arXiv 1404.42632014-04-16hep-th

大规模Jackiw-Rebbi模型

Massive Jackiw-Rebbi Model

Farid Charmchi, Siamak S. Gousheh

备注：: 10 pages, 3 figures
发表信息：: Nucl. Phys. B 883, (2014) p. 256
DOI：: 10.1016/j.nuclphysb.2014.03.021

摘要：

本文分析了一个推广的Jackiw-Rebbi（J-R）模型，在该模型中，一个有质量的费米子作为一个预定的背景场耦合到$λϕ^4$模型的扭折处.我们精确地解析地求解了这个大质量J-R模型，得到了费米子的整个谱，包括束缚态和连续态。费米子的质量项使得解耦的二阶类薛定谔方程的势在两个空间无穷远处的渐近值不同，从而使势不对称.因此，我们遇到了一个不寻常的问题，其中费米子可能存在两种连续谱态：反射态和散射态。然后，我们将所有状态的能量作为扭结参数的函数，即其在空间无穷远处的值（$θ_0$）和其在$x=0$的斜率（$μ$）。作为$θ_0$的函数的能量图，其中描述了束缚态能量和两种连续态，显示出包括三角形形式的能隙的特殊特征，其中不存在束缚态。也就是说，零模式仅对于大于临界值$(θ_0^{\textrm{c}})$的$θ_0$存在。这与通常的（无质量）J-R模型形成鲜明对比，在通常的J-R模型中，零模和因此的基态的费米子数$\pm1/2$总是存在的。这也使得零模的起源非常清楚：它是由$θ_0^{\textrm{c}}$的两个阈值束缚态的联合形成的，在无质量的J-R模型中，$θ_0^{\textrm{c}}$是零。

英文摘要：

In this paper we analyze a generalized Jackiw-Rebbi (J-R) model in which a massive fermion is coupled to the kink of the $λϕ^4$ model as a prescribed background field. We solve this massive J-R model exactly and analytically and obtain the whole spectrum of the fermion, including the bound and continuum states. The mass term of the fermion makes the potential of the decoupled second order Schrodinger-like equations asymmetric in a way that their asymptotic values at two spatial infinities are different. Therefore, we encounter the unusual problem in which two kinds of continuum states are possible for the fermion: reflecting and scattering states. We then show the energies of all the states as a function of the parameters of the kink, i.e. its value at spatial infinity ($θ_0$) and its slope at $x=0$ ($μ$). The graph of the energies as a function of $θ_0$, where the bound state energies and the two kinds of continuum states are depicted, shows peculiar features including an energy gap in the form of a triangle where no bound states exist. That is the zero mode exists only for $θ_0$ larger than a critical value $(θ_0^{\textrm{c}})$. This is in sharp contrast to the usual (massless) J-R model where the zero mode and hence the fermion number $\pm1/2$ for the ground state is ever present. This also makes the origin of the zero mode very clear: It is formed from the union of the two threshold bound states at $θ_0^{\textrm{c}}$, which is zero in the massless J-R model.

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英文摘要：

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