Spin Ice Materials Research Articles

Multiferroicity in spin ice: Towards magnetic crystallography ofTb2Ti2O7in a field

Recently D. I. Khomskii proposed that the elementary excitations in spin ice materials, the magnetic monopoles, also possess electric dipole moments. In this manuscript the properties of such magneto-electric monopoles are explored within a model of spin ice materials that includes nearest-neighbour coupling, Zeeman energy, electric and magnetic dipolar interactions, with a particular emphasis on a prototype spin-liquid pyrochlore Tb${}_{2}$Ti${}_{2}$O${}_{7}$.

Ground State Selection and Spin-Liquid Behaviour in the Classical Heisenberg Model on the Breathing Pyrochlore Lattice

Magnetic pyrochlore oxides, including the spin ice materials, have proved to be a rich field for the study of geometrical frustration in 3 dimensions. Recently, a new family of magnetic oxides has been synthesised in which the half of the tetrahedra in the pyrochlore lattice are inflated relative to the other half, making an alternating array of small and large tetrahedra. These "breathing pyrochlore" materials such as LiGaCr4O8, LiInCr4O8 and Ba3Yb2Zn5O11 provide new opportunities in the study of frustrated magnetism. Here we provide an analytic theory for the ground state phase diagram and spin correlations for the minimal model of magnetism in breathing pyrochlores: a classical nearest neighbour Heisenberg model with different exchange coefficients for the two species of tetrahedra. We find that the phase diagram comprises a Coulombic spin liquid phase, a conventional ferromagnetic phase and an unusual antiferromagnetic phase with lines of soft modes in reciprocal space, stabilised by an order-by-disorder mechanism. We obtain a theory of the spin correlations in this model using the Self Consistent Gaussian Approximation (SCGA) which enables us to discuss the development of correlations in breathing pyrochlores as a function of temperature, and we quantitatively characterise the thermal crossover from the limit of isolated tetrahedra to the strongly correlated limit of the problem. We compare the results of our analysis with the results of recent neutron scattering experiments on LiInCr4O8.

Weyl spin liquids.

The fractionalization of quantum numbers in interacting quantum many-body systems is a central motif in condensed-matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here, we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a Z_{2} gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synthesized hyperhoneycomb structure of β-Li_{2}IrO_{3} when applying a magnetic field. We characterize the topologically protected bulk and surface features of this state, which we dub a Weyl spin liquid, including thermodynamic and transport signatures.

Nonmonotonic residual entropy in diluted spin ice: A comparison between Monte Carlo simulations of diluted dipolar spin ice models and experimental results

Spin ice materials, such as Dy2Ti2O7 and Ho2Ti2O7, have been the subject of much interest for over the past fifteen years. Their low temperature strongly correlated state can be mapped onto the proton disordered state of common water ice and, consequently, spin ices display the same low temperature residual Pauling entropy as water ice. Interestingly, it was found in a previous study [X. Ke {\it et. al.} Phys. Rev. Lett. {\bf 99}, 137203 (2007)] that, upon dilution of the magnetic rare-earth ions (Dy^{3+} and Ho^{3+}) by non-magnetic Yttrium (Y^{3+}) ions, the residual entropy depends {\it non-monotonically} on the concentration of Y^{3+} ions. In the present work, we report results from Monte Carlo simulations of site-diluted microscopic dipolar spin ice models (DSIM) that account quantitatively for the experimental specific heat measurements, and thus also for the residual entropy, as a function of dilution, for both Dy2Ti2O7 and Ho2Ti2O7. The main features of the dilution physics displayed by the magnetic specific heat data are quantitatively captured by the diluted DSIM up to, and including, 85% of the magnetic ions diluted (x=1.7). The previously reported departures in the residual entropy between Dy2Ti2O7 versus Ho2Ti2O7, as well as with a site-dilution variant of Pauling's approximation, are thus rationalized through the site-diluted DSIM. For 90% (x=1.8) and 95% (x=1.9) of the magnetic ions diluted, we find a significant discrepancy between the experimental and Monte Carlo specific heat results. We discuss some possible reasons for this disagreement.

Heat transport of the spin-ice materials [formula omitted] and [formula omitted

The elementary excitations of the spin-ice materials Ho2Ti2O7 and Dy2Ti2O7 in zero field can be described as independent magnetic monopoles. We investigate the influence of these exotic excitations on the heat transport by measuring the magnetic-field dependent thermal conductivity κ. Additional measurements on the highly dilute reference compounds HoYTi2O7 and DyYTi2O7 enable us to separate κ into a sum of phononic (κph) and magnetic (κmag) contributions. For both spin-ice materials, we derive significant zero-field contributions κmag, which are rapidly suppressed in finite magnetic fields. Moreover, κmag sensitively depends on the scattering of phonons by magnetic excitations, which is rather different for the Ho- and the Dy-based materials and, as a further consequence, the respective magnetic-field dependent changes κph(B) are even of opposite signs.

