Pedagogic Demonstration of the Spontaneously Broken Symmetryby Magnetic Compasses
Yoshihiko Saito1 and Kunio Yasue2
1Osaka Museum of Science
4-2-1 Nakanoshima, Kita-ku, Osaka 530-0005, Japane-mail:saito@sci-museum.kita.osaka.jp
2Research Institute for Informatics and Science Notre Dame Seishin University
Okayama 700-8516, Japane-mail:kunio@ndsu.ac.jp
Abstract

A pedagogic daily macroscopic device to demonstrate the spontaneously broken symmetries isdeveloped 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 keywords inrealizing the broken symmetries. However, this ismodern physics is the spontaneously broken

quite a natural situation. A remarkable situation issymmetries (SBS).

Creation of the universe inthat although the Hamiltonian manifests thecosmology, generation of gauge bosons, Higgs

rotational symmetry, the ground state can still

particles, t'Hooft monopoles, instantons and solitons

manifest a directional order. This is SBS. Such anin high energy elementary particle physics, andordered state can be created if the electrons withphenomena

superradiance,spin up condense into one of the ground states, andsuperconductivity, superfluidity, phase transition,an ordered spatial region is formed spontaneously inmagnet and electret in condensed matter physics

which all the spin vectors of the electrons areare all well understood in terms of SBS [1]. Thosealigned in one and the same direction.

phenomena are all known as order-creating

phenomenon of spontaneously broken rotational

phenomena in nature. Symmetry and order are

symmetry or creation of a spin directional order

two mutually complementary concepts: When we

explains the ferromagnetism; the daily familiar

have rotational symmetry, there is no particularmagnetism of magnets made of many internal

direction singled out as being different. When we

magnetic domains.

notify a specific direction, there is directional order,

seen as a magnetic dipole, the system of infinitely

and the rotational symmetry should be lost to create

many spin vectors is nothing but a system ofsuch an order.

infinitely many magnetic dipole vectors interactingoflaser,

In the case some external environment of thesystem imposes certain force to the dynamics of thewitheachThis

As the electron spin can beotherneighborsviamagnetic

dipole-dipole couplings.

system, the Hamiltonian governing the dynamics

Also from the technological point of view ofmanifests no longer the rotational symmetry, thus

modern thin film magnetic devices, a visual real

method to illustrate the formation of stablecertain external force, but by the fact that thelong-range

thesystem itself chooses one and only one ground statequasi-two-dimensional systems of magnetic dipolesamong the infinitely many possible ground statesis needed due to the difficulty in integrating theand the transition of the chosen ground state into

long-range

inother ones cannot be realized. Due to a simple

Such a real method

calculation in quantum field theory [1], the more theillustrates the importance of the long-range

degrees of freedom of the system becomes, the morecharacter of the dipolar interaction in forming andstable the spontaneously chosen ground state is.

stabilizing the macroscopic domain patterns driven

Thus in the case of infinitely many degrees ofby the spontaneously broken. To accurately model

freedom, the ground state with SBS becomes highly

thestable

magneticorder

dipole-dipole

quasi-two-dimensional [2].

behaviorof

anininteraction

ordereddomainwithbutmanifests

thebroken

rotational

spontaneously broken symmetry, one must takesymmetry. The most familiar example of SBS is

into

the ferromagnetism we already sketched briefly.

account

not

just

the

nearest-neighbor

dipole-dipole forces, but also the longer-rangeinteractions.

In any system with SBS, we expect twoHowever, it is quite difficult infeatures: The first is resulting from the originalpractice to compute analytically or numericallyrotational invariance of the dynamics which implies

these long-range interactions. To overcome thesethat the aligned direction of spins can be any

difficulties, a Monte-Carlo type simulation with fastdirection.

summation and periodic boundary conditions isdifferent aligned directions have the same energy,

often used. In our view, however, it is questionable

and so infinitely many ground states can exist

whether these computational approaches are able torealizing the infinitely degenerate ground states.

reproduce all of the nuances and details, which aThe second is resulting from the stability of thereal SBS system can exhibit. Our pedagogic

ground state with SBS. Suppose that a spin vector

demonstration is exactly the real SBS system andin a spatial region of ordered spin vectors in thesuggests many fruitful things as mentioned on thisground state with SBS happens to deviate from thepaper.

aligned one and the same direction of spin vectors.

Namely, all the ground states withThe mechanism of SBS was first demonstrated

In order for the ground state manifesting atheoretically in quantum field theory:[1] In thedirectional order to be stable, this deviation of spin

system of infinitely many degrees of freedom

direction must be detected by other spin vectors

described by the Hamiltonian manifesting thewhich would then have some means of correcting

rotational symmetry, only one state is chosen

the wrong direction of the particular spin vector.

spontaneously among the infinitely degenerateThis requires the presence of certain interaction

ground states as a real ground state of the systemmechanism among the spin vectors in the region ofand the rotational symmetry of the system is broken

the ground state with SBS.

without recourse to any external environment.

mechanism any deviation of local spin from theAlthough SBS is a mechanism characteristic to thealigned one and the same direction of all the spin

system of infinitely many degrees of freedom

vectors can be corrected by the system itself.

