International Workshop on Quantum Spin Ice

In this tutorial, we will review the microscopic aspects of rare-earth magnets relevant for quantum spin ice. We first discuss the single-ion properties of the variety of rare-earth atoms that appear in quantum spin ice candidate materials. Second, we consider the origin of the two-ion exchange interactions, including electric and magnetic multipolar interactions, super-exchange and virtual crystal field mediated interactions. We provide a detailed microscopic basis for the super-exchange interaction and discuss the implications of its multipolar structure for models of rare-earth materials. Finally, we introduce the generic symmetry allowed anisotropic exchange model for rare-earth pyrochlore magnets.

We aim to provide a concise review on theoretical background on emergent quantum electrodynamics in pyrochlore quantum spin ice. We first introduce elementary excitations in quantum spin ice using a simple model and then extend the discussion to more realistic systems. Implications to experiments are also discussed.

Harboring interpenetrating lattices of corner-sharing tetrahedra, materials with the pyrochlore structure type dominate exploration of the physics of quantum spin ices. Recent synthetic advances in the control of point defects, such as in Yb2Ti2O7 and Pr2Zr2O, have demonstrated that even sub-percent changes in the type and/or number of defects radically modulates the low temperature physics of these materials. This sensitivity to disorder is driven not only by the geometrically frustrated nature of the lattice, but also by the propensity of a corner sharing tetrahedral framework to undergo displacive motions, in a manner analogous to that of beta-crystabolite (quartz), either statically, or dynamically (as soft phonons). An outlook for continued improvements in our ability to control the chemistry behind quantum spin ices will also be presented.

The pyrochlore magnet Yb2Ti2O7 has the remarkable property that it orders magnetically, but has no propagating magnons over wide regions of reciprocal space. Using inelastic neutron scattering we observe that at high magnetic fields, in addition to dispersive magnons there is also a two-magnon continuum, which grows in intensity upon reducing field, overlaps with one-magnon states at intermediate fields leading to strong dispersion renormalizations and magnon decays. We re-evaluate the Hamiltonian finding dominant quantum exchange terms, which we propose are responsible for the anomalously strong quantum fluctuation effects observed at low fields.

High quality single crystals of Yb2Ti2O7 synthesized using a traveling solvent floating zone method enable experiments that explore the anisotropic temperature-field phase diagram of this quantum spin ice candidate. We have examined the H//(111) orientation, which is interesting because a phase transition that breaks three fold rotation symmetry is possible in the presence of a magnetic field. Using specific heat, magnetization, and neutron diffraction we find an apparent enhancement of the critical temperature with applied field that is not accounted for by a classical Monte Carlo simulation of the established spin hamiltonian. I shall discuss possible explanations for the discrepancy, which include quantum fluctuations and long range dipolar interactions.

The erbium and ytterbium rare earth pyrochlores exhibit local XY spin anisotropy. Experimental and theoretical investigations of XY pyrochlores have revealed a strong propensity for quantum magnetic phenomena, such as order-by-disorder and the quantum spin ice state. We have conducted a systematic investigation of the family of XY pyrochlores, Yb2B2O7 and Er2B2O7, spanning many non-magnetic B site cations (B = Ge, Ti, Pt, and Sn). We have characterized the magnetism of these XY pyrochlores using heat capacity, muon spin relaxation, neutron diffraction, and inelastic neutron scattering. A diversity of magnetic ground states and behaviours are represented among this family, ordered states ranging from ferromagnetic to antiferromagnetic, and in the case of one material, an absence of magnetic order to at least 100 mK. Moreover, we find that the magnetic ground state properties of these materials are strongly influenced by proximity to competing magnetic phases, consistent with theoretical predictions. We empirically demonstrate the signatures for phase competition in the frustrated XY pyrochlores: multiple heat capacity anomalies, suppressed TN or Tc, sample and pressure dependent ground states, and unconventional spin dynamics.

