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Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7

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Science  16 Oct 2009:
Vol. 326, Issue 5951, pp. 411-414
DOI: 10.1126/science.1178868

Magnetic Monopoles

Magnets come with a north and a south pole. Despite being predicted to exist, searches in astronomy and in high-energy particle physics experiments for magnetic monopoles (either north or south on their own) have defied observation. Theoretical work in condensed-matter systems has predicted that spin-ice structures may harbor such elusive particles (see the Perspective by Gingras). Fennell et al. (p. 415, published online 3 September) and Morris et al. (p. 411, published online 3 September) used polarized neutron scattering to probe the spin structure forming in two spin-ice compounds—Ho2Ti2O7 and Dy2Ti2O7—and present results in support of the presence of magnetic monopoles in both materials.

Abstract

Sources of magnetic fields—magnetic monopoles—have so far proven elusive as elementary particles. Condensed-matter physicists have recently proposed several scenarios of emergent quasiparticles resembling monopoles. A particularly simple proposition pertains to spin ice on the highly frustrated pyrochlore lattice. The spin-ice state is argued to be well described by networks of aligned dipoles resembling solenoidal tubes—classical, and observable, versions of a Dirac string. Where these tubes end, the resulting defects look like magnetic monopoles. We demonstrated, by diffuse neutron scattering, the presence of such strings in the spin ice dysprosium titanate (Dy2Ti2O7). This is achieved by applying a symmetry-breaking magnetic field with which we can manipulate the density and orientation of the strings. In turn, heat capacity is described by a gas of magnetic monopoles interacting via a magnetic Coulomb interaction.

Despite searching within the cosmic radiation, particle colliders, and lunar dust, free magnetic monopoles have not been observed (1, 2). This is particularly disappointing given that unification theories have predicted their existence. Dirac’s original vision for monopoles involves a string singularity carrying magnetic flux, the ends of which act as north and south monopoles. We report the observation of analogous strings and magnetic monopoles in spin ice, magnetic compound Dy2Ti2O7 with a pyrochlore lattice structure. This is a realization of magnetic fractionalization in three dimensions, a separation of north and south monopoles.

Dysprosium titanate contains magnetic 162Dy ions in the highly frustrated pyrochlore lattice that have ferromagnetic exchange and dipolar interactions between the spins. The pyrochlore lattice is a three-dimensional (3D) structure built from corner-sharing tetrahedra (Fig. 1A). Spin ice is realized on this lattice when spins placed on the vertices are constrained to point radially into or out of the tetrahedra and are coupled ferromagnetically—or, as in the case of Dy2Ti2O7, through dipolar coupling (3). This leads to the lowest-energy spin configurations obeying the “ice rules” of two spins pointing into, and two out of, each tetrahedron. This is equivalent to the physics of the proton arrangement in ice, where two protons sit close to each oxygen and two far away—and indeed spin ice exhibits the Pauling ice entropy S ≈ (R/2) ln(3/2) per spin (4, 5), reflecting a huge low-energy density of states in zero magnetic field.

Fig. 1

Gas of deconfined magnetic monopoles. (A) The Ising spins are constrained to point along the direction connecting the centers of the two tetrahedra they belong to. The lowest energy for a tetrahedron is obtained for a two-in-two-out configuration, as illustrated. There are six such configurations with net ferromagnetic moments along one of the six equivalent 〈100〉 directions. The noncollinearity of the Ising axes is the source of the frustration in spin ice. In Dy2Ti2O7 the “Ising” crystal field doublet is separated from other levels by more than 100 K. Applying a field, B || [001], results in a preference for aligning the tetrahedral magnetization with the applied field direction (arrow). In the 3D pyrochlore lattice, Dirac strings of flipped spins terminate on tetrahedra where magnetic monopoles reside. (B) The measured heat capacity per mole of Dy2Ti2O7 at zero field (open squares) is compared with a Debye-Hückel theory for the monopoles (blue line) and the best fit to a single-tetrahedron (Bethe lattice) approximation (red line). The ice-blue background indicates the spin-ice regime; the yellow background indicates the paramagnetic regime.

Each spin can be thought of as a small dipole or solenoid channeling magnetic flux into and out of a tetrahedron. The ice rules are too weak to impose magnetic long-range order, but they do induce dipolar power-law correlations resulting in characteristic pinch-point features in neutron scattering (610).

