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Spin Ice: Microscopic Physics

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Spin Ice

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 197))

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Abstract

This chapter describes the microscopic physics of the rare-earth pyrochlore oxides, focussing on the lattice structure, the strong spin-orbit coupling, the crystal electric field and the large magnetic moment. We explain each of these in turn and how they are connected before examining the types of interaction that may arise in rare-earth pyrochlores. Taking all these ingredients together, we arrive at a relatively straightforward effective model that is known to capture, in quantitative detail, the thermodynamics of the classical spin ice materials.

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Notes

  1. 1.

    Specifically, this is the second Bernal-Fowler ice rule which specifies that in common hexagonal water ice, two protons must be “near” and two protons must be “far” from each oxygen O\(^{2-}\) ion [20, 21]. The first Bernal-Fowler ice rule, which is not relevant to spin ice, states that there must be one and only one proton on each O\(^{2-}\)–O\(^{2-}\) bond.

  2. 2.

    Within the Pauling approximation, the entropy for water ice and spin ice are the same. The small corrections to Pauling’s value for water ice [24] and spin ice [25] have been computed. These corrections are not exactly the same for the two systems since water ice is hexagonal while spin ice is cubic.

  3. 3.

    Both Ho\(^{3+}\) and Dy\(^{3+}\) have more than a half-filled 4f shell and thus \(\mathsf{J}=\mathsf{L}+\mathsf{S}\) is maximized as opposed to being minimized (\(\mathsf{J}=\mathsf{L}-\mathsf{S}\)) for a less than half-filled shell [26].

  4. 4.

    As we are ultimately interested in the collective magnetic properties of these rare-earth pyrochlore materials, it is useful here to comment on some aspects of time-reversal symmetry of the crystal-field states for the rare-earth ions. For even electron systems, the action of the time reversal operator \(\Theta \) is \(\Theta ^2=1\) in contrast to the case of odd electron systems for which \(\Theta ^2=-1\). In general, for the amplitude corresponding to total angular momentum and its projection onto the \({\hat{{\mathbf {z}}}}\) axis \(\left| \mathsf{J},\mathsf{M} \right\rangle \), time reversal acts like \(\Theta c^{\mathsf{J}}_{\mathsf{M}} \left| \mathsf{J},\mathsf{M} \right\rangle = (-)^{\mathsf{J}+\mathsf{M}}(c^{\mathsf{J}}_{{\mathsf{M}}})^\star \left| \mathsf{J},-\mathsf{M} \right\rangle \). Here, \(c^{\mathsf{J}}_{\mathsf{M}}\) are expansion coefficients in the spectral decomposition of the crystal field doublet in terms of \(\left| \mathsf{J},\mathsf{M} \right\rangle \). For a non-Kramers doublet, \(\Theta | \pm \rangle =| \mp \rangle \). The crystal-field Hamiltonian can be chosen to have real matrix elements in the basis of \(\mathsf{J},\mathsf{M}\) and likewise for the eigenstates. Now, consider matrix elements of transverse components of the angular momentum between the two states of the crystal-field ground doublet.

    $$\begin{aligned} \left\langle +\right| \mathsf{J}^+ \left| - \right\rangle = \left\langle +\right| {\Theta }^{\dagger } \Theta \mathsf{J}^{+} {\Theta }^{\dagger } \Theta \left| - \right\rangle = - \left\langle -\right| \mathsf{J}^- \left| + \right\rangle . \end{aligned}$$
    (1.6)

    Furthermore, \(\langle +| \mathsf{J}^+ | -\rangle =\langle -| \mathsf{J}^- | +\rangle ^\star \). It follows that the matrix element vanishes. No such constraint holds for matrix elements between identical states of the doublet. However, we may show that

    $$\begin{aligned} \left\langle +\right| \mathsf{J}^\alpha \left| + \right\rangle = \left\langle +\right| {\Theta }^{\dagger } \Theta \mathsf{J}^\alpha {\Theta }^{\dagger } \Theta \left| + \right\rangle = - \left\langle -\right| \mathsf{J}^\alpha \left| - \right\rangle . \end{aligned}$$
    (1.7)

    The doublet corresponds to an Ising degree of freedom if these matrix elements are nonvanishing.

  5. 5.

    One also expects to find effective exchange terms coupling the transverse components of the pseudo-spin \({S} = 1/2\) describing the ground doublet in these compounds [63, 64] as well as in Tb\(_2\)Ti\(_2\)O\(_7\) [51] and in Yb\(_2\)Ti\(_2\)O\(_7\) [65, 66]. These introduce quantum fluctuations within the degenerate spin ice manifold and may give rise to a quantum spin ice state [67]. Three mechanisms generating effective transverse exchange and quantum dynamics have been discussed in the literature: (i) virtual crystal field fluctuations [51] (e.g. in Tb\(_2\)Ti\(_2\)O\(_7\)), (ii) sufficiently high-rank multipolar interactions in non-Kramers systems [63, 64] (e.g. in Pr\(_2\)(Sn,Zr)\(_2\)O\(_7\)) and (iii) multipolar, including dipolar, interactions in Kramers systems [65, 66, 68, 69] (e.g. in Yb\(_2\)Ti\(_2\)O\(_7\) and Er\(_2\)Ti\(_2\)O\(_7\)) This will be discussed in further detail in subsequent chapters (e.g. see Chap. 9 by Savary and Balents, Chap. 10 by Shannon and Chap. 12 by Ross).

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada

    M. J. P. Gingras

  2. Quantum Materials Program, Canadian Institute for Advanced Research, MaRS Centre, West Tower 661 University Ave., Suite 505, Toronto, ON, M5G 1M1, Canada

    M. J. P. Gingras

  3. Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

    P. A. McClarty

  4. Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4, Canada

    J. G. Rau

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    Masafumi Udagawa

  2. Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, CNRS, Bordeaux, France

    Ludovic Jaubert

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Gingras, M.J.P., McClarty, P.A., Rau, J.G. (2021). Spin Ice: Microscopic Physics. In: Udagawa, M., Jaubert, L. (eds) Spin Ice. Springer Series in Solid-State Sciences, vol 197. Springer, Cham. https://doi.org/10.1007/978-3-030-70860-3_1

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