Long Electron Cyclotron Resonance Plasma Source for Reactive Sputtering

Abstract

The low-energy effective theory of the IIB matrix model developed by H. Aoki et al. is written down explicitly in terms of bosonic variables only. The effective theory is then studied by Monte Carlo simulations in order to investigate the possibility of a spontaneous breakdown of ten-dimensional Lorentz invariance. The imaginary part of the effective action, which causes the so-called sign problem in the simulation, is dropped by hand. The extent of the eigenvalue distribution of the bosonic matrices shows a power-law large-N behavior, consistent with a simple branched-polymer prediction. We observe, however, that the eigenvalue distribution becomes more and more isotropic in the ten-dimensional space-time as we increase N. This suggests that if the spontaneous breakdown of Lorentz invariance really occurs in the IIB matrix model, a crucial rôle must be played by the imaginary part of the effective action.In this paper, we study the IIB matrix model at large N by Monte Carlo simulations. In particular, we investigate the possibility of a spontaneous symmetry breakdown of Lorentz invariance using the low-energy effective theory of the IIB matrix model proposed in ref. [23]. The authors of ref.[23] also study this issue, using an approach where the bosonic and fermionic matrices are both decomposed into diagonal and off-diagonal elements, and the off-diagonal elements are integrated out first perturbatively. Such a perturbative expansion is valid when the bosonic diagonal elements, which can be regarded as coordinates of N points in ten-dimensional flat space-time, are well separated from each other. In other words, what one obtains after integrating over the off-diagonal elements perturbatively can be considered as a low-energy effective theory of the IIB matrix model. Note that this perturbative expansion has nothing to do with the string perturbative expansion with respect to the world-sheet topology. Therefore, even at the one-loop level, the low-energy effective theory is expected to include non-perturbative effects of superstring theory, provided, of course, that the IIB matrix model conjecture is true.In fact, in order to obtain the low-energy effective action only for the bosonic diagonal elements, one still has to perform the integration over the fermionic diagonal elements, which is nontrivial, since their action turns out to be quartic. Although the explicit form of the final low-energy effective theory has not been derived, the theory was shown to be described by some complicated branched-polymer like system, typically involving a "double-tree" structure, in a flat ten-dimensional space time [23]. Thus, even at the one-loop level, 2 the low-energy effective theory contains highly nontrivial dynamics. It was further argued that the double-tree structure of the one-loop low-energy effective theory might cause a collapse of the distribution of the bosonic diagonal elements. The first Monte Carlo results of a branched-polymer system with a double-tree structure was reported in ref. [7].Here, we write down e...

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