Symmetry breaking, definition

Kozlowski, P. M., Rauhut, G., Pulay, P., 1995, Potential Symmetry Breaking, Structure and Definite Vibrational Assignment for Azulene Multiconfigurational and Density Functional Results , J. Chem. Phys., 103, 5650. 

Many problems appear to be ripe for a more quantitative discussion. What is the error involved in the introduction of unstable states as asymptotic states in the frame of the 5-matrix theory 16 What is the role of dissipation in mass symmetry breaking What is the consequence of the new definition of physical states for conservation theorems and invariance properties We hope to report soon about these problems. We would like, however, to conclude this report with some general remarks about the relation between field description and particles. The full dynamical description, as given by the density matrix, involves both p0 and the correlations pv. However, the particle description is expressed in terms of p (see Eq. (50)). Now p has only as many elements as p0. Therefore the... 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

The fact that states with definite parities are almost never observed experimentally cannot be explained alone by the persistence of mixed-parity states. Obviously, symmetry-breaking phenomena are in operation whether or not external perturbations are present. For species in condensed media, symmetry breaking is brought about by intermolecular interactions. In the extreme case of solids, these interactions are so strong that it is proper... 

Another example is the case of the He " (section 1.2.2). Here, the PES has a definite minimum when symmetry constraints are imposed (see Figure 6). However, when the molecule is allowed to vibrate, thereby breaking its symmetry, the minimum turns into a saddle. The same result is obtained with the molecule OL1H2 whose C2v local minimum also turns into a saddle upon symmetry breaking (see section 4.2), as well as with FLiH2 (section 4.3). 

The usual theoretical description of a BE condensate is given in terms of a macroscopic wavefunction with a definite phase, the latter supposedly being produced by a process known as spontaneous symmetry breaking . A definite phase for each member of a pair of condensate wavefimctions is necessary for interference to occur when the condensates overlap. Justification of this description, which is approximate for a finite munber of atoms, has been the subject of much theoretical work (J). 

The most important consequence of the breakdown of the B-O approximation is indisputably the Jahn-Teller (J-T) effect [7], where the structure defined on the basis of this approximation does not in fact hold. The important role of the J-T effect is emphasized in Bersuker s book [8] Moreover, since the J-T effect has been shown to be the only source of spontaneous distortion of high-symmetry configurations, we come to the conclusion that the J-T effect is a unique mechanism of all the symmetry breakings in condensed matter. It is of course well known that problems related to the definition of crystallic structure in solid-state physics are mostly ignored assuming only BO structures. Nevertheless it is often criticised by scientists dedicated to studies of the J-T effect. For instance the proper understanding of superconductors should be evidently based on a solution of the non-adiabatic problem but the impact on the crystallic structure is neither reflected in the Frohlich Hamiltonian [9,10] nor in the BCS theory [11]. 

By definition, cell mitosis results in two daughter cells, identical to each other and to the parent cell. So in order to break symmetry, bit 1 in the state vector is set to 1 in one cell, while it is set to the reverse (0) in the other cell. The first 3 bits of the state vector are prespecified as the ones indicating cell type — in reverse. For example, a cell with state vector = 001110 has type 001-reversed, which is 100 or 4 (in decimal). This entails that the first division inevitably results in two cells of different types with state vectors 000100 and 100100, respectively. 

At a conical intersection, the branching plane is invariant through any unitary transformation within the two electronic states and any such combination of degenerate states is still a solution. Thus, the precise definition of the two vectors in (A.9) or (A. 11) is not unique and depends on an arbitrary rotation within the space of the Cl coefficients (i.e., between the generalized crude adiabatic states), unless the states have different symmetries (then xi is totally symmetric and X2 breaks the symmetry). 

In the latter case, one has to be aware of solutions with broken spatial symmetry. This problem arises also in NCSDFT the (initial) symmetry of a system, as described by a scalar Hamiltonian, is destroyed by the vector field term proportional to as, which, similarly to an external magnetic field, reduces the spatial symmetry of the one-electron Hamiltonian. In spin-polarized calculations including SO interaction, the conventional collinear approach, where only one component of the spin-density s = Tr a p) is used in the definition of the xc energy functional, has the major drawback of breaking the spatial symmetry of the energy functional [18,64]. 

---