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Magnetic-Moment Fragmentation and Monopole Crystallization M Magnetic-Moment Fragmentation and Monopole Crystallization M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke, P. C. W. Holdsworth To cite this version: M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke, P. C. W. Holdsworth. Magnetic-Moment Fragmentation and Monopole Crystallization. Physical Review X, American Phys- ical Society, 2014, 4, pp.011007. 10.1103/PhysRevX.4.011007. hal-01541937 HAL Id: hal-01541937 https://hal.archives-ouvertes.fr/hal-01541937 Submitted on 21 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Magnetic moment fragmentation and monopole crystallization M. E. Brooks-Bartlett,1 S. T. Banks,1 L. D. C. Jaubert,2 A. Harman-Clarke,1, 3 and P. C. W. Holdsworth3, ∗ 1Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom. 2OIST, Okinawa Institute of Science and Technology, Onna-son, Okinawa, 904-0495, Japan 3Laboratoire de Physique, Ecole´ Normale Sup´erieure de Lyon, Universit´ede Lyon, CNRS, 46 All´ee d’Italie, 69364 Lyon Cedex 07, France. The Coulomb phase, with its dipolar correlations and pinch-point scattering patterns, is central to discussions of geometrically frustrated systems, from water ice to binary and mixed-valence alloys, as well as numerous examples of frustrated magnets. To date, the emergent Coulomb phase of lattice based systems has been associated with divergence-free fields and the absence of long-range order. Here we go beyond this paradigm, demonstrating that a Coulomb phase can emerge naturally as a persistent fluctuating background in an otherwise ordered system. To explain this behaviour we introduce the concept of the fragmentation of the field of magnetic moments into two parts, one giving rise to a magnetic monopole crystal, the other a magnetic fluid with all the characteristics of an emergent Coulomb phase. Our theory is backed up by numerical simulations and we discuss its importance with regard to the interpretation of a number of experimental results. I. INTRODUCTION In this paper we demonstrate how the lattice based magnetic Coulomb phase emerges in a considerably wider The interplay between frustration, topological order, class of models than those covered by the conditions fractionalization and emergent physics has been the fo- stipulated above. More specifically, we show that such cus of a rapidly increasing body of work in recent years. a phase exists for a model “magnetolyte” irrespective In themselves these concepts are not new. Frustration of the magnetic monopole density or of monopole or- underpins theories of glassiness and has been much dis- dering. We introduce the concept of magnetic moment cussed since the seminal studies of Anderson and Vil- fragmentation, whereby the magnetic moment field un- lain [1, 2]. Likewise, concepts of topological order and dergoes a novel form of fractionalization into two parts: fractionalization, from the fractional quantum Hall ef- a divergence-full part representing magnetic monopoles fect [3] and quasi-particles in graphene-like systems [4] and a divergence-free part corresponding to the emer- to solitons in one-dimension [5, 6], have been prominent gent Coulomb phase with independent and ergodic spin topics in theroretical and experimental condensed matter fluctuations[12]. Our results apply even for a monopole physics for over thirty years. crystal which is shown to exist in juxtaposition with mo- The subtleties of the inter-dependence between these bile spin degrees of freedom: a previously unseen coexis- concepts has been elucidated only recently, in the con- tence between a spin liquid and long-range order induced text of the so-called emergent Coulomb phase of highly by magnetic moment crystallization. The implications of frustrated magnetic models. However the phenomenol- this field fragmentation are wide-reaching and relevant ogy of the Coulomb phase, with its characteristic dipolar for the interpretation of a number of experiments, as dis- correlations and emergent gauge structure, is not limited cussed below. to realms of frustrated magnetism. Indeed similar be- haviour is observed in water ice [7] and models of heavy fermion behaviour in spinels [8], as well as in models of II. THE MODEL binary and mixed valence alloys [9]. Henley [10] succinctly sets out three requirements for Magnetic monopoles [11] emerge as quasi-particle exci- the emergence of a Coulomb phase on a lattice: (i) each tations from the ground state configurations of the dumb- microscopic variable can be mapped onto a signed flux bell model of spin