We extend the Landau theory of bent-core mesophases and d-wave high-Tc superconductors by considering additional secondary pseudo-proper order parameters. These systems exhibit a remarkable analogy relating their symmetry groups, lists of phases, and an infinite set of physical tensors. This analogy lies upon an internal dual structure shared by the two theories. We study the dual operator transforming rotations into translations in liquid crystals, and gauge symmetries into rotations in superconductors. It is used to classify the bent-core line defects, and to analyze the electronic gap structure of lamellar d-wave superfluids. Full article

Statistical systems, in which spontaneous symmetry breaking can be accompanied by spontaneous local symmetry restoration, are considered. A general approach to describing such systems is formulated, based on the notion of weighted Hilbert spaces and configuration averaging. The approach is illustrated by the example of a ferroelectric with mesoscopic fluctuations of paraelectric phase. The influence of the local symmetry restoration on the system characteristics, such as sound velocity and Debye-Waller factor, is discussed. Full article

Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these equations in their initial and their simplified form are compared, showing in particular that this transformation partially prevents a clear understanding and a full application of the (physically relevant) notion of the so-called step up/down operators. These operators are shown to be recursion operators, related to the Lie point symmetries of the equations, which are also carefully discussed. Full article

The emphasis in the published literature has mostly been on symmetry as the critical source for beauty judgment. In fact, both symmetry and asymmetry serve as highly aesthetic sources of beauty, whether the context is perceptual or conceptual. The human brain is characterized by symbolic cognition and this type of cognition facilitates a range of aesthetic reactions. For example, both art and natural scenery contain asymmetrical elements, which nevertheless render the whole effect beautiful. A further good case in point is, in fact, human faces. Normally, faces are structurally left-right symmetrical content-wise but not size-wise or function-wise. Attractiveness has often been discussed in terms of content-wise full-face symmetry. To test whether or not attractiveness can be gleaned only from the presence of left-right full-faces we tested half faces. Three separate groups of participants viewed and rated the attractiveness of 56 full-faces (women’s and men’s), their 56 vertical left hemi-faces and 56 vertical right hemi-faces. We found no statistically significant differences in the attractiveness ratings of full- and hemi-faces (whether left or right). Instead, we found a strong and significant positive correlation between the ratings of the hemi- and full-faces. These results are consistent with the view that the underpinning of human facial beauty is complex and that bilateral symmetry does not constitute a principle factor in beauty assessment. We discuss that the highly evolved human brain, compared to other animals, as well as symbolic and abstract cognition in humans enable a wide variety of aesthetic reactions. Full article

Information optimization is a centerpiece phenomenon in the universe. It develops from simplicity, then continuously breaks symmetry and cycles through instability to progress to increasingly dense nodes of complexity and diversity. Intelligence has arisen as the information optimization node with the greatest complexity. A contemporary imbalance is presented in that exponentially growing technology could be poised as a potential sole successor to human intelligence. A complex dynamical system is emerging in response, the engineering of life into technology. Numerous network elements are developing which could self-organize into the next node of symmetry, a phase transition in intelligence. Full article

This feature article gives a general introduction to the phenomenon of supramolecular chirogenesis using the most representative examples of different chirogenic assemblies on the basis of ethane-bridged bis-porphyrinoids. Supramolecular chirogenesis is based upon a smart combination of supramolecular chemistry and chirality sciences and deals with various aspects of asymmetry induction, transfer, amplification, and modulation. These chiral processes are governed by numerous noncovalent supramolecular forces thus allowing a judicious, mechanistic, and dynamic control by applying a variety of internal and external influencing factors. Currently, supramolecular chirogenesis is widely used in different fields of fundamental and applied branches of science and modern technology, touching on such important issues as origin of chirality on the Earth, asymmetry sensing, enantioselective catalysis, nonlinear optics, polymer and materials science, pharmacy and medicine, nanotechnology, molecular and supramolecular devices, chiral memory, absolute configuration determination, etc. Full article