Yb2Ti2O7 is a geometrically frustrated magnet that proposed as a quantum spin liquid (QSL) candidate. This would have an emergent U(1) gauge structure, support emergent quasiparticles and a continuum of gapless spin excitations. A cubic power law dependence is expected in the specific heat down to zero temperature. [1,2] Identifying a power law is hindered by the presence of a sharp transition at 0.26K and a Schottky anomaly due to nuclear hyperfine interactions below 0.1K. [3] By preparing an isotopically enriched sample with 174-Yb and 48-Ti, we suppress the Schottky anomaly. This allows us to extend the specific heat to lower temperatures, revealing a polynomial behavior to at least 0.05 K that is suggestive of a quantum spin liquid.

Yb2Ti2O7 and Tb2Ti2O7 share the common point to sit at the edge between different phases. This multiphase competition makes the characterization of their low-temperature physics a challenge and, more generally, brings another degree of complexity to rare-earth pyrochlores. In this talk, we will take two different approaches to study these materials. For Yb2Ti2O7, we explicitly investigate the influence of thermal and quantum fluctuations in this competition, with finite-temperature order by disorder and quantum shifting of the phase boundaries. For Tb2Ti2O7, we will step away from the puzzling zero-field physics, and explain the undisputed order observed experimentally in a high [110] field. This order strongly supports the presence of magneto-electric coupling in Tb2Ti2O7, and offers a rare example of long-range order of magnetic monopoles.

Quantum spin ice, modeled for magnetic rare-earth pyrochlores, has attracted great interest for hosting a U(1) quantum spin liquid, which involves spin-ice monopoles as gapped deconfined spinons, as well as gapless excitations analogous to photons. However, the global phase diagram under a [111] magnetic field remains open. Here we uncover by means of unbiased quantum Monte-Carlo simulations that a supersolid of monopoles, showing both a superfluidity and a partial ionization, intervenes the kagome spin ice and a fully polarized ionic monopole insulator, in contrast to classical spin ice where a direct discontinuous phase transition takes place. We also show that on cooling, kagome spin ice evolves towards a valence bond solid. Possible relevance to experiments is discussed.

This talk will outline recent measurements on the dipolar-octupolar rare earth pyrochlores Nd2Zr2O7 and Nd2Hf2O7. Measurements of their crystal field excitations allows the wavefunction of their ground state Kramer’s doublet to be determined. Both compounds develop long-range magnetic order and their Hamiltonians are extracted by comparing inelastic neutron scattering data to spin-wave theory at low temperatures. The Hamiltonians are used to qualitatively explain AC magnetization measurements as well as neutron data collected in an applied magnetic field. Both system as predicted to lie close to a U(1) spin liquid and the excitation spectrum above the Néel temperature is compared to calculations for bosonic many body quantum spin ice.

By means of neutron scattering measurements, we have observed magnetic fragmentation in the pyrochlore oxide Nd2Zr2O7, in which the Nd3+ ion has a strong Ising character and ferromagnetic interactions are inferred from the positive Curie-Weiss temperature θCW = 195 mK [2]. In this system, an “all in - all out” ordered state, with a reduced magnetic moment, coexists below TN = 285 mK with a fluctuating state [3]. Experimentally, the fragmentation manifests itself via the superposition of magnetic Bragg peaks, characteristic of the antiferromagnetic ordered phase, and a pinch point pattern, characteristic of the Coulomb phase [4]. The finite energy of the pinch point pattern and the dispersive modes that emerge from it, points out the quantum origin of the fragmentation in Nd2Zr2O7 [5], which comes from the very peculiar “dipolar-octopolar” nature of the Kramers Nd3+ doublet.

The true magnetic ground state in thermally equilibrated classical spin ice compounds such as Dy2Ti2O7 remains an important but yet to be settled issue. In this talk, I will present our recent neutron scattering study of static and dynamic magnetic correlations in isotope-enriched 162Dy2Ti2O7 single-crystal samples. Implications within the context of possible quantum effects in dipolar spin ice based on our neutron results will be discussed as well.