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Pedagogic demonstration of spontaneously broken symmetry by magnetic compasses.

A pedagogic daily macroscopic device to demonstrate spontaneously broken symmetries is developed by which both processes of spontaneous breaking of rotational symmetry and of creating the Nambu-Goldstone bosons can be well illustrated.

One of the most important concepts in modern physics is spontaneously broken symmetry (SBS). For example, creation of the universe in cosmology, generation of gauge bosons, Higgs particles, t'Hooft monopoles, instantons and solitons in high energy elementary particle physics, and phenomena of laser action, superradiance, superconductivity, superfluidity, phase transition, magnetism and electret behavior in condensed matter physics are all well understood in terms of SBS. (1) These phenomena are all examples of the spontaneous creation of order in nature. Symmetry and order are two mutually complementary concepts: When we have rotational symmetry, there is no particular direction singled out as being different. When we notify a specific direction, there is directional order, and rotational symmetry should be lost to create such an order.

In the case when the external environment of the system imposes a certain force on the system, the Hamiltonian governing the dynamics no longer manifests a rotational symmetry, thus realizing a "broken symmetry." This is not a particularly interesting situation. However, a remarkable situation is the case when the Hamiltonian manifests rotational symmetry but the ground state manifests a directional order. This is SBS. Such an ordered state can be created if the electrons with spin up condense into one of the ground states, and an ordered spatial region is formed spontaneously in which all the spin vectors of the electrons are aligned in one and the same direction. This phenomenon of spontaneously broken rotational symmetry or creation of a spin directional order explains ferromagnetism-the familiar microscopic magnetism of magnets made of many internal magnetic domains. Because the electron spin can be seen as a magnetic dipole, a system of many spin vectors is nothing but a system of many magnetic dipole vectors interacting with each other via magnetic dipole-dipole couplings.

Also, from the technological point of view of modern thin-film magnetic devices, a visual real method to illustrate the formation of stable long-range magnetic order in the quasi-two-dimensional systems of magnetic dipoles is needed due to the difficulty in integrating the long-range dipole-dipole interaction in quasi-two-dimensional systems. (2) Such a real method illustrates the importance of the long-range character of the dipolar interaction in forming and stabilizing the macroscopic domain patterns driven by the spontaneously broken. To accurately model the behavior of an ordered domain with spontaneously broken symmetry, one must take into account not just the nearest-neighbor dipole-dipole forces, but also the longer-range interactions. However, it is quite difficult in practice to compute analytically or numerically these long-range interactions. To overcome these difficulties, a Monte-Carlo type simulation with fast summation and periodic boundary conditions is often used. In our view, however, it is questionable whether these computational approaches are able to reproduce all of the nuances and details, which a real SBS system can exhibit. Our pedagogic demonstration is exactly the real SBS system and suggests many fruitful things as mentioned in this paper.

The mechanism of SBS was first demonstrated theoretically in quantum field theory: (1) In the system of infinitely many degrees of freedom described by the Hamiltonian manifesting the rotational symmetry, only one state is chosen spontaneously among the infinitely degenerate ground states as a real ground state of the system and the rotational symmetry of the system is broken without recourse to any external environment. Although SBS is a mechanism characteristic to the system of infinitely many degrees of freedom described by quantum field theory, it can equally be applied to a system of finite but many degrees of freedom. "Spontaneous" means the following fact: The rotational symmetry is broken not by imposing a certain external force, but by the fact that the system itself chooses one and only one ground state among the infinitely many possible ground states, and the transition of the chosen ground state into other ones cannot be realized. Due to a simple calculation in quantum field theory, (1) the more degrees of freedom possessed by a system, the more stable its spontaneously chosen ground state. Thus, in the case of infinitely many degrees of freedom, the ground state with SBS becomes highly stable but manifests the broken rotational symmetry. The most familiar example of SBS is ferromagnetism we already sketched briefly.

In any system with SBS we expect the following two features: The first is resulting from the original rotational invariance of the dynamics, which implies that the aligned direction of spins can be any direction. In other words, all the ground states with different aligned directions have the same energy, and so infinitely many ground states can exist realizing the infinitely degenerate ground states. The second factor results from the stability of the ground state with SBS. Suppose that a spin vector in a spatial region of ordered spin vectors in the ground state with SBS happens to deviate slightly from the aligned direction of all the other spin vectors. In order for the ground state manifesting a directional order to be stable, this deviation of spin direction must be detected by other spin vectors, which would then have some means of correcting the wrong direction of the particular spin vector. This requires the presence of a certain interaction mechanism among the spin vectors in the region of the ground state with SBS. With such a mechanism, any deviation of local spin from the aligned direction of all the spin vectors can be corrected by the system itself.

This mechanism is provided by a wave that can propagate over the whole of the ordered region of the ground state with SBS. Because the spin vectors in the ordered region are aligned in one and the same direction, a deviation of a spin vector is corrected but is overcorrected, resulting still in a deviation, thus starting an oscillation around the aligned direction. Since all the spin vectors are strongly coupled, this oscillation is transferred to all the other spin vectors successively. This is a correlation wave propagating throughout the ordered region. Existence of such a long-range correlation wave in the ground state with SBS was demonstrated theoretically in quantum field theory, using what is known today as the Nambu-Goldstone theorem. (1) The long-range correlation wave is often called a Nambu-Goldstone mode. A well known example of the Nambu-Goldstone mode is the spin wave or magnon in a ferromagnet.

