The geometry of spontaneous symmetry breaking

Structure and dynamics of Xn-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory

“…Trivial stratum consists of zero vector only and has the whole G as the isotropy group. Abud-Sartori formulation of Michel's theorem states that, under some general conditions, G-invariant function V will have stationary point on every connected component of every maximally peripheral stratum [9]. We apply this theorem to few examples.…”

“…We will use Abud-Sartori formulation of Michel's theorem [7][8][9]. Stationary points of V are points for which gradient of V is equal to zero.…”

“…Another advantage is that the problem of finding minima of V becomes another case of the spontaneous symmetry breaking phenomenon. This allows us to use the Abud-Sartori-Michel theorem [7][8][9][10] to discuss some stationary points of V. In the next part of the paper we will use symmetries we found to derive lowest order approximate expressions for the function E and to discuss possible shapes of some molecular types that correspond to minima of V. Already in this simplest approximation one successfully obtains some of molecular shapes that exist in nature. Our approach allows finding approximate relations between bond lengths in a molecule and its vibration frequencies.…”

Symmetry Based Fundamentals of Carbon Nanotubes

“…Therefore, the rolled up configuration is not stable, and should be relaxed (slightly changed) to achieve the local minimum of the total energy. Relaxation cannot change the symmetry: according to the famous topological theorem [10], the extremes of the energy functional correspond to the most symmetric configurations. Consequently, equations (2.1) and (2.9) refer to the relaxed structures as well, only the values of D, ϕ 000 , z 000 and a differ.…”

“…Irrespectively of v, according to the topological argument [10], the optimal relative position is among the maximally symmetric ones. In fact, the symmetry group of W@W contains only those transformations which simultaneously leave both walls invariant.…”