Symmetry Breaking Induced Geometric Surfaces with Topological Curves in Quantum and Classical Dynamics of the SU(2) Coupled Oscillators

In the three‐dimensional (3D) transversely symmetric oscillator, there are plentiful degeneracies and gaps in the quantum energy spectrum as a function of the ratio of the transverse to longitudinal frequency. It is theoretically verified that while the SU(2) interaction destroys the original degeneracies, numerous new degeneracies and gaps emerge around the original degeneracies to form a similar fine energy spectrum. The classical trajectories at the emergent degeneracies are analyzed to be localized on the 3D parametric surfaces which are constituted by the topologically invariant curves in the transverse tomography. The quantum coherent states are exploited to develop the wave functions that correspond to the 3D geometric surfaces in classical dynamics. Furthermore, the wave structures of the stationary coherent states at small quantum numbers are explored and found to display peculiar patterns with symmetries related to classical trajectories.     

Atomic refinement with correlated solid-state NMR restraints

The orientation data provided by solid-state NMR can provide a great deal of structural information about membrane proteins. The quality of the information provided is, however, somewhat degraded by sign degeneracies in measurements of the dipolar coupling tensor. This is reflected in the dipolar coupling penalty function used in atomic refinement, which is less capable of properly restraining atoms when dipolar sign degeneracies are present. In this report we generate simulated solid-state NMR data using a variety of procedures, including back-calculation from crystal structures of alpha-helical and beta-sheet membrane proteins. We demonstrate that a large fraction of the dipolar sign degeneracies are resolved if anisotropic dipolar coupling measurements are correlated with anisotropic chemical shift measurements, and that all sign degeneracies can be resolved if three data types are correlated. The advantages of correlating data are demonstrated with atomic refinement of two test membrane proteins. When refinement is performed using correlated dipolar couplings and chemical shifts, perturbed structures converge to conformations with a larger fraction of correct dipolar signs than when data are uncorrelated. In addition, the final structures are closer to the original unperturbed structures when correlated data are used in the refinement. Thus, refinement with correlated data leads to improved atomic structures. The software used to correlate dipolar coupling and chemical shift data and to set up energy functions and their derivatives for refinement, CNS-SS02, is available at our web site.     

Non-relativistic Quantum Theory at Finite Temperature

We propose the non-relativistic finite temperature quantum wave equations for a single particle and multiple particles. We give the relation between energy eigenvalues, eigenfunctions, transition frequency and temperature, and obtain some results: (1) when the degeneracies of two energy levels are same, the transition frequency between the two energy levels is unchanged when the temperature is changed. (2) When the degeneracies of two energy levels are different, the variance of transition frequency at two energy levels is direct proportion to temperature difference.     

Degeneracies in trapped two-component fermi gases

We report on previously unobserved intersystem degeneracies in two-component equal-mass Fermi gases with interspecies zero-range interactions under isotropic harmonic confinement. Over the past 10 years, two-component Fermi gases consisting of n_{1} spin-up and n_{2} spin-down atoms with interspecies zero-range interactions have become a paradigm for modeling condensed matter systems, nuclear matter, and neutron matter. We show that the eigenenergies of the (n_{1}+1,n_{2}-1) system are degenerate with the eigenenergies of the (n_{1},n_{2}) system for any s-wave scattering length a_{s}, including infinitely large, positive, and negative a_{s}. The existence of the intersystem degeneracies is demonstrated explicitly for few-body systems with n_{1}+n_{2}=4, 5, and 6. The degeneracies and associated symmetries are explained within a group theoretical framework.     

Approximate degeneracies of zone boundary phonons in alkali halide crystals

We point out the existence of a pervasive pattern of near degeneracies of phonon frequencies in isobaric alkali halide crystals (NaBr, KCl, RbBr, CsI) which strongly suggests that their dynamical matrices are almost invariant under transformations which exchange positive and negative ions. We extend this hypothesis to a relation between phonon properties of “mirror” alkali halides in which the ions of one crystal are replaced by the oppositely charged isobaric ions of the other, such as RbCl and KBr. Experimental evidence supporting this can also be adduced. Similar near degeneracies universally occurring in NaCl structure alkali halides and alkaline earth oxides are also noted and a possible dynamical basis for understanding these suggested.     

Degeneracy from quadric cones for projective reconstruction from two views

Fundamental matrix encapsulates all the geometric information in two views, and it plays an important role in many applications of three-dimensional computer vision. However, some configurations have an inherent ambiguity, namely, degeneracy, and no matter how many matched image points are used, the fundamental matrix could not be determined uniquely. It is well-known that degeneracies occur when all the space points and both the camera centers belong to a ruled quadric. But it is not so well-known how great the degenerate degrees in different configurations are. In this paper, we discuss the degeneracies caused by a quadric cone and give the corresponding degenerate degrees. We parameterize all the points on a quadric cone by a twisted cubic lying on the cone, and obtain a parametric coefficient matrix of the equations for estimating the fundamental matrix. By analyzing the coefficient matrix, we get the rank of it, i.e. the degenerate degree of the given configuration. It gives a more intuitive degeneracy and the degenerate degrees of the configurations, and reveals the full picture of the degeneracies on a cone.     

