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Experimental Observation of Magnetic Monopoles in Spin Ice

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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

In spin ice, the model of magnetic monopoles arises as a transformation of the dipolar spin ice Hamiltonian developed for materials such as Dy\(_{2}\)Ti\(_{2}\)O\(_{7}\)  and Ho\(_{2}\)Ti\(_{2}\)O\(_{7}\). The treatment of spin ice in terms of an effective theory of emergent monopoles presents both theoretical and experimental challenges. In this chapter we give ‘monopole theory’ a precise definition which allows us to critically assess the extent to which magnetic monopoles have been observed in experiment. We start by answering some basic questions: what magnetic monopoles are, whether or not they are quasiparticles, to what extent they form a ‘magnetic electrolyte’ and what it means to observe them. We then introduce the main experimental techniques and their relation with the monopole theory, before comparing experimental results on the canonical spin ices Ho\(_{2}\)Ti\(_{2}\)O\(_{7}\) and Dy\(_{2}\)Ti\(_{2}\)O\(_{7}\) with theoretical expectations. We conclude with some comments on different viewpoints on magnetic monopoles, different definitions and disagreements in the literature. Our main conclusion is that the monopole theory is strongly supported by experiment. The Chapter is organised as follows:

  • 8.1 Introduction What are magnetic monopoles in spin ice? Magnetic monopoles as quasiparticles. Confirmation or falsification of monopole theory. Spin ice as a magnetic electrolyte. Direct observation of magnetic monopoles.

  • 8.2 Quantities available to experiment Equilibrium thermodynamics. Linear response and non-equlibrium thermodynamics.

  • 8.3 Experiments in weak applied fields Magnetisation correlations measured by neutron scattering. Specific heat. dc-Susceptibility. ac-Susceptibility. Summary: success and failures of the monopole theory in the weak field regime.

  • 8.4 Experiments in strong applied fields Monopole condensation with applied field along [111]. Strong field correlations. Strong field sweeps and quenches.

  • 8.5 Monopole derived properties Thermal Conductivity. Field distribution at point probes. Dielectric response.

  • 8.6 Future directions for monopole observation Plasmas. Phonons. New materials. Quantum spin ice.

  • 8.7 Conclusions Different viewpoints. Definitions and disagreements. Final word.

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Notes

  1. 1.

    Strictly the auxiliary variable is \({\boldsymbol{\varOmega }}(\mathbf{r},t) = \mathbf{M}(\mathbf{r},t)/Q\), the configuration vector [13].

  2. 2.

    In chemical thermodynamics, ideal solution equations involving the concentration or mole fraction x of a species, are applied to non-ideal solutions by replacing the mole fraction with activity a. The activity coefficient is defined as \(\gamma = a/x\). Note that \(\gamma \) is always precisely defined and measurable: the theoretical challenge is to calculate it.

  3. 3.

    Defined here as ratio of monopole drift velocity to local (H-) field at zero concentration gradient.

  4. 4.

    The ‘quantum’ of charge is material-specific because it depends on the single-ion magnetic moment and lattice spacing of the spin ice under study [10].

  5. 5.

    In principle, it is weakly temperature dependent as a result of thermal expansion, but high precision neutron Larmor diffraction experiments cannot resolve any thermal expansion in the relevant temperature range for either Dy\(_{2}\)Ti\(_{2}\)O\(_{7}\) or Ho\(_{2}\)Ti\(_{2}\)O\(_{7}\) [36].

  6. 6.

    This behaviour is of interest for another reason: it is a signature of ‘topological sector fluctuations’ in a harmonic field [37].

  7. 7.

    Unfortunately both entropy and correlation function are invariably denoted by S, the context should make clear which we are referring to.

  8. 8.

    If the incident neutron polarisation is perpendicular to the scattering plane then spin flip scattering (which flips the neutron spin) isolates the in-plane component of the fluctuating magnetisation, as in Fig. 8.2, while non-spin flip scattering isolates the out-of-plane component.

  9. 9.

    In Ho\(_{2}\)Ti\(_{2}\)O\(_{7}\) the chemical potential is \(\mu /k = 5.7\) K and the Coulomb energy per monopole is 1.5 K; Debye-Hückel screening at high temperature lowers the effective chemical potential to slightly less than \(\mu _\mathrm{eff}/k = 5.7 -1.5\) K or approximately 4 K—hence \(\sqrt{n}\) varies with an exponential amplitude of \(\sim 2\) K.

  10. 10.

    Note that in [53] the parameter \(\xi _\mathrm{ice}\) needs to be divided by \(2\pi \) to get \(l_\mathrm{diff}\).

  11. 11.

    Note the different convention with respect to \(2\pi \) in the works of Snyder et al. [86].

  12. 12.

    Another spin model with a type of ice rule constraint on the kagome lattice was orginally studied by Wills et al. [122] and referred to as kagome spin ice. It is topologically distinct to kagome ice in spin ice, as it lacks the constraint on the ice rule introduced by the interlayer spins [123]. However, it is the prototype of many artificial spin ices.

  13. 13.

    Deflagration is “combustion which propagates through a gas or across the surface of an explosive at subsonic speeds, driven by the transfer of heat.”

  14. 14.

    For example the detailed theory of pinch points [18, 60] post-dated experiments [53, 58] on them.

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  1. Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, London, UK

    Steven T. Bramwell

  2. Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, Villigen, Switzerland

    T. Fennell

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    Masafumi Udagawa

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Bramwell, S.T., Fennell, T. (2021). Experimental Observation of Magnetic Monopoles in Spin Ice. 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_8

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