A spin flip violates the ice rule in two tetrahedra, at a cost of ~2 K per tetrahedron in Dy2Ti2O7. It was proposed that to a good approximation this can be viewed as the formation of a pair of monopoles of opposite sign in adjacent tetrahedra (11). These monopoles are deconfined (Fig. 1A); they can separate and move essentially independently. Thus, the equilibrium defect density is determined not by the cost of a spin flip but by the properties of the gas of interacting monopoles. In Fig. 1B, we compare the measured heat capacity to Debye-Hückel theory (12), which describes a gas of monopoles with Coulomb interactions. This theory is appropriate to low temperatures, where the monopoles are sparse, and it captures the heat capacity quantitatively. At higher temperatures, spin ice turns into a more conventional paramagnet and the monopole description breaks down (13). Together with a recent analysis of dynamic susceptibility (14), this lends strong support to the monopole picture of the low-temperature phase of spin ice.

Monopole deconfinement is reflected in the spin configurations: As the two monopoles of opposite sign separate, they leave a tensionless string of reversed spins connecting them. These strings of reversed flux between the monopoles can be viewed as a classical analog of a Dirac string. In the theory of Dirac (15), these are infinitely narrow, unobservable solenoidal tubes carrying magnetic flux density (B-field) emanating from the monopoles. Here, the strings are real and observable thanks to the preformed dipoles of the spins; strings can change length and shape at no cost in energy other than the magnetic Coulomb interaction between their endpoints.

Because the strings consist of magnetic dipoles, the method of choice for imaging them is magnetic neutron scattering. As a first step, using diffuse neutron-scattering techniques, we measured 3D correlation functions in Dy2Ti2O7. Figure 2A shows the results with no applied magnetic field. One important property of the correlation functions in zero field is the existence of pinch points, a signature of the spin-ice state, which can be seen in the experiment as 3D singularities (fig. S1). For comparison with the neutron-scattering measurements, we performed a large-N (self-consistent mean-field) calculation (6, 16), supplemented with the relevant geometric factors for neutron-scattering experiments and the magnetic form factor of Dy (Fig. 2B). Good agreement between theory and experiment [see also (17)] demonstrates that the correlations do indeed follow the predicted dipolar form, even though direct observation of the pinch points is not possible because they are covered by Bragg peaks.

Fig. 2

3D rendering of the dipolar correlations in reciprocal space (hkl) of spin ice at 0.7 K. (A) Neutron diffraction data taken at 0.7 K and 0.0 T on the flat-cone diffractometer E2, at HZB, with the diffuse peaks at (030) and minima in (3/2 5/2 0) positions (r.l.u., reciprocal lattice units). Bragg peaks (red spots) lie on top of the pinch points. A secondary Bragg peak is from a smaller crystallite. (B) Pinch points are found in the correlation functions and these are 3D in nature, with the diffuse scattering constricting at the reciprocal lattice point (020) and its equivalents. Diffuse peak and scattering minima positions agree with the data.

In the Dirac string picture, the spin-ice ground state satisfying the ice rules can be considered as a dense network of interwoven strings that are either closed (i.e., loops) or terminate at the surface of the sample. If the applied field is zero, the strings have no privileged orientation and they describe isotropic, intertwined 3D random walks of arbitrary length. Excitations correspond to monopoles at the end of Dirac strings (broken loops) in the bulk. The cost of lengthening such Dirac strings is solely against the weak attractive force between the monopoles at their ends (11). Indeed, as the strings fluctuate between different configurations consistent with a given distribution of monopoles, there is not even a unique way of tracing their paths.

However, there exists an elegant remedy: The application of a large magnetic field along one of the principal axes (here we choose the [001] direction) orients all spins. The resulting ground state is unique and free of monopoles; the ice rules are observed everywhere, and each tetrahedron is magnetized in the [001] direction. Upon lowering the field, sparse strings of flipped spins appear against the background of this fully magnetized ground state. In the absence of monopoles, such strings must span the length of the sample and terminate on the surface; otherwise, they can terminate on magnetic monopoles in the bulk (as explained above).

The presence or absence of the strings at a given temperature and field is determined by a balance between the energy cost of producing them and the gain in entropy due to their presence. As pointed out in (18, 19), each link in the string will involve a spin being reversed against the field (note that this still maintains the two-in-two-out ice rules along the string). Each spin flip costs a Zeeman energy of Embedded Image, where h = gmB is the strength of the field applied along [001], g is the Landé g-factor, m = 10μB is the magnetic moment per Dy ion, and B is the applied field. As there are two possibilities to choose for the continuation of the string, there is an associated entropy per link of s = kB ln(2), where kB is the Boltzmann constant. The free energy per link, as the string becomes large, isEmbedded Image(1)where u is the Zeeman energy and T is absolute temperature. For fields above Embedded Image, the number of strings goes to zero as the free energy of formation is macroscopic and positive. However, at the Kasteleyn field hK, a transition occurs where strings spontaneously form as the free energy becomes favorable and the entropy of string formation wins. This transition is a 3D example of a Kasteleyn transition (18), a highly unusual topological phase transition.

