Matrix factorizations for nonaffine LG-models

We propose a natural definition of a category of matrix factorizations for nonaffine Landau-Ginzburg models. For any LG-model we construct a fully faithful functor from the category of matrix factorizations defined in this way to the triangulated category of singularities of the corresponding fiber. We also show that this functor is an equivalence if the total space of the LG-model is smooth.Comment: 12 pages, minor corrections of TEX fil

Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities

In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W=0.Comment: 26 pp., LaTe

Derived categories of Burniat surfaces and exceptional collections

We construct an exceptional collection $\Upsilon$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal A$ of $\Upsilon$ is an "almost phantom" category: it has trivial Hochschild homology, and K_0(\mathcal A)=\bZ_2^6.Comment: 15 pages, 1 figure; further remarks expande

Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation

The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.Comment: 18 pages, 4 figure

Single-electron latch with granular film charge leakage suppressor

A single-electron latch is a device that can be used as a building block for Quantum-dot Cellular Automata (QCA) circuits. It consists of three nanoscale metal "dots" connected in series by tunnel junctions; charging of the dots is controlled by three electrostatic gates. One very important feature of a single-electron latch is its ability to store ("latch") information represented by the location of a single electron within the three dots. To obtain latching, the undesired leakage of charge during the retention time must be suppressed. Previously, to achieve this goal, multiple tunnel junctions were used to connect the three dots. However, this method of charge leakage suppression requires an additional compensation of the background charges affecting each parasitic dot in the array of junctions. We report a single-electron latch where a granular metal film is used to fabricate the middle dot in the latch which concurrently acts as a charge leakage suppressor. This latch has no parasitic dots, therefore the background charge compensation procedure is greatly simplified. We discuss the origins of charge leakage suppression and possible applications of granular metal dots for various single-electron circuits.Comment: 21 pages, 4 figure

Bound, virtual and resonance SS-matrix poles from the Schr\"odinger equation

A general method, which we call the potential $S$-matrix pole method, is developed for obtaining the $S$-matrix pole parameters for bound, virtual and resonant states based on numerical solutions of the Schr\"odinger equation. This method is well-known for bound states. In this work we generalize it for resonant and virtual states, although the corresponding solutions increase exponentially when $r\to\infty$. Concrete calculations are performed for the $1^+$ ground and the $0^+$ first excited states of $^{14}\rm{N}$, the resonance $^{15}\rm{F}$ states ($1/2^+$, $5/2^+$), low-lying states of $^{11}\rm{Be}$ and $^{11}\rm{N}$, and the subthreshold resonances in the proton-proton system. We also demonstrate that in the case the broad resonances their energy and width can be found from the fitting of the experimental phase shifts using the analytical expression for the elastic scattering $S$-matrix. We compare the $S$-matrix pole and the $R$-matrix for broad $s_{1/2}$ resonance in ${}^{15}{\rm F}$Comment: 14 pages, 5 figures (figures 3 and 4 consist of two figures each) and 4 table