We study 3d O(N ) symmetric scalar field theories using Polchinski's renormalisation group. In the infinite N limit the model is solved exactly including at strong coupling. At short distances the theory is described by a line of asymptotically safe ultraviolet fixed points bounded by asymptotic freedom at weak, and the Bardeen-Moshe-Bander phenomenon at strong sextic coupling. The Wilson-Fisher fixed point arises as an isolated low-energy fixed point. Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions. We substantiate a duality between Polchinski's and Wetterich's versions of the functional renormalisation group, also showing that that eigenperturbations are identical at any fixed point. At a critical sextic coupling, the duality is worked out in detail to explain the spontaneous breaking of scale symmetry responsible for the generation of a light dilaton. Implications for asymptotic safety in other theories are indicated. Contents
Modern heterostructure devices as esgc heterojunctions, sing.le quantum wells, and s+erlattices are often isotropic in one plane s o that effectively a onedimensional problem remains only. Therefore exact solutions of the one-dimensional SchrBdinger equation and their properties are of fundamental interest. In recent times the continuum states play an essential role, especially in resonant tunneling structures /1 t o 4/. In these calculations the infinite space is usually preferred rather than the large box quantization because the handling of the former one is easier. But, on the other hand, the powerful results concerning (regular ) Sturm-Liouville problems /5/ a r e not applicable, and one has t o resort to the more elaborate theories of Weyl, Titchmarsh, and Kodaira /6 t o 8/. But the additional suppositions on the potential (continuity o r absolute integrability) often made in these theories are sometimes not satisfied by the model systems which are studied in the applications mentioned above.Because there were controversial discussions regarding the normalization, orthogonality, and completeness of eigenfunctions in the infinite space /2, 4, 9/ we will treat these problems for a finite step potential explicitly. The restriction t o this potential is caused by brevity; a generalization t o more complicated quantum well o r quantum b a r r i e r structures and/or the inclusion of Ben Daniel-Duke Hamiltonians /lo/ with a stepwise constant potential is straightforward. (For instance, one can prove completeness for the potential (6) of /4/ with only slight modifications of the procedure described in the following.) F o r potentials with bound states one has t o take into account these onea in the contour integration (see below) in such a way as in /ll/-Our model system is Hy&)= %*(d 1) PSF 327, DDR-6300 Ilmenau, GDR.
During the last few years an increasing number of ideas concerning new heterostructure devices in micro-and optoelectronics was published (see, e. g. 11 to 41). For device modelling a detailed knowledge of the electronic parameters as band edges, charge densities, and subband energies is necessary. To describe more complicated structures, e. g. single and multiple quantum wells or adequate barriers, within this scheme one has to generalize the LDA. In this note we will show the principle way to construct the LDS according to an arbitrary potential. After formulating the general equations the proposed procedure will be illustrated by two examples.Firstly, we suppose some requirements on the appropriate LDS which have t o be realized from a general point of view. (ii) Because the potential V(?) is an electrostatic one in most applications, one has to demand naturally gauge invariance, that means D ( E , f ; V(?)) = D ( E + const, t; V(?) + const) . l ) PSF 327, DDR-6300 Ilmenau, GDR.
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