Dumping topological charges on neighbors: ice manifolds for colloids and vortices

We investigate the recently reported analogies between pinned vortices in nano-structured superconductors or colloids in optical traps, and spin ice materials. It has been found experimentally and numerically that both colloids and vortices exhibit ice or quasi-ice manifolds. However, the frustration of colloids and vortices differs essentially from spin ice at the vertex level. We show that the effective vertex energetics of the colloidal/vortex systems is made identical to that of spin ice materials by the contribution of an emergent field associated to the topological charge of the vertex. The similarity extends to the local low-energy dynamics of the ice manifold, where the effect of geometric hard constraints can be subsumed into the spatial modulation of the emergent field, which mediates an entropic interaction between topological charges. There, as in spin ice materials, genuine ice manifolds enter a Coulomb phase, whereas quasi-ice manifolds posses a well defined screening length, provided by a plasma of embedded topological charges. We also show that such similarities break down in lattices of mixed coordination because of topological charge transfer between sub-latices. This opens interesting perspective for extensions beyond physics, to social and economical networks.

Low-energy electrodynamics of novel spin excitations in the quantum spin ice Yb2Ti2O7

In condensed matter systems, formation of long-range order (LRO) is often accompanied by new excitations. However, in many geometrically frustrated magnetic systems, conventional LRO is suppressed, while non-trivial spin correlations are still observed. A natural question to ask is then what is the nature of the excitations in this highly correlated state without broken symmetry? Frequently, applying a symmetry breaking field stabilizes excitations whose properties reflect certain aspects of the anomalous state without LRO. Here we report a THz spectroscopy study of novel excitations in quantum spin ice Yb2Ti2O7 under a <001> directed magnetic field. At large positive fields, both right- and left-handed magnon and two-magnon-like excitations are observed. The g-factors of these excitations are dramatically enhanced in the low-field limit, showing a crossover of these states into features consistent with the quantum string-like excitations proposed to exist in quantum spin ice in small <001> fields.

ChemInform Abstract: Quantum Spin Ice: A Search for Gapless Quantum Spin Liquids in Pyrochlore Magnets

AbstractReview: 203 refs.

Ultrasonic investigations of the spin icesDy2Ti2O7andHo2Ti2O7in and out of equilibrium

We report ultrasound studies of spin-lattice and single-ion effects in the spin-ice materials Dy$_2$Ti$_2$O$_7$ (DTO) and Ho$_2$Ti$_2$O$_7$ (HTO) across a broad field range up to 60 T, covering phase transformations, interactions with low-energy magnetic excitations, as well as single-ion effects. In particular, a sharp dip observed in the sound attenuation in DTO at the gas-liquid transition of the magnetic monopoles is explained based on an approach involving negative relaxation processes. Furthermore, quasi-periodic peaks in the acoustic properties of DTO due to non-equilibrium processes are found to be strongly affected by {\em macroscopic} thermal-coupling conditions: the thermal runaway observed in previous studies in DTO can be suppressed altogether by immersing the sample in liquid helium. Crystal-electric-field effects having higher energy scale lead to a renormalization of the sound velocity and sound attenuation at very high magnetic fields. We analyze our observations using an approach based on an analysis of exchange-striction couplings and single-ion effects.

Thermal fluctuations in artificial spin ice.

Artificial spin ice systems have been proposed as a playground for the study of monopole-like magnetic excitations, similar to those observed in pyrochlore spin ice materials. Currents of magnetic monopole excitations have been observed, demonstrating the possibility for the realization of magnetic-charge-based circuitry. Artificial spin ice systems that support thermal fluctuations can serve as an ideal setting for observing dynamical effects such as monopole propagation and as a potential medium for magnetricity investigations. Here, we report on the transition from a frozen to a dynamic state in artificial spin ice with a square lattice. Magnetic imaging is used to determine the magnetic state of the islands in thermal equilibrium. The temperature-induced onset of magnetic fluctuations and excitation populations are shown to depend on the lattice spacing and related interaction strength between islands. The excitations are described by Boltzmann distributions with their factors in the frozen state relating to the blocking temperatures of the array. Our results provide insight into the design of thermal artificial spin ice arrays where the magnetic charge density and response to external fields can be studied in thermal equilibrium.

Vacancy defects and monopole dynamics in oxygen-deficient pyrochlores.

