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2Physics Quote:
"The exchange character of identical particles plays an important role in physics. For bosons, such an exchange leaves their quantum state the same, while a single exchange between two fermions gives a minus sign multiplying their wave function. A single exchange between two Abelian anyons gives rise to a phase factor that can be different than 1 or -1, that corresponds to bosons or fermions, respectively. More exotic exchanging character are possible, namely non-Abelian anyons. These particles have their quantum state change more dramatically, when an exchange between them takes place, to a possibly different state." -- Jin-Shi Xu, Kai Sun, Yong-Jian Han, Chuan-Feng Li, Jiannis K. Pachos, Guang-Can Guo
(Read Full Article: "Experimental Simulation of the Exchange of Majorana Zero Modes"

Sunday, November 01, 2009

Observation of Magnetic Monopoles in Spin Ice

Hiroaki Kadowaki, Yuji Aoki and Naohiro Doi of Tokyo Metropolitan University

[This is an invited article based on recently published work of the authors -- 2Physics.com]

Authors: H. Kadowaki1, Y. Aoki1, T. J. Sato2, J. W. Lynn3

Affiliations: 1
Department of Physics, Tokyo Metropolitan University, Tokyo, Japan,
NSL, Institute for Solid State Physics, University of Tokyo, Tokai, Japan,
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, USA

From the symmetry of Maxwell's equations of electromagnetism, magnetic charges or monopoles would be expected to exist in parallel with electric charges. About 80 years ago, a quantum mechanical hypothesis of the existence of magnetic monopoles was proposed by Dirac [1]. Since then, many experimental searches have been performed, ranging from a monopole search in rocks of the moon to experiments using high energy accelerators [2]. But none of them was successful, and the monopole is an open question in experimental physics. Theoretically, monopoles are predicted in grand unified theories as topological defects in the energy range of the order 1016 GeV [2]. However these enormous energies preclude all hope of creating them in laboratory experiments.

Taku J. Sato of University of Tokyo

Alternatively, recent theories predict that tractable analogs of the magnetic monopole might be found in condensed matter systems [3,4,5]. One prediction [4] is for an emergent elementary excitation in the spin ice compound Dy2Ti2O7 [6], where the strongly competing magnetic interactions exhibit the same type of frustration as water ice [7]. In addition to macroscopically degenerate ground states [6], the excitations from these states are topological in nature and mathematically equivalent to the Dirac monopoles [1,4]. We have successfully observed [8] the signature of magnetic monopoles in the spin ice Dy2Ti2O7 using neutron scattering, and find that they interact via the magnetic inverse-square Coulomb force. In addition, specific heat measurements show that the density of monopoles can be controlled by temperature and magnetic field, with the density following the expected Arrhenius law.

Jeffrey W. Lynn of NIST, USA

In Fig. 1 we illustrate creation of a magnetic monopole and antimonopole pair in spin ice under applied magnetic field along a [111] direction. This excitation is generated by flipping a spin, which results in ice-rule-breaking "3-in, 1-out" and "1-in, 3-out" tetrahedral neighbors, simulating magnetic monopoles, with net positive and negative charges sitting on the centers of tetrahedra. The monopoles can move and separate by consecutively flipping spins in the kagome lattice.

Fig. 1. Spins of Dy2Ti2O7 occupy a cubic pyrochlore lattice, which is a corner -sharing network of tetrahedra, and consists of a stacking of triangular and kagome lattices. The competing magnetic interaction brings about a geometrical constraint where 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. (A) By applying a small magnetic field along a [111] direction, the spins on the triangular lattices are parallel to the field, while those on the kagome lattices retain disorder under the same ice rules. This is referred to as the kagome ice state [9]. (B) Creation of a magnetic monopole (blue sphere) and antimonopole (red sphere) pair in the kagome ice state.

A straightforward signature of monopole-pair creation is an Arrhenius law in the temperature (T) dependence of the specific heat (C). This Arrhenius law of C(T) is clearly seen in Fig. 2 at low temperatures, indicating that monopole-antimonopole pairs are thermally activated from the ground state, and that the number of monopoles can be tuned by changing temperature and magnetic field.

