Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice

Abstract

At low temperatures, a spin ice enters a Coulomb phase—a state with algebraic correlations and topologically constrained spin configurations. We show how analytical and numerical approaches for model spin-ice systems reveal a crossover between two Curie laws. One of these laws characterizes the high-temperature paramagnetic regime, while the other, which we call the “spin-liquid Curie law,” characterizes the low-temperature Coulomb-phase regime, which provides implicit evidence that the topological sector fluctuates. We compare our theory with experiment for Ho2Ti2O7 , where this process leads to a nonstandard temperature evolution of the bulk susceptibility and the wave-vector-dependent magnetic susceptibility, as measured by neutron scattering. Theory and experiment agree for bulk quantities and at large scattering wave vectors, but differences at small wave vectors indicate that the classical spin-ice states are not equally populated at low temperatures. More generally, the crossover appears to be a generic property of the emergent gauge field for a classical spin liquid, and it sheds light on the experimental difficulty of measuring a precise Curie-Weiss temperature in frustrated materials. The susceptibility at finite wave vectors is shown to be a local probe of fluctuations among topological sectors on varying length scales.

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Plain Language Summary

In many situations in life, frustration is counterproductive and best avoided. In physics, however, frustration—the presence of competing forces that cannot be simultaneously satisfied—is often the source of exotic physical properties of materials systems. The exotic spin-liquid behavior in frustrated magnets provides just such an example. Defying the conventional expectation that at low temperatures individual microscopic spins in a material should become ordered in their orientations, spins in a frustrated magnet are thought to show no apparent orientational order. Rather, their orientations fluctuate among a large number of configurations where the fluctuations, though order-destroying, are actually highly correlated. Adding more kindling to the fire of interest in this so-called spin-liquid state are the tantalizing possibilities that spin liquids can host a range of very exotic phenomena, not least, topological phases and artificial gauge fields. In this article, we report a combined theoretical and experimental study of a frustrated magnet (Ho2Ti2O7 ) that points to the presence of a low-temperature spin-liquid state in which the correlations in the spin fluctuations are topological in nature.

Ho2Ti2O7 is a well-known frustrated magnet. In this material, two competing forces frustrate each other: While the physical interaction between the individual microscopic spins requires the spins to align in the same direction, the geometry of the crystalline lattice the spins are located in makes it impossible for this requirement to be met for all spins. The result is a local spin-lattice structure that resembles the local coordinated oxygen framework in water, ice, and a large number of low-energy, but energetically equivalent states of globally correlated spin configurations. These configurations can be classified into different sectors by a topological invariant. We are able to show that experimental techniques such as neutron scattering and magnetic susceptibility measurements, which probe spin-spin correlations, can be used to gauge the topological properties. Indeed, we have succeeded in identifying an abnormal temperature dependence in the spin-spin correlations as the emergence of correlated fluctuations among the many topological sectors at very low temperatures, indicative of a spin-liquid state characterized by topological fluctuations.

We believe that our findings are general, in that they should apply to a vast family of models and materials for frustrated magnets.