An Example of Symmetry Breaking in Nonlinear Optics
Symmetry breaking is becoming one of the most studied phenomena common in many areas of physics. The notion extends from gauge field theories1 to solid-state physics2 and beyond. What is usually meant by symmetry breaking is the following. The equation governing the behaviour of a given system has a number of symmetry properties but the states of the system do not exhibit all of these symmetries. A well known example is provided by ferromagnetic materials2: above the Curie temperature the materials have no magnetization and the states exhibit all symmetry properties of the Hamiltonian (rotational symmetry in particular). Below the Curie temperature the materials acquire a finite magnetization vector which points in a well-defined spatial direction. The rotational symmetry is obviously broken in this case. In some sense, however, the system still remembers its symmetry since the direction of the magnetization is completely random. It is not predetermined by the system itself but by external perturbations which are always present in a realistic system. Furthermore, in most cases symmetry breaking occurs in some abstract space (Hilbert space of the states or phase space of the system, etc.) with only a few exceptions, of which ferromagnetism is again one of the most significant examples. Here the breaking of the symmetry takes place in real space. Very few other examples are known where symmetry breaking occurs in the three dimensional coordinate space.
KeywordsSymmetry Breaking Dispersion Curve Nonlinear Medium Eigenvalue Equation Total Internal Reflection
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