The idea of magnetic monopoles in spin ice has enjoyed much success at intermediate temperatures, but at low temperatures a description in terms of monopole dynamics alone is insufficient. Recently, numerical simulations were used to argue that magnetic impurities account for this discrepancy by introducing a magnetic equivalent of residual resistance in the system. Here we propose that oxygen deficiency is the leading cause of magnetic impurities in as-grown samples, and we determine the defect structure and magnetism in Y2Ti2O7-δ using diffuse neutron scattering and magnetization measurements. These defects are eliminated by oxygen annealing. The introduction of oxygen vacancies causes Ti(4+) to transform to magnetic Ti(3+) with quenched orbital magnetism, but the concentration is anomalously low. In the spin-ice material Dy2Ti2O7 we find that the same oxygen-vacancy defects suppress moments on neighbouring rare-earth sites, and that these magnetic distortions markedly slow down the long-time monopole dynamics at sub-Kelvin temperatures.

Dy2−x Y x Ti2O7: phonon vibration and magnetization with dilution

In this paper, the dilution effects of non-magnetic Y ions on spin-ice compound Dy2Ti2O7 by infrared and Raman spectra and magnetization measurements were investigated. An anomalous phonon softening with temperature decreasing is found in both the parent and diluted compounds, and Y doping can relax the softening of phonons except that of the IR mode near 233 cm−1, indicating a strong phonon–phonon coupling in the spin-ice material. The magnetization measurements reveal that the non-magnetic impurities do not severely influence the spin-ice rules in the ground state when the level of dilution is not very high. However, a large amount of dilution enhance the disorder and break the spin-ice state because the collective spin-flip clusters are no longer available.

Realizing three-dimensional artificial spin ice by stacking planar nano-arrays

Artificial spin ice is a frustrated magnetic two-dimensional nano-material, recently employed to study variety of tailor-designed unusual collective behaviours. Recently proposed extensions to three dimensions are based on self-assembly techniques and allow little control over geometry and disorder. We present a viable design for the realization of a three-dimensional artificial spin ice with the same level of precision and control allowed by lithographic nano-fabrication of the popular two-dimensional case. Our geometry is based on layering already available two-dimensional artificial spin ice and leads to an arrangement of ice-rule-frustrated units which is topologically equivalent to that of the tetrahedra in a pyrochlore lattice. Consequently, we show, it exhibits a genuine ice phase and its excitations are, as in natural spin ice materials, magnetic monopoles interacting via Coulomb law.

Colloquium: Artificial spin ice: Designing and imaging magnetic frustration

Frustration in the presence of competing interactions is ubiquitous in the physical sciences and is a source of degeneracy and disorder, giving rise to new and interesting physical phenomena. Perhaps nowhere does it occur more simply than in correlated spin systems, where it has been studied in the most detail. In the past few years, a new perspective has opened in the study of frustration through the creation of artificial frustrated magnetic systems. These materials consist of arrays of lithographically fabricated single-domain ferromagnetic nanostructures that behave like giant Ising spins. The nanostructures' interactions can be controlled through appropriate choices of their geometric properties and arrangement on a (frustrated) lattice. The degrees of freedom of the material can not only be directly tuned, but also individually observed. Experimental studies have unearthed intriguing connections to the out-of-equilibrium physics of disordered systems and non-thermal granular materials, while revealing strong analogies to (spin) ice materials and their fractionalized magnetic monopole excitations, lending the enterprise a distinctly interdisciplinary flavor. The experimental results have also been closely coupled to theoretical and computational analyses, facilitated by connections to classic models of frustrated magnetism, whose hitherto unobserved aspects have here found an experimental realization. We review the considerable experimental and theoretical progress in this field, including connections to other frustrated phenomena, and we outline future vistas for progress in this rapidly expanding field.

Crystal shape-dependent magnetic susceptibility and Curie law crossover in the spin ices Dy2Ti2O7 and Ho2Ti2O7

We present an experimental determination of the isothermal magnetic susceptibility of the spin ice materials Dy2Ti2O7 and Ho2Ti2O7 in the temperature range 1.8–300 K. The use of spherical crystals has allowed accurate correction for demagnetizing fields and allowed the true bulk isothermal susceptibility χT(T) to be estimated. This has been compared against a theoretical expression based on a Husimi tree approximation to the spin ice model. Agreement between experiment and theory is excellent at T > 10 K, but systematic deviations occur below that temperature. Our results largely resolve an apparent disagreement between neutron scattering and bulk measurements that has been previously noted. They also show that the use of non-spherical crystals in magnetization studies of spin ice may introduce very significant systematic errors, although we note some interesting—and possibly new—systematics concerning the demagnetizing factor in cuboidal samples. Finally, our results show how experimental susceptibility measurements on spin ices may be used to extract the characteristic energy scale of the system and the corresponding chemical potential for emergent magnetic monopoles.