With such a

described by quantum field theory, it can equally beThis mechanism is provided by a wave thatapplied to the system of finite but many degrees ofcan propagate over the whole of the ordered region

freedom. “Spontaneous” means the following fact:of the ground state with SBS. Because the spin

The rotational symmetry is broken not by imposing

vectors in the ordered region are aligned in one andthe same direction, a deviation of a spin vector isspherical plastic container of diameter 3.0 cm andcorrected but overcorrected resulting still in arotatable freely around the vertical axis. Thisdeviation, thus start oscillating around the aligned

magnetic compass was designed originally as andirection. Since all the spin vectors are strongly

automobile accessory, and we marked a visible

coupled, this oscillation is transferred to all the otherarrow on the rotator by luminous paint parallel tospin vectors successively. This is a correlation wave

the magnetic dipole of the ferrite. (Fig. 1)

propagating in an infinite range. Existence of sucha long-range correlation wave in the ground statewith SBS was demonstrated theoretically inquantum field theory, known today as the
Nambu-Goldstone theorem [1].

The long-range

correlation wave is often called a Nambu-Goldstone
mode.

A

well

known

exampleofthe

Nambu-Goldstone mode is the spin wave or magnon
in a ferromagnet.
Although SBS plays the most important role tounderstand theoretically various order creating
phenomena in nature, its concept is highly advanced
and requires a mathematical framework of modern
quantum field theory such as the Nambu-GoldstoneFig. 2 Picture of a typical ordered domain structure oftheorem.

350 magnetic compasses distributed in a dense

The absence of an intuitive visualdemonstration of SBS has been considered to be a

two-dimensional triangle lattice structure.

week point of modern advanced framework ofquantum field theory. One of the authors (YS)
developed a daily macroscopic device to demonstrate
SBS and Nambu-Goldstone modes: The device looks
simple, 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
15 Gauss at the maximum point on this device
floating and confined in oil in a transparent

Fig. 3 Picture of a typical ordered domain structure of
350 magnetic compasses distributed in a dense nematic
non-lattice structure of spin glass.
Machinery of the device is simple, too. We put
Fig. 1 Picture of a magnetic compass which has a

the 350 magnetic compasses densely one by one andplastic rotator of diameter 3.0 cm .

line by line as a crystal growth onto a rectangle tray

obtaining the two-dimensional triangle lattice (Fig.
2), and 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 ordereddomain in which all the magnetic dipoles point one
and the same direction independent to the Earth(A)magnetic field direction. We can see such a typicalmagnetic domain structure as in the ferromagnetic
domain structure in the typical theoretical model offerromagnetism. 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 weneed some considerable vibrational disturbance tothe system of magnetic compasses confined on therectangle tray, thus demonstrating intuitively therole of temperature in the ferromagnetic phasetransition.

By observing the formation of the(B)ordered domain structure in this device, one canunderstand visually the essential aspect of SBS
without recours to the teary advanced mathematics
of quantum field theory.

(C)
Fig. 5 Pictures showing the transformation of ordereddomains of the magnetic compasses.

The ordereddomains can be found at the triangle lattice arrangement(A). If the arrangement is transformed to the rectangle
lattice one by the quasi-static process, then the ordereddomains are maintained (B).

If the set of magnetic

Fig. 4 Picture showing no ordered domain structure ofcompasses is perturbed, then the ordered domains

350 magnetic compasses distributed in a strict

disappear (C).

two-dimensional

rectangle

latticestructure.

We put the 350 magnetic compasses also into theOsaka Museum of Science for their helpful advice

strict two-dimensional rectangle lattice, and foundand support. The second author likes to dedicate

no ordered domains just as was predicted

this note to the late Prof. Hiroomi Umezawa whotheoretically (Fig. 4).

We, however, found thatdeveloped a new world-view based on physics ofordered domains are impressed on the case ofSBS. This research was partly supported by therectangle lattice if a crystal structure of theScientific Research Fund of the Ministry ofmagnetic compasses is transformed from theEducation, Science, and Culture under grant

triangle lattice arrangement to the rectangle one

No.12914022.

by the quasi-static process (Fig.5A, Fig.5B). This isa metastable state familiar in such as supercooling

References

or supersaturation phenomena. The domains go[1] H. Umezawa, in: Advanced Field Theory - Micro,out and never come out, if we vibrate the set ofMacro, and Thermal Physics -, Americancompasses (Fig.5C).

Institute of Physics, New York, 1993.

Furthermore, if one disturbs a magnetic[2] K. De’Bell, A. B. Maclsaac, J. P. Whitehead,compass in the ordered domain by putting a small“Dipolar effects in magnetic thin films andmagnet closer by hand, propagation of the local

quasi-two dimensional systems” Rev. Mod. Phys.

directional disturbance to the whole of the ordered

72,(2000) 225.

domain can be observed. This demonstrates theNambu-Goldstone theorem, that is, the creation ofthe long-range correlation wave in the system withSBS. (Since a still picture is not adequate forshowing the propagation of the local directional
disturbance, that is, the Nambu-Goldstone mode,
we prepared a digitized motion picture (700 KB)
available on the World-Wide Web site:
http://www.sci-museum.kita.osaka.jp/news/graphic/mag.mpg

To create a successful demonstration ofmagnetic domain structure, it is not sufficient tomerely place many ordinary magnetic compasses ina tray. This is because typical small compasses
have a very small magnetic dipole moment, andconsequently their mutual magnetic interaction ismuch smaller than the Earth's ambient magnetic
field. Our demonstration solves this problem byusing 350 compasses with a very large magnetic
dipole moment, to ensure that their mutualinteraction is large compared to the Earth's field.

Acknowledgments
The first author (YS) thanks Prof. Takaki Shichiriof Osaka City University and his colleagues in the