具有虚磁场的BCS模型中的超导相。III。非零自由色散关系
Superconducting phase in the BCS model with imaginary magnetic field. III. Non-vanishing free dispersion relations
摘要：
分析了一类在正温度下虚磁场作用下自由色散关系为非零的BCS模型。负耦合常数的大小必须很小，但允许与温度和虚磁场无关。自由能量密度的无限体积极限的特点。证明了自发对称性破缺和非对角长程序只在高温下发生。这是因为该模型中的间隙方程只有在温度高于临界值时才有正解。证明是基于格拉斯曼积分公式的双尺度积分。在该方案中，我们首先与红外协方差积分，然后与紫外协方差积分。作为另一个重点，我们研究了相边界的几何性质，这是一个封闭的曲线在二维空间的温度和实时变量的周期性副本。在这里，我们通过考虑其在正温度下的动态相变的当代物理学中的相关性，采用实时变量代替温度乘以虚磁场。作为主要结果，我们表明，对于任何选择的一个非零的自由色散关系的相边界的代表曲线只有一个局部极小点，或者换句话说，相边界不随温度振荡，当且仅当最小的自由色散关系的幅度超过最大值大于临界值$\sqrt{17-12\sqrt{2}}$。
英文摘要：
We analyze a class of the BCS model, whose free dispersion relation is non-vanishing, under the influence of imaginary magnetic field at positive temperature. The magnitude of the negative coupling constant must be small but is allowed to be independent of the temperature and the imaginary magnetic field. The infinite-volume limit of the free energy density is characterized. A spontaneous symmetry breaking and an off-diagonal long range order are proved to occur only in high temperatures. This is because the gap equation in this model has a positive solution only if the temperature is higher than a critical value. The proof is based on a double-scale integration of the Grassmann integral formulation. In this scheme we integrate with the infrared covariance first and with the ultra-violet covariance afterwards. As the other focus, we study geometric properties of the phase boundaries, which are periodic copies of a closed curve in the two-dimensional space of the temperature and the real time variable. Here we adopt the real time variable in place of the temperature times the imaginary magnetic field by considering its relevance within contemporary physics of dynamical phase transition at positive temperature. As the main result, we show that for any choice of a non-vanishing free dispersion relation the representative curve of the phase boundaries has only one local minimum point, or in other words the phase boundaries do not oscillate with temperature, if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than the critical value $\sqrt{17-12\sqrt{2}}$.
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