Inversion symmetry, breaking

Sai N, Meyer B, Vanderbilt D (2000) Compositional inversion symmetry breaking in ferroelectric perovskites. Phys Rev Lett 84 5636... 

Kimura, T., Lashley, J.C., Ramirez, A.R Inversion-symmetry breaking in the noncollinear magnetic phase of the triangular-lattice antiferromagnet CuEe02. Rhys. Rev. B 73,... 

Fang, W., Kong, J., Dresselhaus, M.S., and Kalbac, M. (2013) Mass-related inversion symmetry breaking and phonon self-energy renormalization in isotopically labeled AB-stacked bilayer graphene. Sci. Rep., 3, 2061. 

We start with some elementary information about anisotropic intermolec-ular interactions in liquid crystals and molecular factors that influence the smectic behaviour. The various types of molecular models and commonly accepted concepts reproducing the smectic behaviour are evaluated. Then we discuss in more detail the breaking of head-to-tail inversion symmetry in smectic layers formed by polar and (or) sterically asymmetric molecules and formation of particular phases with one and two dimensional periodicity. We then proceed with the description of the structure and phase behaviour of terminally fluorinated and polyphilic mesogens and specific polar properties of the achiral chevron structures. Finally, different possibilities for bridging the gap between smectic and columnar phases are considered. 

Materials that have a nonzero second-order susceptibility will produce light at twice the incident frequency. The magnitude of this effect is small, and has been a practical consideration only since the advent of lasers. If the symmetry of a crystal or other medium is such that it has a center of inversion, no SHG effect will be observed. However, surfaces by their very nature break this inversion symmetry. Hence, an SHG signal may arise at the electrode-solution interface even though both bulk phases may be considered centrosymmetric [66], The magnitude of the SHG signal is sensitive to surface conditions (e.g., electrode potential, ionic or molecular adsorption, etc.). Surface spectroscopy is also feasible since the SHG signal will be enhanced if either the incident frequency (to) or SHG (2co) corresponds to an electronic absorption of a surface species [66]. 

One of the most important theoretical contributions of the 1970s was the work of Rudnick and Stern [26] which considered the microscopic sources of second harmonic production at metal surfaces and predicted sensitivity to surface effects. This work was a significant departure from previous theories which only considered quadrupole-type contributions from the rapid variation of the normal component of the electric field at the surface. Rudnick and Stern found that currents produced from the breaking of the inversion symmetry at the cubic metal surface were of equal magnitude and must be considered. Using a free electron model, they calculated the surface and bulk currents for second harmonic generation and introduced two phenomenological parameters, a and b , to describe the effects of the surface details on the perpendicular and parallel surface nonlinear currents. In related theoretical work, Bower [27] extended the early quantum mechanical calculation of Jha [23] to include interband transitions near their resonances as well as the effects of surface states. 

Fig. 3. Conventional scheme of consequent temperature-dependent symmetry breakings triggered by JT or pseudo JT effects. G = R(3)/n(N)Ci is the symmetry of an atomic gas, where R(3) is the group of rotations of the free atom, 77(A) is the group of permutation and Q is inversion the primed values have the same meaning for the gas of molecules. Crystal I and crystal II denote two crystal phases with decreasing symmetry, respectively. QM, gL, Gc> Gc> and QC" are the separate and independent coordinates of symmetry breaking, while the temperature scale is in common.

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

As successful as the loop algorithm is, it is restricted - as most classical cluster algorithms - to models with spin inversion symmetry. Terms in the Hamiltonian which break this spin-inversion symmetry, such as a magnetic field, are not taken into account during loop construction. Instead they enter through the acceptance rate of the loop flip, which can be exponentially small at low temperatures. 

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