Knots,BPS States,and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.     

Runge-Lenz vector, accidental SU(2) symmetry, and unusual multiplets for motion on a cone

We consider a particle moving on a cone and bound to its tip by 1/r or harmonic oscillator potentials. When the deficit angle of the cone divided by 2π is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some of the corresponding multiplets have fractional “spin” and unusual degeneracies.     

On the relativistic calculation of spontaneous emission

The Einstein A coefficients are calculated, in all cases of degeneracies of the Dirac transition currents, by means of the energy balance method.     

On the mass degeneracies in type I open superstring theory

We calculate some massive two-point amplitudes in type-I open superstring theory. The results suggest that the mass degeneracies are not maintained at the one-loop level.     

Three Solvable Matrix Models of a Quantum Catastrophe

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.     

Distribution of degeneracies in simple quantum systems

In some simple quantum mechanical systems, the degeneracy of typical energy levels grows as a power of the energy or size. We ask whether, after dividing out this average growth, there is a well defined probability distribution of scaled degeneracies in the limit of large size or energy. The answer is yes, for a free particle in a sphere or cube. For the sphere, the distribution of scaled degeneracies is shown to follow a circular law. For the cube, a numerical study shows that the distribution rises linearly for low values of the scaled degeneracy and decays exponentially for large values.     

Shubnikov de Haas oscillations in n-type inversion layers on (110) and (111) surfaces of Si

Oscillatory magnetocunductance of electrons on (110) and (111) surfaces of Si have been observed. The ground-state degeneracies were determined and possible energy level schemes are proposed.     

Highly Efficient Transfer of Quantum State and Robust Generation of Entanglement State Around Exceptional Lines

Exceptional points (EPs) are non-Hermitian degeneracies or branch points where eigenvalues and their corresponding eigenvectors coalesce. Due to the complex non-trivial topology of Riemann surfaces associated with non-Hermitian Hamiltonians, the dynamical encirclement or proximity of EPs in parameter space has been shown to lead to topological mode conversions and some novel physical phenomena. In fact, degeneracies can also form continuous line geometries, which are called exceptional lines (ELs). The problem is whether the state transfer around the ELs can show different characteristics from the EPs, which is less explored. Here, novel properties of quantum state transfer around the ELs based on a quantum walk platform are explored. It is found that the evolutionary state around the ELs is independent of the initial state and evolution direction, and the transfer of quantum state is more efficient than the case around the EPs. Furthermore, based on such a property, an entangled state generation insensitive to the incident state is realized experimentally. The work opens up a new way for the application of non-Hermitian physics in the field of quantum information.     

Non-abelian spin-singlet quantum Hall states: wave functions and quasihole state counting

We investigate a class of non-abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain (k+1)-body interaction Hamiltonians. The k=1 cases are the familiar Halperin abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k>1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states are non-degenerate, while the quasihole excitations over these states show characteristic degeneracies, which give rise to non-abelian braid statistics. Using conformal field theory methods, we derive counting rules that determine the degeneracies in a spherical geometry. The results are checked against explicit numerical diagonalization studies for small numbers of particles on the sphere.     

Geometric phases and multiple degeneracies in harmonic resonators

In a recent experiment Lauber et al. have deformed cyclically a microwave resonator and have measured the adiabatic normal-mode wave functions for each shape along the path of deformation. The nontrivial observed cyclic phases around a threefold degeneracy were accounted for by Manolopoulos and Child within an approximate theory. However, open-path geometrical phases disagree with experiment. By solving exactly the problem, we find unsuspected extra degeneracies around the multiple one that account for the measured phase changes throughout the path. It turns out that proliferation of additional degeneracies around a multiple one is a common feature of quantum mechanics.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries: I. One,Two, and Three Particles

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the N particle case.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries. II. N Particles

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly–unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.     

Cooperative phenomena in superionic conductors within generalized lattice gas models

We study the equilibrium properties of a generalized lattice gas as applied to the cooperative phenomena of superionic conductors. For a model describing interacting Frenkel defects with various degeneracies we discuss the competing effects of degeneracy and interaction. The phase diagram is computed within mean field theory and the possibility of first order, second order and continuous transitions is shown. For a one dimensional model with two sites and different degeneracies we derive explicit exact results for the site occupancies as functions of temperature. Finally we consider the effect of long range (coulomb) interactions for the site occupancy correlation in a partly filled ionic channel. This effect is shown to be relevant and not reproducible by effective short range interactions.     

Eigenspaces of symmetric graphs are not typically irreducible

We construct rich families of Schrödinger operators on symmetric graphs, both quantum and combinatorial, whose spectral degeneracies are persistently larger than the maximal dimension of an irreducible representation of the symmetry group.