Magnetic monopoles in spin ice.docx

1、Magnetic monopoles in spin iceNature451, 42-45 (3 January 2008) |doi:10.1038/nature06433; Received 27 June 2007; Accepted 29 October 2007Magnetic monopoles in spin iceC. Castelnovo1, R. Moessner1,2& S. L. Sondhi31. Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, UK2

2、. Max-Planck-Institut fr Physik komplexer Systeme, 01187 Dresden, Germany3. PCTP and Department of Physics, Princeton University, Princeton, New Jersey 08544, USACorrespondence to: C. Castelnovo1Correspondence and requests for materials should be addressed to C.C. (Email:castelphysics.ox.ac.uk).Topo

3、f pageAbstractElectrically charged particles, such as the electron, are ubiquitous. In contrast, no elementary particles with a net magnetic charge have ever been observed, despite intensive and prolonged searches (see ref.1for example). We pursue an alternative strategy, namely that of realizing th

4、em not as elementary but rather as emergent particlesthat is, as manifestations of the correlations present in a strongly interacting many-body system. The most prominent examples of emergent quasiparticles are the ones with fractional electric chargee/3 in quantum Hall physics2. Here we propose tha

5、t magnetic monopoles emerge in a class of exotic magnets known collectively as spin ice3,4,5: the dipole moment of the underlying electronic degrees of freedom fractionalises into monopoles. This would account for a mysterious phase transition observed experimentally in spin ice in a magnetic field6

6、,7, which is a liquidgas transition of the magnetic monopoles. These monopoles can also be detected by other means, for example, in an experiment modelled after the Stanford magnetic monopole search8.Spin-ice materials are characterized by the presence of magnetic momentsiresiding on the sitesiof a

7、pyrochlore lattice (depicted inFig. 1). These moments are constrained to point along their respective local Ising axes(the diamond lattice bonds inFig. 1), and they can be modelled as Ising spinsi=Si, whereSi=1 and. For the spin-ice compounds discussed here, Dy2Ti2O7and Ho2Ti2O7, (where Dy is dyspro

8、sium and Ho is holmium) the magnitudeof the magnetic moments equals approximately ten Bohr magnetons (10B). The thermodynamic properties of these compounds are known to be described with good accuracy by an energy term that accounts for the nearest-neighbour exchange and the long-range dipolar inter

9、actions9,10(for a review of spin ice, see ref.4):Figure 1:The pyrochlore and diamond lattices.The magnetic moments in spin ice reside on the sites of the pyrochlore lattice, which consists of corner-sharing tetrahedra. These are at the same time the midpoints of the bonds of the diamond lattice (bla

10、ck) formed by the centres of the tetrahedra. The ratio of the lattice constant of the diamond and pyrochlore lattices is. The Ising axes are the local 111 directions, which point along the respective diamond lattice bonds.High resolution image and legend (99K)The distance between spins isrij, anda3.

11、54 is the pyrochlore nearest-neighbour distance.D=02/(4a3) = 1.41K is the coupling constant of the dipolar interaction.Spin ice was identified as a very unusual magnet when it was noted that it does not order to the lowest temperaturesTeven though it appeared to have ferromagnetic interactions3. Ind

12、eed, spin ice was found to have a residual entropy at lowT(ref.5), which is well-approximated by the Pauling entropy for water ice,SSP= (1/2)log(3/2) per spin. Paulings entropy measures the huge ground-state degeneracy arising from the so-called ice rules. In the context of spin ice, its observation

13、 implies a macroscopically degenerate ground state manifold obeying the ice rule that two spins point into each vertex of the diamond lattice, and two out.We contend that excitations above this ground-state manifoldthat is, defects that locally violate the ice ruleare magnetic monopoles with the nec

14、essary long-distance properties. From the perspective of the seemingly local physics of the ice rule, the emergence of monopoles at first seems rather surprising. We will probe deeper into how the long-range magnetic interactions contained in equation (1) give rise to the ice rule in the first place

15、. We then incorporate insights from recent progress in understanding the entropic physics of spin ice, and the physics of fractionalization in high dimensions11,12,13,14,15, of which our monopoles will prove to be a classical instance.We consider a modest deformation of equation (1), loosely inspire

16、d by Nagles work16on the unit model description of water ice: we replace the interaction energy of the magnetic dipoles living on pyrochlore sites with the interaction energy of dumbbells consisting of equal and opposite magnetic charges that live at the ends of the diamond bonds (seeFig. 2). The two ways of assigning charges on each diamond bond