A mechanistic study of the bimolecular nucleophilic substitution (S_N2) reaction for halomethane CH₃X (X = Cl, Br, or I) is approached by using symmetry principles and molecular orbital theory. The electrophilicity of the functionalized sp³–carbon is attributable to a 2p-orbital-based antibonding MO along the C–X bond. This antibonding MO, upon accepting an electron pair from a nucleophile, gives rise to dissociation of the C–X bond and formation of a new Nuc–C bond. Correlations are made between the molecular orbitals of reactants (Nuc^_- and CH₃X) and products (NucCH₃ and X^_-). Similar symmetry analysis has been applied to mechanistic study of the bimolecular b-elimination (E2) reactions of haloalkanes. It well explains the necessity of an anti-coplanar arrangement of the C_α–X and C_β–H bonds for an E2 reaction (anti-elimination). Having this structural arrangement, the bonding C_α–X (σ_C-X) and antibonding C_β–H (σ_C-H*) orbitals become symmetry–match. They can partially overlap resulting in increase in electron density in σ_C-H*, which weakens and polarizes the C_β–H bond making the β-H acidic. An E2 reaction can readily take place in the presence of a base. The applications of symmetry analysis to the S_N2 and E2 reactions represent a new approach to studying organic mechanisms. Full article

Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness by considering the time reversal operator, as far as the Hamiltonian is invariant with respect to the inversion of the direction of motion. This can explain the degeneracy of pairs of vibrational and rotational states spanning the so-called separably degenerate irreducible representations, in symmetric tops of low symmetry, and Kramers degeneracy in odd electron molecules in the absence of magnetic fields. An extension with account of time reversal is also useful to determine relative phase conventions on vibration-rotation wavefunctions, which render all vibration-rotation matrix elements real. An extension of a molecular symmetry group may be required for molecules which can attain different geometries by large amplitude periodical motions, if such motions are hindered and are not completely free. Special cases involving the internal rotation are discussed in detail. It is observed that the symmetry classification of vibrational modes involving displacements normal to the internal rotation axis is not univocal, but can be done in several ways, which actually correspond to different conventions on the separation of vibration and internal rotation in the adopted basis functions. The symmetry species of the separate vibrational and torsional factors of these functions depend on the adopted convention. Full article

I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the Planck-scale/quantum-gravity realm. I also give a true/false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the қ-Poincaré Hopf algebra, a scenario with deformed Poincaré symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators. Full article

Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface in spacetime). Such a time-symmetric perspective of BCs, as applied to classical fields, leads to interesting parallels with quantum theory. This paper will map out some of the consequences of this counter-intuitive premise, as applied to covariant classical fields. The most notable result is the contextuality of fields constrained in this manner, naturally bypassing the usual arguments against so-called “realistic” interpretations of quantum phenomena. Full article

The NICS_zz-scan curves of aromatic organic, inorganic and “all-metal” molecules in conjunction with symmetry-based selection rules provide efficient diagnostic tools of the σ-, π- and/or double (σ + π)-aromaticity. The NICS_zz-scan curves of σ-aromatic molecules are symmetric around the z-axis, having half-band widths approximately less than 3 Å with the induced diatropic ring current arising from T_x,y-allowed transitions involving exclusively σ-type molecular orbitals. Broad NICS_zz-scan curves (half-band width approximately higher than 3 Å) characterize double (σ + π)-aromaticity, the chief contribution to the induced diatropic ring current arising from T_x,y-allowed transitions involving both σ- and π-type molecular orbitals. NICS_zz-scan curves exhibiting two maxima at a certain distance above and below the molecular plane are typical for (σ + π)-aromatics where the π-diatropic ring current overwhelms the σ-type one. In the absence of any contribution from the σ-diatropic ring current, the NICS_zz(0) value is close to zero and the molecule exhibits pure π-aromaticity. Full article

In this paper, we study the influence of hard supersymmetry breaking terms in a N = 1, d = 4 supersymmetric model, in S¹ × R³ spacetime topology. It is shown that when the radius of the compact dimension is large supersymmetry is unbroken, and dynamically breaks as the radius decreases. We point out that this resembles the inverse symmetry breaking of continuous symmetries at finite temperature (however, in the case of supersymmetry, the role of the temperature is played by the compact dimension’s radius). Furthermore, we also find a universality in the dependence of the critical length L_c as a function of a coupling g₃, after comparing all cases. Full article

