Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice
C. Paulsen S. R. Giblin E. Lhotel D. Prabhakaran K. Matsuhira G. Balakrishnan S. T. Bramwell



[]carley.paulsen@neel.cnrs.fr
Institut Néel, CNRS and Université Grenoble Alpes, 38042 Grenoble, France

School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom

Institut Néel, CNRS and Université Grenoble Alpes, 38042 Grenoble, France

Clarendon Laboratory, Physics Department, Oxford University, Oxford, OX1 3PU, United Kingdom

Kyushu Institute of Technology, Kitakyushu 804-8550, Japan

Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom

London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London, WC1H 0AJ, United Kingdom





Extensive work on single molecule magnets has identified a fundamental mode of relaxation arising from the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic dipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of purely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were conducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring the magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical coupling with the hyperfine fields bring the electronic spins associated with magnetic monopoles to resonance, allowing the monopoles to hop and transport magnetic charge. Our result shows how the coupling of electronic spins with nuclear spins may be used to control the monopole current. It broadens the relevance of the assisted quantum tunnelling mechanism from single molecular spins to emergent excitations in a strongly correlated system.



Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice
    S. T. Bramwell
    December 30, 2023
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§ INTRODUCTION


In the canonical dipolar spin ice materials (, ) <cit.>, rare earth ions with total angular momentum J = 15/2 (Dy^3+) and J = 8 (Ho^3+) are densely packed on a cubic pyrochlore lattice of corner-linked tetrahedra. The ions experience a very strong ⟨111⟩ crystal field, resulting in two effective spin states (M_ J = ± J)  that define a local Ising-like anisotropy. At the millikelvin temperatures discussed here (0.08  K<T<0.2 K), a lattice array of such large and closely-spaced spins would normally be ordered by the dipole-dipole interaction <cit.>, but the pyrochlore geometry of spin ice frustrates the dipole interaction and suppresses long-range order. Instead, the system is controlled by an ice-rule, 
that maps to the Pauling model of water ice <cit.>. In 
the effective ground state, the spins describe a flux with closed-loop topology and critical correlations, that may be described by a local gauge symmetry rather than by a traditional broken symmetry <cit.>. This strongly correlated spin ice state is stabilised by a remarkable self-screening of the dipole interaction <cit.>. Excitations out of the spin ice state fractionalise to form effective magnetic monopoles <cit.>, but the excited states are no longer self-screened and this manifests as an effective Coulomb interaction between monopoles. The static properties of spin ice are accurately described by the monopole model <cit.>.  The dynamic properties can also be described by assuming an effective monopole mobility <cit.>, but there have been few studies of the microscopic origin of the monopole motion <cit.>.



The field and energy scales involved in monopole motion are illustrated in Fig. 1, a-e.  When a monopole hops to a neighbouring site a spin is flipped (Fig. 1a). For an isolated monopole (far from any others) this spin flip takes place at nominally zero energy cost (Fig. 1b) because contributions from near-neighbour antiferromagnetic superexchange and ferromagnetic dipole–dipole coupling individually cancel. The cancellation of the field contribution relies on the dipolar self-screening <cit.> that maps the long-range interacting system <cit.> to the degenerate Pauling manifold of the near neighbour spin ice model <cit.>. This surprising cancellation is a key result of the many–body physics of spin ice. 
In practice, a  monopole hop may also involve a finite energy change arising from longitudinal fields at the spin site: the main source of fields is nearby monopoles <cit.> (Fig. 1b), while further contributions arise from corrections to the mapping, which give a finite energy spread to the Pauling manifold <cit.> (here of order ∼0.1 K <cit.>).
The mechanism of the hop is believed to be quantum tunnelling and several key signatures of this have been observed in the high temperature regime between 2 K and 10 K <cit.>.

At lower temperatures (T < 0.6 K), spin ice starts to freeze <cit.>. This is due in part to the rarefaction of the monopole gas whose density n(T) varies as ∼ e^-|μ|/T  where the chemical potential |μ| = 4.35 and 5.7 K for  and   respectively <cit.>, and also in part to geometrical constraints that create noncontractable, monopole-antimonopole pairs that cannot easily annihilate <cit.>. These factors, which are independent of the monopole hopping mechanism, suggest that the 
relaxation rate ν(T) ∝ n(T) will fall to exponentially small values at low temperature (T <  0.35 K). 

Previous thermal quenching experiments have demonstrated monopole populations well  below the nominal freezing temperature that are both long lived and able to mediate magnetic relaxation <cit.>. This paradoxical frozen but dynamical character of the system suggests the relevance of resonant magnetic tunnelling, where magnetisation reversal can only occur when the longitudinal field is smaller than the tunnelling matrix element Δ E. The monopolar fields may add a longitudinal component that takes the spin off the resonance condition (Fig. 1c,d) but in addition may add a transverse component that amplifies Δ E: together these lead to a suppression and dispersion of the monopole mobility. 

In the following, we will demonstrate experimentally that hyperfine interactions (Fig. 1e) play a significant role in bringing monopoles back to their resonance condition, enabling dynamics at very low temperatures (T < 0.35 K).



§ RESULTS



Samples 
