Frustrated magnets are materials in which localized magnetic moments, or spins, interact through competing exchange interactions that cannot be simultaneously satisfied, giving rise to a large degeneracy of the system ground state. Under certain conditions, this can lead to the formation of fluid-like states of matter, so-called spin liquids, in which the constituent spins are highly correlated but still fluctuate strongly down to a temperature of absolute zero. The fluctuations of the spins in a spin liquid can be classical or quantum and show remarkable collective phenomena such as emergent gauge fields and fractional particle excitations. This exotic behaviour is now being uncovered in the laboratory, providing insight into the properties of spin liquids and challenges to the theoretical description of these materials.

Spin liquids in frustrated magnets

A frustrated system is one whose symmetry precludes the possibility that every pairwise interaction (“bond”) in the system can be satisfied at the same time. Such systems are common in all areas of physical and biological science. In the most extreme cases, they can have a disordered ground state with “macroscopic” degeneracy; that is, one that comprises a huge number of equivalent states of the same energy. Pauling9s description of the low-temperature proton disorder in water ice was perhaps the first recognition of this phenomenon and remains the paradigm. In recent years, a new class of magnetic substance has been characterized, in which the disorder of the magnetic moments at low temperatures is precisely analogous to the proton disorder in water ice. These substances, known as spin ice materials, are perhaps the “cleanest” examples of such highly frustrated systems yet discovered. They offer an unparalleled opportunity for the study of frustration in magnetic systems at both an experimental and a theoretical level. This article describes the essential physics of spin ice, as it is currently understood, and identifies new avenues for future research on related materials and models.

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Spin ice state in frustrated magnetic pyrochlore materials.

We are familiar with elementary particles that carry either negative or positive electric charge, such as electrons and protons, but there is no evidence of elementary particles with a net magnetic charge. Magnets tend to come with inseparable north and south poles, and there are no known magnetic monopoles despite concerted efforts to find them. But an intriguing theoretical study now proposes that magnetic monopoles may exist, not as elementary particles, but as emergent particles in exotic condensed matter magnetic systems such as 'spin ice'. The theory, based on an analogy to fractional electric charges seen, for example, in quantum Hall systems in two dimensions, can explain a mysterious phase transition that has been observed experimentally in spin ice. The cover, by Alessandro Canossa, depicts a magnetic monopole (red sphere) emerging from break-up of the dipole moment (arrows) of the underlying electronic degrees of freedom in spin ice. A theoretical study proposes that magnetic monopoles may appear not as elementary but as emergent particles in complex, strongly-correlated magnetic systems such as spin ice, in analogy to fractional electric charges in quantum Hall systems. This theory explains a mysterious phase transition in spin ice that has been observed experimentally. Electrically charged particles, such as the electron, are ubiquitous. In contrast, no elementary particles with a net magnetic charge have ever been observed, despite intensive and prolonged searches (see ref. 1 for example). We pursue an alternative strategy, namely that of realizing them not as elementary but rather as emergent particles—that is, as manifestations of the correlations present in a strongly interacting many-body system. The most prominent examples of emergent quasiparticles are the ones with fractional electric charge e/3 in quantum Hall physics2. Here we propose that magnetic monopoles emerge in a class of exotic magnets known collectively as spin ice3,4,5: the dipole moment of the underlying electronic degrees of freedom fractionalises into monopoles. This would account for a mysterious phase transition observed experimentally in spin ice in a magnetic field6,7, which is a liquid–gas transition of the magnetic monopoles. These monopoles can also be detected by other means, for example, in an experiment modelled after the Stanford magnetic monopole search8.

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Magnetic monopoles in spin ice

Within the past 20 years or so, there has occurred an explosion of interest in the magnetic behavior of pyrochlore oxides of the type $A_{2}^{3+}$$B_{2}^{4+}$O$_{7}$ where $A$ is a rare-earth ion and $B$ is usually a transition metal. Both the $A$ and $B$ sites form a network of corner-sharing tetrahedra which is the quintessential framework for a geometrically frustrated magnet. In these systems the natural tendency to form long range ordered ground states in accord with the Third Law is frustrated, resulting in some novel short range ordered alternatives such as spin glasses, spin ices and spin liquids and much new physics. This article attempts to review the myriad of properties found in pyrochlore oxides, mainly from a materials perspective, but with an appropriate theoretical context.