Thus, by measuring close to this transition, we can dial up a regime where the strings are sparse and oriented against the field direction. This allows us to check, qualitatively and quantitatively, the properties of these strings; in particular, we find that they lead to a qualitative signature in the neutron scattering, which can be manipulated by tilting the field. In the following, we first locate the Kasteleyn transition and hence hK, and then discuss the neutron-scattering data in detail by comparing them to a theoretical model. Evidence of this transition can be seen in the magnetization as a function of temperature and field along [001] (Fig. 3, A and B). Whereas the magnetization in response to field changes at constant temperature only equilibrates above 0.6 K, spin ice can remain closer to its thermodynamic equilibrium behavior when cooled in zero magnetic field. Above this temperature, the system shows a transition to saturation at a field hS(T) consistent with that expected for the 3D Kasteleyn transition. Indeed, in the ergodic region of the phase diagram the saturation field hS coincides with the Kasteleyn field hK, where a kink in the magnetization appears as it reaches its saturation value (18). However, below ~0.6 K, equilibration times become so long that the system starts freezing (18), and the measured magnetization is no longer an equilibrium property. Another signature of this freezing (Fig. 3B) is that the saturation field hS (dotted white line) becomes temperature-independent.

Fig. 3

Magnetization and diffuse neutron scattering with field applied along [001]. (A) Magnetization plotted versus [001] field over temperature for fixed temperatures of 0.4, 0.5, 0.6, 0.7, 0.85, 0.9, and 1.0 K, showing a clear departure from h/T scaling at 0.6 K and below. Inset: Surface of magnetization as a function of temperature and field as constructed from more than 70 field and temperature sweeps. (B) Contour plot of the magnetization as a function of field and temperature. The dotted white line shows hS, which is seen to freeze below 0.6 K. (C) 3D representation of the single-crystal neutron diffraction data from E2, HZB, at 5/7hS and 0.7 K, showing a cone of scattering coming from the (020) Bragg peak. (D) Calculation of diffuse scattering characteristic of the weakly biased random-walk correlations with bias of 0.53:0.47 and Bint || [001].

Here, we restrict ourselves to equilibrium phenomena: All the neutron measurements in this study were undertaken above 0.6 K. These experimental temperatures are high enough, relative to the creation energy of monopoles, that the assumption of perfect compliance to the ice rules is no longer valid and the transition is rounded: The (low) thermal density of monopoles leads to strings of finite length.

Figure 3C shows reciprocal space slices at a field near saturation of h = 5/7hS. Instead of the two lobes coming down to a pinch point in zero field, cone-like scattering emanates from what was the position of the pinch point. As the field is decreased, the diffuse scattering smoothly deforms back to the zero-field form.

As described above, the strings execute a random walk; when their density is small, interactions between them can be neglected to a first approximation, so that the spin correlations are those of a diffusion process with the z coordinate assuming the role usually played by time:

Embedded Image(2)

where γ is a geometric constant (13).

To capture lattice effects, we simulate random walks on the pyrochlore lattice. The correlations we find are essentially unchanged if we include interactions in the form of hard-core exclusion (13). Because there is a finite thermal population of monopoles and defects in the material, we expect the strings to be finite in length. For the field of h = 5/7hS and 0.7 K, a string length on the order of 50 sites is required for agreement with the data. Indeed, for this temperature in zero field, the density of monopoles in numerical simulations is found to be very low—well below 1% of all tetrahedra—in keeping with a large string length.

The scattering from a large ensemble of such hard-core walks has been calculated including all the geometrical factors for the neutron-scattering cross section. As can be seen from the side-by-side comparison of the data and modeling, the string configurations account very well for the data and reproduce the cone of scattering observed.

We repeated the experiment with an effective field tilted ~10° toward the [011] direction to induce a net tilt in the meandering of the strings. The cone of diffuse scattering collapses into sheets of scattering at an angle of 45°, matching the opening angle of the original cone. This sharp sheet in reciprocal space widens with decreasing field (Fig. 4C). Within the random-walk model, tilting the applied magnetic field changes the relative probabilities of each step (thus generating a biased random walk) as inequivalent spin flips incur different energy costs. The field and temperature-dependent Boltzmann factor for the ratio of probabilities of stepping from tetrahedral site 1 to site 3, versus site 1 to site 4, is Embedded Image(3)where θ1 and θ2 are the angles between the magnetic moments on the two final sites and the magnetic field h. Because of the ferromagnetism induced in spin ice, demagnetization effects must be carefully accounted for in the modeling. A bias of 0.8:0.2 at 4/7hS and 0.64:0.36 at 2/7hS is anticipated from Eq. 3. Using these weighting factors and modeling the new ensemble of Dirac strings, the tilts and widths of the scattering are well reproduced.