Fig. 2. Specific heat of Dy2Ti2O7 under [111] magnetic fields is plotted as a function of 1/T. In intermediate temperature ranges these data are well represented by the Arrhenius law denoted by solid lines.

A microscopic experimental method of observing monopoles is to perform magnetic neutron scattering using the neutron's dipole moment as the probe. One challenge to the experiments is to distinguish the relatively weak scattering from the monopoles from the very strong magnetic scattering of the ground state. By choosing appropriate field-temperature values, we have successfully observed scattering by magnetic monopoles, diffuse scattering close to the (2,-2,0) reflections, and that by the ground state (Fig. 3) [8].

Fig. 3. Intensity maps of neutron scattering at T = Tc + 0.05 K in the scattering plane perpendicular to the [111] field are shown for H = 0.5 T and H = Hc. The kagome ice state at H = 0.5 T (A) compared with the MC simulation (C). The weakened kagome-ice state scattering plus the diffuse monopole scattering (B) at H = Hc agree with the MC simulation (D).

Typical elementary excitations in condensed matter, such as acoustic phonons and (gapless) magnons, are Nambu-Goldstone modes where a continuous symmetry is spontaneously broken when the ordered state is formed. This contrasts with the monopoles in spin ice, which are point defects that can be fractionalized in the frustrated ground states. Such excitations are unprecedented in condensed matter, and now enable conceptually new emergent phenomena to be explored experimentally [10].

[1] "Quantised singularities in the electromagnetic field",
P. A. M. Dirac, Proc. R. Soc. A 133, 60 (1931). Article.
[2] "Theoretical and experimental status of magnetic monopoles",
K. A. Milton, Rep. Prog. Phys. 69, 1637 (2006).
[3] "The anomalous Hall effect and magnetic monopoles in momentum space", Zhong Fang, Naoto Nagaosa, Kei S. Takahashi, Atsushi Asamitsu, Roland Mathieu, Takeshi Ogasawara, Hiroyuki Yamada, Masashi Kawasaki, Yoshinori Tokura, Kiyoyuki Terakura, Science 302, 92 (2003).
[4] "Magnetic monopoles in spin ice"
C. Castelnovo, R. Moessner, S. L. Sondhi, Nature 451, 42 (2008).
[5] "Inducing a magnetic monopole with topological surface states"
X-L. Qi, R. Li, J. Zang, S-C. Zhang, Science 323, 1184 (2009).
[6] "Spin ice state in frustrated magnetic pyrochlore materials"
S. T. Bramwell, M. J. P. Gingras, Science 294, 1495 (2001).
[7] "The structure and entropy of ice and of other crystals with some randomness of atomic arrangement" , L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935).
[8] "Observation of Magnetic Monopoles in Spin Ice", H. Kadowaki, N. Doi, Y. Aoki, Y. Tabata, T. J. Sato, J. W. Lynn, K. Matsuhira, Z. Hiroi, J. Phys. Soc. Jpn. 78, 103706 (2009).
[9] "A new macroscopically degenerate ground state in the spin ice compound Dy2Ti2O7 under a magnetic field" K. Matsuhira, Z. Hiroi, T. Tayama, S. Takagi and T. Sakakibara, J. Phys. Condens. Matter 14, L559 (2002).
Article; "Kagome ice State in the dipolar spin ice Dy2Ti2O7" Y. Tabata, H. Kadowaki, K. Matsuhira, Z. Hiroi, N. Aso, E. Ressouche, and B. Fåk, Phys. Rev. Lett. 97, 257205 (2006). Abstract.
[10] In Oct. 2009, in addition to [8], three experimental papers on the magnetic monopoles in spin ice have been published: "Measurement of the charge and current of magnetic monopoles in spin ice" S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran & T. Fennell
, Nature 461, 956 (2009), Abstract; "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7" D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, R. S. Perry, Science 326, 411 (2009) Abstract; "Magnetic Coulomb Phase in the Spin Ice Ho2Ti2O7" T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prabhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow, S. T. Bramwell, Science 326, 415 (2009). Abstract.

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