Spontaneous Crack of symmetry

The concept of spontaneous crack of symmetry plays a big role in Physique of the particles and Physique of the condensed matter.

Definition

At high temperature (or high energy), the matter forms a gas or a plasma and this state have all symmetries of the equations describing the movement of the particles. At low temperature, the matter can be in a state which does not have all symmetries of the microscopic equations, but only one sub-group of the complete group of symmetry. This phenomenon is called spontaneous crack of symmetry. A concrete example of state of the matter having a spontaneous crack of symmetry is the state crystalline solid. A crystal is indeed invariant only under the action of one discrete group of symmetry including/understanding of the discrete translations, of the reflections and of rotations of 60°, 90°, 120°, 180° around plan or of particular axes, whereas the equation of Schrödinger which describes the movement of the electrons and of the cores which constitute this crystal is invariant under any translation, rotation or reflection.

There exist in physics of the condensed matter of numerous other examples of spontaneous cracks of symmetries.

Examples

In the magnetic systems (ferromagnetic, antiferromagnetic, ferrimagnetic), symmetry of rotation SO (3) of the magnetic moments and invariance by inversion of time are broken spontaneously.

In the nematic Liquid crystals, invariance by rotation SO (3) is also in one symmetry of rotation around an axis called director. If $\ vec {N}$ is an unit vector parallel with this axis, the transformation of $\ vec {N}$ into $- \ vec {N}$ does not change obviously pas la direction of this axis, and one can thus choose $n$ or $-n$ indifferently to describe the same nematic state. That constitutes an important difference with magnetism where the change of the sign of magnetization does not give again the initial thermodynamic state.

In smectite liquid crystals has, the molecules organize themselves in layers separated by a distance determined. However, inside the layers, the molecules do not have an order with long distance. In this example, the symmetry of translation is broken only in the perpendicular direction with the layers, and the symmetry of rotation is reduced to rotations around an orthogonal axis with the layers.

In crystalline metal alloys, one can observe ordered states where one of the atoms forming alloy occupies a site of the network preferentially. It results a lowering from it from symmetry compared to that of the state to high temperature in which the site could be occupied by an atom of one or the other species. In this case, the crack of symmetry reduces a discrete group of translation to under group, whereas in the preceding examples symmetry was broken in a continuous group.

Broken symmetry is not always associated with geometrical transformations. For example, in the superconductive and the superfluid , broken symmetry is a Symétrie of abelian gauge continuous U (1). It is also the case of the cracks of symmetries studied in physics high energies which correspond to cracks of symmetries for nonabelian groups.

Physical consequences of the cracks of symmetries

When a symmetry is broken spontaneously in systems physique, there exists a new parameter who is invariant under the action of the sub-group leaving invariant the state with broken symmetry but not under the action of the complete group leaving invariant the microscopic equations of the movement. This parameter is called the Paramètre of order and it measures the importance of the crack of symmetry. The orbit of this parameter of order under the action of the complete group of symmetry of gives the unit states physically not equivalents in which symmetry is broken. To give an example concrete, in the case of magnetism, the parameter of order is the vector magnetization, and the orbit parameter of order under the action of the group of symmetry is the sphere $S^2$ . The existence of a parameter of order is at the base of the theory of Pram of the transitions from phase. In a state with broken symmetry, there exist forces which tend to impose a uniform value parameter of order in all the system. This phenomenon is called generalized rigidity. The most banal example is the existence of a modulus of rigidity in a solid in addition to the module of compression. In the case of a superconductor, it is the existence of a generalized rigidity which makes it possible to explain the circulation of a current without dissipation. The generalized rigidity of the superconductor combined with the electromagnetic invariance of gauge is also responsible for the Meissner effect, like discussed it P.W. Anderson. In physics of the particles, a similar phenomenon, the phenomenon of Higgs, makes it possible to explain the formation of a mass for the particles of gauge (see it deliver C. Itzykson and J.B. Zuber).