This work expands the results and derivations presented in a recent letter. It is argued that symmetry breaking Hartree-Fock (HF) solutions of a simple model of the Cu-O planes in La₂CuO₄, are able to describe the insulator and antiferromagnetic characters of this material. Then, this classical primer of a Mott insulator is alternatively obtained here as an exact Slater insulator within the simplest of the first principles schemes. Moreover, pseudogap HF states are also predicted. The maximal energy gap of 100 meV over the Fermi surface of this wavefunction, reasonably well matches the ARPES upper pseudogap measurements for La₂CuO₄ in the zero doping limit. These surprising results followed after eliminating spin and crystal symmetry constraints usually imposed on the HF orbitals. The discussion helps to clarify the role of the antiferromagnetism and pseudogaps in the physics of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures for the description of the transition metal oxides. Full article

Through-space isotropic NMR shielding values of a perpendicular diatomic hydrogen probe moved in a 0.5 Å grid 2.5 Å above several polycyclic aromatic/antiaromatic ring and aromatic/aromatic hydrocarbons were computed with Gaussian 03 at the GIAO HF/6-31G(d,p) level. Combinations of benzene fused with cyclobutadiene, with the tropylium ion, and with the cyclopentadienyl anion were investigated. Subtraction of the isolated H₂ isotropic value gave shielding increments(Δσ), which, when plotted against Cartesian coordinates of the probe over each hydrocarbon, gave representations of three-dimensional isotropic shielding increment surfaces. The results are related to the degree of bond length alternation, the extent of electron delocalization, and (for the ions) the NPA charge distribution. The shielding increment data are compared to NICS(1) values computed at the same level; both indicate the degree of aromaticity or antiaromaticity of the component rings. Full article

Recent results on the optical absorption and symmetry of the Np(V) complexes with dicarboxylate and diamide ligands are reviewed. The importance of recognizing the “silent” feature of centrosymmetric Np(V) species in analyzing the absorption spectra and calculating the thermodynamic constants of Np(V) complexes is emphasized. Full article

Cyclooctatetraene (COT), the first 4nπ-electron system to be studied, adopts an inherently nonplanar tub-shaped geometry of D_2d symmetry with alternating single and double bonds, and hence behaves as a nonaromatic polyene rather than an antiaromatic compound. Recently, however, considerable 8π-antiaromatic paratropicity has been shown to be generated in planar COT rings even with the bond alternated D_4h structure. In this review, we highlight recent theoretical and experimental studies on the antiaromaticity of hypothetical and actual planar COT. In addition, theoretically predicted triplet aromaticity and stacked aromaticity of planar COT are also briefly described. Full article

We report the mechanism and scope of “preferential enrichment”, which is an unusual symmetry-breaking enantiomeric resolution phenomenon that is initiated by the solvent-assisted solid-to-solid transformation of a metastable polymorphic form into a thermodynamically stable one during crystallization from the supersaturated solution of certain kinds of racemic mixed crystals (i.e., solid solutions or pseudoracemates) composed of two enantiomers. The mechanism can well be interpreted in terms of a symmetrybreaking complexity phenomenon involving multistage processes that affect each other. Full article

When liquid molecules are confined in a narrow gap between smooth surfaces, their dynamic properties are completely different from those of the bulk. The molecular motions are highly restricted and the system exhibits solid-like responses when sheared slowly. This solidification behavior is very dependent on the molecular geometry (shape) of liquids because the solidification is induced by the packing of molecules into ordered structures in confinement. This paper reviews the measurements of confined structures and friction of symmetric and asymmetric liquid lubricants using the surface forces apparatus. The results show subtle and complex friction mechanisms at the molecular scale. Full article

Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators are discussed. Next, group theoretical approach to generating functions that have unitarity are discussed. Full article