*Magnetic Monopoles and Spin Liquids
*

From
the symmetry of Maxwell's equations of electromagnetism, as well as field
theoretical arguments, magnetic charges or monopoles would be expected to exist
and produce magnetic fields, and moving magnetic monopoles electric fields. But
no magnetic monopoles have ever been observed despite longstanding experimental
searches^{1} since the hypothesis of the Dirac monopole^{2}.
Recently, attention has turned to condensed matter systems where tractable
analogs of magnetic monopoles might be found, and one prediction^{3} is
for an emergent elementary excitation in the spin ice compound^{
}Dy_{2}Ti_{2}O_{7}.
'Spin Ice' is a special magnetic system where the Dy spins occupy a cubic
pyrochlore lattice, which is a corner sharing network of tetrahedra with the
spins located at the vertices of the tetrahedra (see figure below). The
magnetic anisotropy requires the spins to point along a line to the center of
the tetrahedron, either parallel or antiparallel, while neighboring spins
interact via an effective ferromagnetic coupling. The interactions between
all the spins in the system cannot be optimized simultaneous, making the
magnetic spins 'frustrated', and preventing a conventional ordered magnetic
state to form even at the very lowest temperatures achievable. Instead of
a conventional
ground state with long range magnetic order, the lowest energy spin
configurations on each tetrahedron follow the ice rule, in which two spins point
inward and two point outward on each tetrahedron. This "2-in & 2-out" ground
state spin structure has a six-fold degeneracy (see Figure below), and the possible
ground states of the entire tetrahedral network are macroscopically degenerate
in the same way as the famous example of disordered protons in H_{2}O ice. In
solid water, the protons are disordered even at absolute zero temperature
and thus retain finite entropy, and
the same disordered
ground state occurs for the magnetic moments in Dy_{2}Ti_{2}O_{7},--hence
the name 'Spin Ice'.

Neutron scattering research has detected the equivalent of "magnetic monopoles" in a low temperature condensed matter magnetic system called 'spin ice'. These monopoles are analogs to the elusive magnetic monopole particles theorized in 1931 by Paul Dirac - but never actually found. The red arrows locate the scattering associated with the creation of magnetic monopoles.

In
addition to this remarkable behavior, it has been recently theorized^{3}
that the excitations from these highly degenerate ground states are topological
in nature and mathematically equivalent to magnetic monopoles.
We have carried out neutron scattering experiments that confirm the existence of
monopoles
in spin ice, and that they interact via the magnetic
Coulomb force.^{4} The establishment of magnetic monopoles in this condensed matter
system represents a fundamentally new type of collective excitation, and opens
the door for studies of aspects of magnetic monopoles that include pair creation
and interaction, individual motion, correlations, and cooperative phenomena.

These cubic pyrochlore
materials, with their competing interactions and magnetic frustration, exhibit
many other interesting emergent properties. Including the quantum spin
fluctuations inherent at these very low temperatures ( < 1 K), no magnetic order
at all may occur, leading to a hidden order state of electric quadrupole
moments, neighboring a spin liquid that is likely a U(1) quantum spin liquid
state.^{5,6}

1. For a review, see K. A. Milton, Theoretical and experimental status of magnetic monopoles. Rep. Prog. Phys. 69, 1637-1712 (2006).

2. P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc. A 133, 60-72 (1931).

3. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42-45 (2008).

4. Observation of Magnetic Monopoles in Spin Ice, Hiroaki Kadowaki, Naohiro Doi, Yuji Aoki, Yoshikazu Tabata, Taku J. Sato, J. W. Lynn, K. Matsuhira, and Z. Hiroi, J. Phys. Soc. Japan 78, 103706 (2009).

**5. Quantum Spin Fluctuations in the Spin Liquid State of Tb**_{2}**Ti**_{2}**O**_{7},
Hiroshi Takatsu, Hiroaki Kadowaki, Taku J. Sato,
Jeffrey W. Lynn,
Yoshikazu Tabata,
Terno Yamakaki, and Kazuyuki Matsuhira,
*J.
Phys. Cond. Matr. *
**24**, 052201 (2012).

**6. Quadrupole Order in
the Frustrated Pyrochlore Tb**_{2+x}**Ti**_{2−}_{x}**O**_{7+y},
H. Takatsu,
S. Onoda,
S. Kittaka, A. Kasahara, Y. Kono, T. Sakakibara, Y. Kato, B. Fåk, J. Ollivier, J. W. Lynn, T.
Taniguchi, M. Wakita, and H. Kadowaki, *
Phys. Rev.
Lett. 116,
217201 (2016).*

See the Invited Talk
(C9.2) entitled "Observation
of Magnetic Monopoles in the Spin Ice Dy_{2}Ti_{2}O_{7}"
presented at the
International Conference on Neutron Scattering (May, 2009).

Illustration of spin ice, kagomé ice and
magnetic monopoles. a, Magnetic moments
(spins) of Dy_{2}Ti_{2}O_{7} reside on the corners of
tetrahedra of the cubic pyrochlore lattice. They are represented by arrows
pointing inward or outward from centres of the tetrahedra, depicted equivalently
by black and white spheres. At low temperatures, four magnetic moments on each
tetrahedron obey the ice rule (2-in & 2-out structure) to minimize the effective
ferromagnetic interaction between neighbouring spins. The resulting spin ice
state consists of a macroscopic number of disordered configurations, an example
of which is shown in a.
The
pyrochlore lattice consists of stacked triangular and kagomé lattices, shown by
green and blue lines, respectively, along a [111] direction. (b)
Under small [111] magnetic fields, spins on the kagomé lattice remain in
the disordered kagomé ice state.
(c) An excited state is induced by
flipping a spin from (b), enclosed by a dashed circle, where neighboring
tetrahedra have 3-in, 1-out and 1-in, 3-out configurations. These
ice-rule-breaking tetrahedra are represented by magnetic monopoles with opposite
charges depicted by spheres. (d) By consecutively flipping two spins from
(c), the monopoles are fractionalized. (e) As the magnetic field is
increased, H >> Hc, spins realize a fully ordered, staggered arrangement of
monopoles

An article about the work is available at 2Physics.com:

in the
November 2009 archives
of 'Condensed
Matter' or
'Elementary
Particles'.

For a popular account, see

NIST Tech Beat

http://www.nist.gov/ncnr/monopole_100609.cfm