Although SBS plays the most important role in understanding theoretically the various order-creating phenomena in nature, it is a highly advanced concept and requires the mathematical framework of modern quantum field theory such as the Nambu-Goldstone theorem. The absence of an intuitive visual demonstration of SBS has been considered to be a weak point of quantum field theory. One of the authors (YS) developed a macroscopic device to demonstrate SBS and Nambu-Goldstone modes. The device looks simple: it consists of 350 ball-shaped magnetic compasses fabricated by Sumitomo Special Metals Co., Ltd., each of which has a plastic rotator containing a ferrite magnet of magnetic flux density 15Gauss at the maximum point on this device floating and confined in oil in a transparent spherical plastic container of diameter 3.0 cm and rotatable freely around the vertical axis. This magnetic compass was designed originally as an automobile accessory, and we marked a visible arrow on the rotator by luminous paint parallel to the magnetic dipole of the ferrite. (Fig. 1)


Operation of the device is simple, too. We put the 350 magnetic compasses densely one by one and line by line (as in crystal growth) onto a rectangular tray, obtaining the two-dimensional triangle lattice (Fig. 2). We then poured them densely into the tray obtaining the nematic non-lattice structure like the spin glass (Fig. 3). In both cases of the two-dimensional distribution of 350 magnetic compasses, about 10 to 30 neighboring compasses tend to form an ordered domain in which all the magnetic dipoles point in one and the same direction independent of the Earth's magnetic field. We can see a similiar magnetic domain structure in the theoretical model of ferromagnetism. In the case of Fig.2, the domain shape and the dipole direction tend to depend on the crystal structure. The domain structure of those 350 magnetic compasses are stable in the sense that we need considerable vibrational disturbance to the system of magnetic compasses confined on the tray, thus demonstrating intuitively the role of temperature in the ferromagnetic phase transition. By observing the formation of the ordered domain structure in this device, one can understand visually the essential aspect of SBS without recourse to abstract mathematics.


We put the 350 magnetic compasses also into the strict two-dimensional rectangle lattice, and found no ordered domains just as was predicted theoretically (Fig. 4). However, we found that ordered domains can be impressed on the case of rectangular lattice if the assembly of magnetic compasses is transformed from the triangular lattice arrangement to the rectangular one by a quasi-static process (Fig.5A, Fig.5B). This is a metastable state analogous to supercooling or supersaturation. The domains vanish and do not reappear if we vibrate the set of compasses (Fig.5C).


Furthermore, if one disturbs one magnetic compass in the ordered domain by moving a small magnet closer by hand, propagation of the local directional disturbance to the whole of the ordered domain can be observed. This demonstrates the Nambu-Goldstone theorem, that is, the creation of the long-range correlation wave in a system with SBS. (Since a still picture is not adequate for showing the propagation of the local directional disturbance, that is, the Nambu-Goldstone mode, we prepared a digitized motion picture (700 KB) available on the following World-Wide Web site: http://www news/graphic/mag.mpg).

To create a successful demonstration of magnetic domain structure, it is not sufficient to merely place many ordinary magnetic compasses in a tray. This is because typical small compasses have a very small magnetic dipole moment, and consequently their mutual magnetic interaction is much smaller than the Earth's ambient magnetic field. Our demonstration solves this problem by using 350 compasses with a very large magnetic dipole moment, to ensure that their mutual interaction is large compared to the Earth's field.


Yashihiko Saito thanks Prof. Takaki Shichiri of Osaka City University and his colleagues in the Osaka Museum of Science for their helpful advice and support. Kunio Yasue would like to dedicate this note to the late Prof. Hiroomi Umezawa who developed a new world-view based on physics of SBS. This research was partly supported by the Scientific Research Fund of the Ministry of Education, Science, and Culture under grant No.12914022.


(1.) Umezawa, H. (1993). Advanced field theory-micro, macro, and thermal physics. New York: American Institute of Physics.

(2.) De'Bell, K., Maclsaac, A. B., and Whitehead, J. P. (2000). Dipolar effects in magnetic thin films and quasi-two-dimensional systems. Rev. Mod. Phys., 72,225.

Yoshihiko Saito (1) and Kunio Yasue (2)

(1) Osaka Museum of Science 4-2-1 Nakanoshima, Kita-ku, Osaka 530-0005, Japan

(2) Research Institute for Informatics and Science Notre Dame Seishin University Okayama 700-8516, Japan
COPYRIGHT 2001 Temple University - of the Commonwealth System of Higher Education, through its Center for Frontier Sciences
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2001 Gale, Cengage Learning. All rights reserved.

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Author:Saito, Yoshihiko; Yasue, Kunio
Publication:Frontier Perspectives
Date:Mar 22, 2001
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