Erratum: Volume operator for spin networks with planar or cylindrical symmetry [Phys. Rev. D73, 124004 (2006)]

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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Einstein Equations in the Case of Static Cylindrical Symmetry and the Diagonal Stress-Energy Tensor with Mutually Proportional Components

Ukrainian Journal of Physics ◽

10.15407/ujpe58.09.0894 ◽

2013 ◽

Vol 58 (9) ◽

pp. 894-904 ◽

Cited By ~ 1

Author(s):

S.B. Grigoryev ◽

◽

A.B. Leonov ◽

Keyword(s):

Cylindrical Symmetry ◽

Einstein Equations ◽

Energy Tensor ◽

Stress Energy Tensor ◽

Stress Energy

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The Volume of the Quiver Vortex Moduli Space

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptab012 ◽

2021 ◽

Author(s):

Kazutoshi Ohta ◽

Norisuke Sakai

Keyword(s):

Gauge Theory ◽

Moduli Space ◽

Riemann Surfaces ◽

Gauge Theories ◽

Target Space ◽

Contour Integral ◽

Quiver Gauge Theory ◽

Volume Formula ◽

Volume Operator ◽

Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

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Magnetisation Processes in Geometrically Frustrated Spin Networks with Self-Assembled Cliques

Entropy ◽

10.3390/e22030336 ◽

2020 ◽

Vol 22 (3) ◽

pp. 336 ◽

Cited By ~ 1

Author(s):

Bosiljka Tadić ◽

Miroslav Andjelković ◽

Milovan Šuvakov ◽

Geoff J. Rodgers

Keyword(s):

Domain Walls ◽

Spin Dynamics ◽

Higher Order ◽

Spin Networks ◽

Graph Theoretic ◽

Geometrically Frustrated ◽

Geometric Frustration ◽

Antiferromagnetic Materials ◽

Geometrically Constrained ◽

Ferromagnetic Interactions

Functional designs of nanostructured materials seek to exploit the potential of complex morphologies and disorder. In this context, the spin dynamics in disordered antiferromagnetic materials present a significant challenge due to induced geometric frustration. Here we analyse the processes of magnetisation reversal driven by an external field in generalised spin networks with higher-order connectivity and antiferromagnetic defects. Using the model in (Tadić et al. Arxiv:1912.02433), we grow nanonetworks with geometrically constrained self-assemblies of simplexes (cliques) of a given size n, and with probability p each simplex possesses a defect edge affecting its binding, leading to a tree-like pattern of defects. The Ising spins are attached to vertices and have ferromagnetic interactions, while antiferromagnetic couplings apply between pairs of spins along each defect edge. Thus, a defect edge induces n − 2 frustrated triangles per n-clique participating in a larger-scale complex. We determine several topological, entropic, and graph-theoretic measures to characterise the structures of these assemblies. Further, we show how the sizes of simplexes building the aggregates with a given pattern of defects affects the magnetisation curves, the length of the domain walls and the shape of the hysteresis loop. The hysteresis shows a sequence of plateaus of fractional magnetisation and multiscale fluctuations in the passage between them. For fully antiferromagnetic interactions, the loop splits into two parts only in mono-disperse assemblies of cliques consisting of an odd number of vertices n. At the same time, remnant magnetisation occurs when n is even, and in poly-disperse assemblies of cliques in the range n ∈ [ 2 , 10 ] . These results shed light on spin dynamics in complex nanomagnetic assemblies in which geometric frustration arises in the interplay of higher-order connectivity and antiferromagnetic interactions.

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