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Magnetic Pyrochlore Oxides

The study of paramagnetic metal-ion aggregates has been of increasing interest since the observation that such molecules can exhibit magnetic memory effects. Termed singlemolecule magnets or SMMs, the important factors leading to such properties derive from the combination of a large ground-state spin and a large magnetic anisotropy of the Ising (easy-axis) type. Studies have largely been based on transition-metal compounds since they typically exhibit both of the aforementioned features. The incorporation of lanthanides into these complexes has been investigated to take advantage of the potentially large number of unpaired f-electrons available. However, very little work has been done to date on purely lanthanide-based systems. The origin of SMM behavior in lanthanide-containing compounds is more complicated than that of d-block transition-metal ions since there is likely to be a significant orbital component. In the lanthanide-containing phthalocyanine complexes reported in the literature the ligand environment induces a large splitting of the ground Jmanifold, whereas in SMMs large-spin ground states arising from magnetic interactions between the metal centers of the cluster can enhance the weaker single-ion

Dysprosium triangles showing single-molecule magnet behavior of thermally excited spin states

Common water ice (ice I-h) is an unusual solid-the oxygen atoms form a periodic structure but the hydrogen atoms are highly disordered due to there being two inequivalent O-H bond lengths'. Pauling showed that the presence of these two bond lengths leads to a macroscopic degeneracy of possible ground states(2,3), such that the system has finite entropy as the temperature tends towards zero. The dynamics associated with this degeneracy are experimentally inaccessible, however, as ice melts and the hydrogen dynamics cannot be studied independently of oxygen motion(4). An analogous system(5) in which this degeneracy can be studied is a magnet with the pyrochlore structure-termed 'spin ice'-where spin orientation plays a similar role to that of the hydrogen position in ice I-h. Here we present specific-heat data for one such system, Dy2Ti2O7, from which we infer a total spin entropy of 0.67Rln2. This is similar to the value, 0.71Rln2, determined for ice I-h, SO confirming the validity of the correspondence. We also find, through application of a magnetic field, behaviour not accessible in water ice-restoration of much of the ground-state entropy and new transitions involving transverse spin degrees of freedom.

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Zero-point entropy in ‘spin ice’

Pyrochlore magnets are candidates for what Harris et al. [Phys. Rev. Lett. 79, 2554 (1997)] call ``spin-ice'' behavior. We present theoretical simulations of relevance for the pyrochlore family ${R}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ ( $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{rare}\mathrm{earth}$) supported by magnetothermal measurements on selected systems. By considering long-ranged dipole-dipole as well as short-ranged superexchange interactions, we get three distinct behaviors: (i) an ordered doubly degenerate state, (ii) a highly disordered state with a broad transition to paramagnetism, and (iii) a partially ordered state with a sharp transition to paramagnetism. Closely corresponding behavior is seen in the real compounds.

Ising Pyrochlore Magnets: Low-Temperature Properties, "Ice Rules," and Beyond

Neutron time-of-flight spectroscopy has been employed to study the crystal-field interaction in the pyrochlore titanate Ho2Ti2O7. The crystal-field parameters and corresponding energy-level scheme have been determined from a profile fit to the observed neutron spectra. The ground state is a well separated Eg doublet with a strong Ising-like anisotropy, which can give rise to frustration in the pyrochlore lattice. Using the crystal-field parameters determined for the Ho compound as an estimate of the crystal-field potential in other pyrochlore magnets, we also find the Ising type behavior for Dy. In contrast, the almost planar anisotropy found for Er and Yb prevents frustration, because of the continuous range of possible spin orientations in this case.

https://digital.library.unt.edu/ark:/67531/metadc711960/m2/1/high_res_d/750480.pdf

Crystal-field interaction in the pyrochlore magnet Ho2Ti2O7

Field-driven phase transitions generally arise from competition between Zeeman energy and exchange or crystal-field anisotropy. Here we present the phase diagram of a frustrated pyrochlore magnet Gd2Ti2O7, where crystal-field splitting is small compared to the dipolar energy. We find good agreement between zero-temperature critical fields and those obtained from a mean-field model. Here, dipolar interactions couple real space and spin space, so the transitions in Gd2Ti2O7 arise from field-induced "cooperative anisotropy," reflecting the broken spatial symmetries of the pyrochlore lattice.

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A. Hayashi

Papers

Zero-point entropy in ‘spin ice’

Ising Pyrochlore Magnets: Low-Temperature Properties, "Ice Rules," and Beyond

Crystal-field interaction in the pyrochlore magnet Ho2Ti2O7

Multiple field-induced phase transitions in the geometrically frustrated dipolar magnet: Gd2Ti2O7