Fig. 4

Biased random walks in a tilted field. (A) Neutron diffraction data from E2, HZB, showing the (hk0), (hk1), and (0kl) planes taken at 0.7 K and a field of 4/7hS. The red spots are Bragg peaks; the peaks at (2.7, –1.8, 0), (1.3, –2.3, 1), (3.8, –0.9, 1), and (3.5, 2.5, 1) are from a smaller second crystallite. (B) Random-walk string model with biasing 0.8:0.2. (C) Field dependence of the diffuse scattering and calculations in the (1,k,l) plane. Spin-ice scattering collapses into walls of scattering at higher fields. Further understanding of the Dirac string and monopole physics may improve our modeling of the data. (D) Data versus calculation for the (h,2η+l,l) diffuse wall of scattering, where η is an integer (here η = 0).

There still remains the issue of interstring correlations. Comparison with the hard-core string model (Fig. 4, C and D) shows that the simple random-walk approach does not capture the intensity distribution so well. Figure 4D shows a cut through the walls of scattering in the (h,2η+l,l) plane, where η is an integer. The intensity within the sheets is indicative of correlations between strings. This may indicate short-range ordering of the strings, and the increased intensities around (2, 2/3, 2/3) would suggest a local hexagonal patterning. Further calculations are needed, including interactions between strings, to clarify this in detail.

Our study of the spin-ice state in zero field and under an applied magnetic field along [001] lends support to the strongly correlated and degenerate nature of the ground state, as well as the resulting long-range dipolar correlations. The low-energy excitations of such a complex ground state are remarkable in their simplicity and can largely be accounted for by weakly interacting point-like quasiparticles (the magnetic monopoles) connected by extended objects (the Dirac strings of reversed spins). In zero field, a description based on a gas of monopoles accounts for the measured low-temperature specific heat. Under fields applied along [001], the picture is that of Dirac strings of reversed spins meandering along the direction of the applied field and terminating on monopoles. This picture accounts very well for the spin correlations observed through neutron scattering. The behavior of the magnetization as a function of temperature and field near saturation, where all the strings are expelled, is an example of a (thermally rounded) 3D Kasteleyn transition. This description is rather robust and gives a simple picture of the spin configuration under tilted fields in terms of biases in the string direction. Our main result consists of the experimental identification of these string-like spin excitations in a gas of magnetic monopoles. These constitute hardy and practical building blocks for the understanding of the low-energy behavior of spin ice. Perhaps the most intriguing open issue is the precise connection between these building blocks and the low-temperature freezing observed in the spin-ice compounds (14, 20).

Our work constitutes direct evidence of Dirac strings. It provides compelling evidence for the dissociation of north and south poles—the splitting of the dipole—and the identification of spin ice as the first fractionalized magnet in three dimensions. The emergence of such striking states is profoundly important in physics, both as a manifestation of new and singular properties of matter and as a route to potential technologies. Examples of fractionalization are extremely rare and almost exclusively pertain to one and two dimensions, and so the 3D pyrochlore lattice offers a promising direction for future exploration in both magnets and exotic metals.

Our findings are of relevance not only from a fundamental physics aspect—we have evidenced a set of quasiparticles that have no elementary cousins—but also because they imply a new type of degree of freedom in magnetism, namely an object with both local (point-like monopole) and extended (tensionless Dirac string) properties. Dy2Ti2O7 is an exceptionally clean material, and with the full array of powerful experimental techniques and pulsed fields, equilibrium and non-equilibrium properties can be comprehensively addressed, although this will present a substantial statistical physics and dynamical systems challenge. The results of such studies may shed light on other systems where string-like objects can appear—for instance, in the study of polymers or nanoclusters—but where freezing of solvents and inhomogeneities can restrict access to all the physics. Spin ice promises to open up new and complementary insights on both the emergence of fractionalized states and the physics of ensembles of strings in and out of equilibrium.

Supporting Online Material

www.sciencemag.org/cgi/content/full/1178868/DC1

Materials and Methods

Figs. S1 to S13

Table S1

References

References and Notes

  1. See supporting material on Science Online.
  2. We thank S. L. Sondhi for help and encouragement, J. T. Chalker for insights into the expected behavior in field of the correlation functions, K. Siemensmeyer for help with sample cutting and preparation, and J. Heinrich for help with the COMSOL Multiphysics computer package used for the demagnetization calculations. Supported by the Royal Society (S.A.G.) and by Engineering and Physical Sciences Research Council (UK) grant GR/R83712/01 (C.Ca.).
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