Nematic, vector–multipole, and plateau–liquid states in t he classical O (3) pyrochlore antiferromagnet with biquadratic interactions in applied magnetic field Nic Shannon, 1 Karlo Penc, 2 and Yukitoshi Motome 3 $^{1}$H. H. Wills Physics Laboratory, University of Bristol, Tyndall Av, BS8–1TL, UK. 2 Research Institute for Solid State Physics and Optics, H–15 25 Budapest, P.O.B. 49, Hungary $^{3}$Department of Applied Physics, University of Tokyo, Bunkyo–ku, Tokyo 113–8656, Japan (Dated: November 6, 2018) The classical bilinear–biquadratic nearest–neighbor Hei senberg antiferromagnet on the pyrochlore lattice does not exhibit conventional N´eel–type magnetic order atany temperature or magnetic field. Instead spin correlations decay algebraically over length scales r ≲ ξ c ∼ √$_{T}$$_{, behavior characteristic of a Coulomb phase}$ arising from a strong local constraint. Despite this, its thermodynamic properties remain largely unchanged if N´eel order is restored by the addition of a degeneracy–li fting perturbation, e.g., further neighbor interac- tions. Here we show how these apparent contradictions can be resolved by a proper understanding of way in which long–range N´eel order emerges out of well–formed loc al correlations, and identify nematic and vector– multipole orders hidden in the different Coulomb phases of the model. So far as experiment is concerned, our results suggest that where long range interactions are unimportant, the magnetic properties of Cr spinels which exhibit half–magnetization plateaux may be largely indepe ndent of the type of magnetic order present. PACS numbers: 75.10.-b, 75.10.Hk 75.80.+q I. INTRODUCTION Frustrated magnets have long been studied as a paradigm for complex behavior in condensed matter and statistical physics$^{1}$. The most widely studied systems are frustrated an- tiferromagnets (AF), where competing interactions suppress classical N´eel order. In some highly frustrated magnets, spins do not order at any temperature, and the ground state retains only very short–ranged spin–spin correlations. The result ing state is generally termed a “spin liquid”. However it is also possible to invert this paradigm and think of highly frustrated magnets as systems where local “order” is robust enough to survive, even where long range order has been obliterated by fluctuations. Conventional N´eel ordercan then easily be restored — albeit with a relatively low critical temperature — by any perturbation which forces long–range coherence on this preformed local order. Moreover, where quantities such as heat capacity and magnetic susceptibility are controlled by local fluctuations, the thermodynamic pro p- erties of the globally–disordered spin–liquid phase may be practically indistinguishable from those of the magnetically ordered phase. In this paper we explore this “bottom–up” formulation of frustration, showing how the different multipolar and spin liq- uid states of a simple classical frustrated antiferromagnet in applied magnetic field already contain the seeds of long rang e N´eel order. The model which we consider is the antiferro- magnetic nearest–neighbor Heisenberg model with addition al biquadratic interactions b H = J$_{1}$ ∑ 〈 ij 〉 1 [ S$_{i}$ · S$_{j}$ − b ( S$_{i}$ · S$_{j}$ ) $^{2}$] − h · ∑ i S$_{i}$ , (1) matic half–magnetization plateau observed in Cr spinels 3–8 . In this case the biquadratic interaction b originates in a strong coupling to the lattice. However such terms can also be of electronic origin, and quite generally they can be taken to characterize the effects of quantum and/or thermal fluctua- tions in a frustrated magnet 9–11 . Thus we anticipate many of our results will also be relevant for the quantum model. Re- cent results for the S = 3 / 2 XXZ pyrochlore AF in applied magnetic field suggest that this is indeed the case $^{12,13}$. FIG. 1: (Color online) 16-site cubic unit cell of the pyrochlore lattice — a network of corner–sharing tetrahedra. Exchange interac tions J$_{1}$ are associated with the first neighbor bonds 〈 ij 〉 $_{1}$, and J$_{3}$ with the two (inequivalent) types of third–neighbor bond 〈 ij 〉 $_{3}$. It is well known that the classical Heisenberg model with the nearest–neighbor bilinear couplings only does not exhi bit N´eel–type magnetic order on the pyrochlore lattice at any temperature$^{16,17}$. As we shall see, these arguments are es- sentially unchanged by the introduction of magnetic field, o r by nearest–neighbor biquadratic interactions b . The system can however be brought to order by introducing an interaction where the sum 〈 ij 〉 1 runs over the nearest neighbor bonds of a pyrochlore lattice (Fig. 1). All the energy scales including h ≡ | h | and temperature T are measured in units of J$_{1}$ hereafter. This model was introduced in Ref. [2] to explain the dra- arXiv:1001.0270v1[cond-mat.str-el]4 Jan 2010