ice. Here, the point dipoles of the dipo- directed along a bond in a bipartite lattice; (ii) the sum lar spin ice model [13] are extended to infinitesimally thin of the incoming fluxes at each lattice vertex is zero and magnetic needles lying along the axes linking the cen- (iii) the system has no long-range order. Elementary ex- tres of adjoining tetrahedra of the pyrochlore lattice (see citations out of the divergence free manifold are seen to Fig. 1) which constitute a diamond lattice of nodes for fractionalize, giving rise to effective magnetic monopoles magnetic charge [14]. The needles touch at the diamond that interact via a Coulomb potential [11] and (in three lattice sites so that, by construction, the long range part dimensions) may be brought to infinite separation with of the dipolar interactions are perfectly screened for the finite energy cost. ensemble of ice rules states in which two needles point in and two out of each tetrahedron [15]. These ground states form a vacuum from which monopoles are excited by reversing the orientation of a needle, breaking the ice ∗ [email protected] rules on a pair of neighbouring sites. As vacuums go 2 this one is rather exceptional, as it is far from empty. gives Qi = 0 while three-in-one-out (three-out-one-in) Rather, the magnetic moments constitute the curl of a gives Qi = +(−)2m/a [11]. lattice gauge field [16], the Coulomb phase, with manifest The fields for an isolated three-in-one-out vertex can be experimental consequences for spin ice materials. These split into divergence-full and divergence-free parts sub- include diffuse neutron scattering patterns showing the ject to the constraint that the amplitude of each field sharp pinch point features [17] characteristic of dipolar element is |Mij | = m/a: correlations, and the generation of large internal mag- netic fields despite the status of vacuum [18, 19]. a [Mij ] = (−1, −1, −1, 1) (2) In an analogy with electrostatics, the monopole charge m 1 1 1 1 1 1 1 3 distribution obeys Gauss’ law for the magnetic field, = − , − , − , − + − , − , − , . ∇~ .H~ = ρ , where ρ is the monopole density. The 2 2 2 2 2 2 2 2 m m monopole number is not conserved and the energetics of the dumbbell model at low temperature correspond to a The first set of fields satisfy Gauss’ law for the charge at Coulomb gas in the grand canonical ensemble, in which the origin; the second set satisfy a discrete divergence- the phase space of monopole configurations is constrained free condition and constitute a residual dipolar field by the underlying ice rules. One can define a Landau en- dressing the monopole (see Fig. 1c). Decomposition fur- ther away from the charge could be made by solving the ergy, U˜ = UC − µN, where UC is the Coulomb energy of the neutral gas of N = monopoles and µ is the chemical lattice Poisson equation to find the field sets belonging ~ potential such that −2µ is the energy cost of introducing to Mm which would be subtracted from the Mij to find a neutral pair of monopoles an infinite distance apart. the discrete elements of M~ d. Singly charged monopoles For temperatures of order −µ the Coulomb gas picture leave a residual contribution to M~ d at each vertex and it of spin ice must be extended to include double charged is only when the vertex is occupied by a double charged monopoles (see Appendix A). monopole for, which [Mij ]= ±(m/a)(1, 1, 1, 1), that the ~ The divergence in H is related to the breaking of the contribution to M~ d is totally suppressed. Hence, a fluid ice rules through M~ , the magnetic moment density which of singly charged monopoles should be accompanied by itself obeys Gauss’ law, ∇~ .M~ = −ρm. This does not how- a correlated random dipolar field whose detailed struc- ever, define the entire moment density which can have ture is updated by the monopole dynamics and only de- two contributions: the first, M~ m, is the gradient of a stroyed on the temperature scale at which doubly charged scalar potential and provides the magnetic charge; the monopoles proliferate. Indeed the pinch points in diffuse neutron scattering from spin ice materials are maintained second, M~ d, a dipolar field, can be represented as the curl of a vector potential and is divergence-free [20, 21] up to surprisingly high temperatures [17, 23], indicating the presence of such a dipolar field. The emergence of the dipolar field is further illustrated in the Appendix C M~ = M~ m + M~ d = ∇~ ψ(r)+ ∇~ ∧ Q.~ (1) where we show how the [Mij ] are divided around a pair of isolated nearest neighbour charges of the same sign.
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