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arXiv 1209.4490hep-thcond-mat.other
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耦合Fermion-Kink系统的Casimir能及其稳定性

Casimir Energy for a Coupled Fermion-Kink System and its stability

Siamak Sadat Gousheh, Azadeh Mohammadi, Leila Shahkarami

备注：
revtex4, 10 pages, 9 figures
发表信息：
Physical Review D 87, 045017 (2013)
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摘要：

本文计算了一个由费米子和一个具有指定扭结形式的赝标场组成的系统的Casimir能量。这个模型是不完全可解的，我们使用相移方法来计算Casimir能量。我们用松弛法求束缚态，用Runge-Kutta-Fehlberg法求出了整个$x$区间内费米子的散射波函数.由此产生的相移是一致的弱和强形式的列文森定理。然后，我们计算并绘制Casimir能量作为赝标场参数的函数，即$ϕ(x)$在x = 0处的斜率（$μ$）和$ϕ(x)$在无穷远处的值（$θ_0$）。在卡西米尔能量作为$μ$的函数的图中，当费米子束缚态能量穿过E = 0的线时，出现尖锐的最大值。此外，该曲线图显示，当$θ_0$是$π$的整数倍时，对于$μ\rightarrow 0$和$μ\rightarrow \infty$，Casimir能量变为零。此外，Casimir能量作为$θ_0$的函数的曲线图表明，该能量平均是$θ_0$的增函数，并且每当存在零费米子模时具有尖点。最后计算了由价费米子组成的系统在基态的总能量。最重要的是，我们表明，这个能量（卡西米尔能量和费米子的能量的总和）是最小的，当背景场的绕组数为1，独立的背景配置文件的细节。在整个文件中，我们比较我们的结果与一个简单的完全可解模型，其中一个分段线性轮廓近似扭结。我们发现，扭结是一个几乎无反射的障碍费米子，在我们的模型的背景下。

英文摘要：

We compute the Casimir energy for a system consisting of a fermion and a pseudoscalar field in the form of a prescribed kink. This model is not exactly solvable and we use the phase shift method to compute the Casimir energy. We use the relaxation method to find the bound states and the Runge-Kutta-Fehlberg method to obtain the scattering wavefunctions of the fermion in the whole interval of $x$. The resulting phase shifts are consistent with the weak and strong forms of the Levinson theorem. Then, we compute and plot the Casimir energy as a function of the parameters of the pseudoscalar field, i.e. the slope of $ϕ(x)$ at x=0 ($μ$) and the value of $ϕ(x)$ at infinity ($θ_0$). In the graph of the Casimir energy as a function of $μ$ there is a sharp maximum occurring when the fermion bound state energy crosses the line of E=0. Furthermore, this graph shows that the Casimir energy goes to zero for $μ\rightarrow 0$, and also for $μ\rightarrow \infty$ when $θ_0$ is an integer multiple of $π$. Moreover, the graph of the Casimir energy as a function of $θ_0$ shows that this energy is on the average an increasing function of $θ_0$ and has a cusp whenever there is a zero fermionic mode. We finally compute the total energy of a system consisting of a valence fermion in the ground state. Most importantly, we show that this energy (the sum of the Casimir energy and the energy of the fermion) is minimum when the background field has winding number one, independent of the details of the background profile. Throughout the paper we compare our results with those of a simple exactly solvable model, where a piece-wise linear profile approximates the kink. We find that the kink is an almost reflectionless barrier for the fermions, within the context of our model.

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