Anisotropic heat transport via monopoles in the spin-ice compound Dy2Ti2O7

We report a study of the thermal conductivity $\kappa$ of the spin-ice material Dy$_2$Ti$_2$O$_7$. From the anisotropic magnetic-field dependence of \kappa$ and by additional measurements on the phononic reference compounds Y$_2$Ti$_2$O$_7$ and DyYTi$_2$O$_7$, we are able to separate the phononic and the magnetic contributions to the total heat transport, i.e. $\kappa_{ph}$ and $\kappa_{mag}$, respectively, which both depend on the magnetic field. The field dependent $\kappa_{ph}$ arises from lattice distortions due to magnetic-field induced torques on the non-collinear magnetic moments of the Dy ions. For $\kappa_{mag}$, we observe a highly anisotropic magnetic-field dependence, which correlates with the corresponding magnetization data reflecting the different magnetic-field induced spin-ice ground states. The magnitude of $\kappa_{mag}$ increases with the degree of the ground-state degeneracy. This anisotropic field dependence as well as various hysteresis effects suggest that $\kappa_{mag}$ is essentially determined by the mobility of the magnetic monopole excitations in spin ice.

Thermal Conductivity ofHo2Ti2O7along the [111] Direction

Thermal transport measurements have been made on the spin-ice material Ho(2)Ti(2)O(7) in an applied magnetic field with both the heat current and the field parallel to the [111] direction for temperatures from 50 mK to 1.2 K. A large magnetic field >6 T is applied to suppress the magnetic contribution to the thermal conductivity in order to extract the lattice conductivity. The low field thermal conductivity thus reveals a magnetic field dependent contribution to the conductivity which both transfers heat and scatters phonons. We interpret these magnetic excitations as monopolelike excitations and describe their behavior via existing Debye-Hückel theory.

Spin liquid regimes at nonzero temperature in quantum spin ice

Quantum spin liquids are highly entangled ground states of quantum systems with an emergent gauge structure, fractionalized spinon excitations, and other unusual properties. While these features clearly distinguish quantum spin liquids from conventional, mean-field-like states at zero temperature ($T$), their status at $T&gt;0$ is less clear. Strictly speaking, it is known that most quantum spin liquids lose their identity at nonzero temperature, being then adiabatically transformable into a trivial paramagnet. This is the case for the $U(1)$ quantum spin liquid states recently proposed to occur in the quantum spin ice pyrochlores. Here we propose, however, that in practical terms, the latter quantum spin liquids can be regarded as phases distinct from the high-temperature paramagnet. Through a combination of gauge mean-field theory calculations and physical reasoning, we argue that these systems sustain both quantum spin liquid and thermal spin liquid phases, dominated by quantum fluctuations and entropy, respectively. These phases are separated by a first-order ``thermal confinement'' transition such that, for temperatures below the transition, spinons and emergent photons are coherently propagating excitations, and above it the dynamics is classical. Even for parameters for which the ground state is magnetically ordered and not a quantum spin liquid, this strong first-order transition occurs, preempting conventional Landau-type criticality. We argue that this picture explains the anomalously low-temperature phase transition observed in the quantum spin ice material Yb${}_{2}$Ti${}_{2}$O${}_{7}$.

Brownian motion and quantum dynamics of magnetic monopoles in spin ice

Spin ice illustrates many unusual magnetic properties, including zero point entropy, emergent monopoles and a quasi liquid–gas transition. To reveal the quantum spin dynamics that underpin these phenomena is an experimental challenge. Here we show how crucial information is contained in the frequency dependence of the magnetic susceptibility and in its high frequency or adiabatic limit. The typical response of Dy2Ti2O7 spin ice indicates that monopole diffusion is Brownian but is underpinned by spin tunnelling and is influenced by collective monopole interactions. The adiabatic response reveals evidence of driven monopole plasma oscillations in weak applied field, and unconventional critical behaviour in strong applied field. Our results clarify the origin of the relatively high frequency response in spin ice. They disclose unexpected physics and establish adiabatic susceptibility as a revealing characteristic of exotic spin systems.

Ice rule correlations in stuffed spin ice

Stuffed spin ice is a chemical variation of a spin ice material like Ho2Ti2O7 in which extra magnetic ions are inserted into the crystal structure. Previous studies have shown that the degree of stuffing has very little effect on the residual entropy in the system, which takes a value very close to the spin ice entropy. We argue, however, that the observation of this entropy does not imply long range coherence of the ice rules, that determine the local spin configurations. We have characterized deviations from the ice rules by means of a polarized neutron diffraction study of a single crystal of Ho2+δTi2−δO7−δ/2 with δ = 0.3. Our results demonstrate that the ice rules in stuffed spin ice are strictly valid only over a relatively short range, and that at longer range stuffed spin ice exhibits some characteristics of a ‘cluster glass’, with a tendency to more conventional ferromagnetic correlations.