spin-ice materials, Dy2Ti2O7 and Ho2Ti2O7, due to random disorder and varied chemical pressure of B-site (GaSb tetrahedra) ions.
The ground state ensemble of the highly frustrated pyrochlore-lattice antiferromagnet can be mapped to a coarse-grained ``polarization'' field satisfying a zero-divergence condition From this it follows that the correlations of this field, as well as the actual spin correlations, decay with separation like a dipole-dipole interaction ($1/|R|^3$). Furthermore, a lattice version of the derivation gives an approximate formula for spin correlations, with several features that agree well with simulations and neutron-diffraction measurements of diffuse scattering, in particular the pinch-point (pseu…
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 investigate the transport properties of itinerant electrons interacting with a background of localized spins in a correlated paramagnetic phase of the pyrochlore lattice. We find a residual resistivity at zero temperature due to the scattering of electrons by the static dipolar spin-spin correlation that characterizes the metallic Coulomb phase. As temperature increases, thermally excited topological defects, also known as magnetic monopoles, reduce the spin correlation, hence suppressing electron scattering. Combined with the usual scattering processes in metals at higher temperatures, thi…
Er$_2$Sn$_2$O$_7$ remains a puzzling case among the extensively studied frustrated compounds of the rare-earth pyrochlore family. Indeed, while a first order transition towards a long-range antiferromagnetic state with the so-called Palmer-Chalker structure is theoretically predicted, it has not been observed yet, leaving the issue, as to whether it is a spin-liquid candidate, open. We report on neutron scattering and magnetization measurements which evidence a second order transition towards this Palmer-Chalker ordered state around 108 mK. Extreme care was taken to ensure a proper thermalizat…
Spin ice, a peculiar thermal state of a frustrated ferromagnet on the pyrochlore lattice, has a finite entropy density and excitations carrying magnetic charge. By combining analytical arguments and Monte Carlo simulations, we show that spin ice on the two-dimensional kagome lattice orders in two stages. The intermediate phase has ordered magnetic charges and is separated from the paramagnetic phase by an Ising transition. The transition to the low-temperature phase is of the three-state Potts or Kosterlitz-Thouless type, depending on the presence of defects in charge order.