The description of the states of a system having a crack of symmetry in terms of a parameter of order also allows to classify the defects which can appear in a system by means of the methods algebraic topology. Indeed, a state where the parameter of order varies in space can be described by means of an application of $\ Bbb {R} ^3$ in space quotient of the parameter of order. For example, an inhomogenous state of a magnetic system can be represented like an application of $\ Bbb {R} ^3$ in the sphere $S^2$ , and an inhomogenous state of a superconductor like an application of $\ Bbb {R} ^3$ in the sphere $S^1$ . It can exist stable defects of the parameter of order only if the application cannot be deformed continuously in a constant application. In a more precise way, the theory of homotopy makes it possible to affirm that if the group $\ pi_0$ is noncommonplace, the defects of the wall type are stable, if it group $\ pi_1$ is stable the defects of the line type are stable, and if the group $\ pi_2$ is not-commonplace, the defects of the type not are stable. In particular, like $\ pi_2 (S^2) = \ Bbb {Z}$ , a magnetic system can present defects of the type not, but like $\ pi_1 (S^2) =0$ , it cannot have defects of the line type. In the case of the superconductor, $\ pi_1 (S^1) = \ Bbb {Z}$ , therefore the lines are defects topologically stable. These defects correspond obviously to the vortices. The topological theory also allows to predict that these vortices have a quantified load.

Theorem of Goldstone

When broken symmetry is a continuous symmetry, there exists a theorem of with J. Goldstone according to which it must appear new excitations with low energy. Generally, these excitations are bosons. In the case of a solid, the Phonon S transverses (waves of shearing) can be regarded as bosons of Goldstone resulting from the crack of the continuous symmetry of translation. In the case of magnetism, the crack of symmetry of rotation involves the apparation of Magnon S with low temperature. In the case of the superfluid ones, the crack of symmetry U (1) results in the appearance of one phonon method with a superfluid density proportional to the superfluid density. In the case superconductors, P.W. Anderson showed that the Coulomb interaction with long range prevented appearance of the modes of Goldstone with low energy.

The theorem of Goldstone makes it possible to build basic theories energy describing only them bosons of Goldstone. These effective theories are discussed by C.P. Burgess.

Theorem of Mermin-Wagner-Hohenberg-Coleman

In 1966, NR. D. Mermin and H. Wagner established a theorem showing that a spontaneous crack of a symmetry continues was impossible in a two-dimensional system. A demonstration of this theorem, using only the inequality of Bogoliubov can be found in the book of C. Itzykson and J. Mr. Drouffe. In 1967, PC Hohenberg extended this theorem to superfluid and the superconductors. This theorem was reformulated by Sidney Coleman in 1973 within the framework of the quantum theory of the fields which showed that the hypothetical theory of the fields which would describe bosons of Goldstone in the case of a spontaneous crack of symmetry in dimension (1+1) could not satisfy the axioms of Wightman. The theorem of Mermin-Wagner-Hohenberg-Coleman has as consequence that the models O (NR) with NR > 1 in two dimensions cannot present order to long distance.

In the case NR = 2, there is the Modèle XY which has the transition from Berezinskii-Kosterlitz-Thouless, between a complete disorder at high temperature and an quasi-order with long distance at low temperature. For $N \ Ge 3$ , the model O (NR) is in a disordered phase at any temperature. In its version theory of the fields, the theorem of Mermin-Wagner-Hohenberg-Coleman makes it possible to affirm that an antiferromagnetic chain of spins cannot have a state of Néel even to the absolute zero.

Theorem of Elitzur

Up to now, symmetries which we discussed were only total symmetries. One can to wonder local symmetries whether, as symmetries of gauge can be also broken. A theorem due to S. Elitzur makes it possible to answer this question by the negative one.

References

C. Itzykson and J.B. Zuber, Quantum Field Theory (McGrawHill)
C. Itzykson and J. Mr. Drouffe, statistical Theory of the fields (CNRS-Intereditions)
P.W. Anderson, BASIC Concepts off Condensed Matter Physics (Addison-Wesley)
C.P. Burgess, year Ode to Effective Lagrangians
Id. , “'' Goldstone and Pseudo-Goldstone Bosons in Nuclear, Particle and Condensed-Matter Physics ''”

See too

Crack of symmetry
Mechanism of Higgs
Field of Higgs

Category: Physics of the particles Category: Statistical physics

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