Magnetic monopoles in spin ice

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Magnetic Monopoles in Spin Ice
C. Castelnovo1, R. Moessner1,2 , and S. L. Sondhi3

Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, UK 2 Max-Planck-Institut f¨ r Physik komplexer Systeme, 01187 Dresden, Germany and u 3 Department of Physics, Princeton University, Princeton, NJ 08544 (Dated: October 31, 2007)

Electrically charged particles, such as theelectron, are ubiquitous. By contrast, no elementary particles with a net magnetic charge have ever been observed, despite intensive and prolonged searches1 . We pursue an alternative strategy, namely that of realising them not as elementary but rather as emergent particles, i.e., as manifestations of the correlations present in a strongly interacting many-body system. The most prominent examplesof emergent quasiparticles are the ones with fractional electric charge e/3 in quantum Hall physics2 . Here we show that magnetic monopoles do emerge in a class of exotic magnets known collectively as spin ice3–5 : the dipole moment of the underlying electronic degrees of freedom fractionalises into monopoles. This enables us to account for a mysterious phase transition observed experimentally inspin ice in a magnetic field6,7 , which is a liquid-gas transition of the magnetic monopoles. These monopoles can also be detected by other means, e.g., in an experiment modelled after the celebrated Stanford magnetic monopole search8 .

Spin-ice materials are characterised by the presence of magnetic moments µi residing on the sites of a pyrochlore lattice (depicted in Fig. 1). These momentsare constrained to point along their respective local Ising axes ei (the diˆ amond lattice bonds in Fig. 1), and they can be modelled as Ising spins µi = µSi , where Si = ±1 and µ = |µi |. For the spin ice compounds discussed here, Dy2 Ti2 O7 and Ho2 Ti2 O7 , the magnitude µ of the magnetic moments equals approximately 10 Bohr magnetons (µ = 10µB ). The thermo-

dynamic properties of thesecompounds are known to be described with good accuracy by an energy term that accounts for the nearest neighbour exchange and the long ranged dipolar interactions4,9,10 : H = J 3 Si Sj

+ Da3

ei · ej ˆ ˆ ˆ 3 ei · rij ej · rij ˆ − 3 5 |rij | |rij |

Si Sj . (1)

FIG. 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 (black) formed by the centres of the tetrahedra. The ratio of the p lattice constant of the diamond and pyrochlore lattices is ad /a = 3/2. The Ising axes are the local [111] directions, which point along the respective diamond lattice bonds.

˚ The distance between spinsis rij , and a ≃ 3.54 A is the py2 rochlore nearest-neighbour distance. D = µ0 µ /(4πa3 ) = 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 temperatures, T , even though it appeared to have ferromagnetic interactions3. Indeed, spin ice was found to have a residual entropy at low T 5 ,which is well-approximated by the famous Pauling entropy for (water) ice, S ≈ SP = (1/2) log(3/2) per spin. Pauling’s entropy measures the huge ground state degeneracy arising from the so-called ice rules. In the context of spin ice, its observation implies a macroscopically degenerate ground state manifold obeying the “ice rule” that two spins point into each vertex of the diamond lattice, andtwo out. We contend that excitations above this ground state manifold, i.e., defects that locally violate the ice rule, are magnetic monopoles with the necessary long distance properties. From the perspective of the seemingly local physics of the ice rule, the emergence of monopoles would at first sight seem rather surprising. To demystify it we will probe deeper into how the long range magnetic...
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