Multiseparability and superintegrability in three dimensions

In complex two-dimensional Euclidean space, the Hamilton-Jacobi or Schrödinger equation with a given “nondegenerate” potential is maximally superintegrable if and only if it is separated in more than one coordinate system. A similar statement for three dimensions is not known. In this paper, a start will be made on this problem by investigating the known separable Hamilton-Jacobi and Schrödinger systems to find those that are superintegrable.     

On the superintegrability of TTW model

The superintegrability of so-called Tremblay–Turbiner–Winternitz (TTW) model has been conjectured on the basis of the fact that all its trajectories are closed. This conjecture has been proven using the method based on solving the partial differential equations for two functions having the same Poisson bracket with the Hamiltonian. In the present short Letter we show that superintegrability of TTW model can be established by using well-known elegant techniques of analytical mechanics. Moreover, the resulting expression (after an appropriate ordering) can be generalized to the quantum-mechanical case.     

Lie symmetries and superintegrability in quantum mechanics

Starting from the structure of the higher order Lie symmetries of the Schrödinger equation in the Euclidean plane E2, we establish, in the case of first-and second-order symmetries, the relations between separation of variables and superintegrable systems in quantum mechanics.     

Equilibrium polymerization in two dimensions

A mean-field theory of equilibrium polymerization on a square lattice is presented. A novel feature of the model is the treatment of the whole range of rigidity of polymers (from flexible to stiff) as well as a simultaneous account of both polymer rings and excluded volume. As a result a line of tricritical points is found.     

Non-abelian bosonization in two dimensions

A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.Supported in part by NSF Grant PHY-80-19754     

Quantum matrices in two dimensions

Quantum matrices in two dimensions, admitting left and right quantum spaces, are classified: they fall into two families, the 2-parametric family GL_p,q(2) and a 1-parametric family GL _inf ^sup J(2). Phenomena previously found for GL_p,q(2) hold in this general situation: (a) powers of quantum matrices are again quantum and (b) entries of the logarithm of a two-dimensional quantum matrix form a Lie algebra.     

Classical molecules in two dimensions

The ground-state configurations and spectrum of two parallel two-dimensional classical atoms are obtained as a function of the inter-atomic distance ( ). The classical particles are confined by parabolic potentials and repel each other through a Coulomb potential. As a function of we find several configurational transitions which are of first or second order. For first- (second-) order transitions the first (second) derivative of the energy with respect to is discontinuous, the radial position of the particles changes discontinuously (continuously) and the frequency of the eigenmodes exhibit a jump (softening of a mode). In the limit of an infinite number of electrons the Wigner bilayer system is recovered which moves through five different stable crystalline phases as a function of . For unequal strength of parabolic confinement we find that the number of configurational transitions increases.     

Quantum censorship in two dimensions

It is pointed out that increasingly attractive interactions, represented by partially concave local potential in the Lagrangian, may lead to the degeneracy of the blocked, renormalized action at the gliding cutoff scale by tree-level renormalization. A quantum counterpart of this mechanism is presented in the two-dimensional sine-Gordon model. The presence of Quantum Censorship is conjectured which makes the loop contributions pile up during the renormalization and thereby realize an approximate semiclassical effect.     

Non-translation-invariant states in two dimensions

We construct a class of models with translation-invariant interaction for which in dimension two there already exist non-periodic Gibbs states at low temperatures.     

Dilaton gravity in two dimensions

The study of general two-dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade, new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject.     

Anomalous hydrodynamics in two dimensions

A new approach is presented to discuss two-dimensional hydrodynamics with gauge and gravitational anomalies. Exact constitutive relations for the stress tensor and charge current are obtained. Also, a connection between response parameters and anomaly coefficients is discussed. These are new results which, in the absence of the gauge sector, reproduce the results found by the gradient expansion approach.     

Cooperative dynamics in two dimensions

We report results from molecular dynamics simulations of cooperative motion in a quasi-two-dimensional system of colloid particles. We find that the onset of the deviation of the single-particle displacement distribution from Gaussian form starts in the liquid phase and extends, with increasing magnitude, through the hexatic phase into the crystalline phase. The time for which the deviation is maximum increases exponentially with the density. As the density increases toward the hexatic phase a third dynamical relaxation mode emerges. We argue that the collective motion is generated by superpositions of instantaneous normal mode vibrations, with lifetimes that increase with the density, along paths with strong bond-orientation correlation.     

Critical interface in two dimensions

The interface separating two-dimensional phases close to a critical point is observed in the ferroelectric BaZr_xTi_1−xO₃ by transmission electron microscopy. The temperature dependence of the interfacial thickness is determined and compared with theoretical predictions.