Wigner distributions for finite dimensional quantum systems: An algebraic approach

We discuss questions pertaining to the definition of `momentum', `momentum space', `phase space', and `Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of `momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.Comment: Latex, 13 page

Comment on "Scaling of the quasiparticle spectrum for d-wave superconductors"

In a recent Letter Simon and Lee suggested a scaling law for thermodynamic and kinetic properties of superconductors with lines of gap nodes. However their crossover parameter between the bulk dominated regime and the vortex dominated regime is different from that found in our paper (N.B. Kopnin and G.E. Volovik, JETP Lett., {\bf 64}, 690 (1996); see also cond-mat/9702093). We discuss the origin of the disagreement.Comment: submitted to Physical Review Letters as "Comment" to the paper by S.H. Simon and P.A. Lee, Phys. Rev. Lett., 78 (1997) 1548 (cond-mat/9611133

Phase-space descriptions of operators and the Wigner distribution in quantum mechanics II. The finite dimensional case

A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.Comment: 14 pages, typos corrected, para and references added in introduction, submitted to Jour. Phys.

Topological and geometric decomposition of nematic textures

Directional media, such as nematic liquid crystals and ferromagnets, are characterized by their topologically stabilized defects in directional order. In nematics, boundary conditions and surface-treated inclusions often create complex structures, which are difficult to classify. Topological charge of point defects in nematics has ambiguously defined sign and its additivity cannot be ensured when defects are observed separately. We demonstrate how the topological charge of complex defect structures can be determined by identifying and counting parts of the texture that satisfy simple geometric rules. We introduce a parameter called the defect rank and show that it corresponds to what is intuitively perceived as a point charge based on the properties of the director field. Finally, we discuss the role of free energy constraints in validity of the classification with the defect rank.Comment: 16 pages, 5 figure

Testing for Majorana Zero Modes in a Px+iPy Superconductor at High Temperature by Tunneling Spectroscopy

Directly observing a zero energy Majorana state in the vortex core of a chiral superconductor by tunneling spectroscopy requires energy resolution better than the spacing between core states $\Delta^2/eF$. We show that nevertheless, its existence can be decisively tested by comparing the temperature broadened tunneling conductance of a vortex with that of an antivortex even at temperatures $T >> \Delta^2/eF$.Comment: 5 pages, 4 figure