Burkhardt T W, Eisenriegler E
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
Theoretical Soft Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany.
Phys Rev E. 2021 Jan;103(1-1):012120. doi: 10.1103/PhysRevE.103.012120.
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.
通过共形不变性方法，伯克哈特、吉姆和薛研究了临界伊辛模型，该模型定义在上半平面(y>0)上，在负(x)轴和正(x)轴上具有不同的边界条件(a)和(b)。对于(ab = - +)和(f +)，他们确定了自旋(\sigma)和能量(\varepsilon)的单点和两点平均值。这里(+)、(-)和(f)分别代表自旋向上、自旋向下和自由自旋边界。还分析了(+\ - +\ - +\cdots)的情况，其中边界条件在(x)轴上的任意点(\zeta_1)、(\zeta_2)、(\cdots)处(+)和(-)之间切换。在本文的前半部分，对交替边界条件(+f + f +\cdots)以及三种不同边界条件的(-f +)情况进行了类似的研究。通过共形不变性方法得出了(\sigma)、(\varepsilon)和应力张量(T)的单点和两点平均值的精确结果。从(\langle T\rangle)的结果中，推导出了混合边界条件下楔形夹杂物边界的临界卡西米尔相互作用。在本文的后半部分，用边界算子展开法分析了具有混合边界条件的任意二维临界系统。考虑了两种不同类型的展开——远离边界条件切换点的展开和在切换点处的展开。利用这些展开式，我们根据单点平均值来表示边界附近两点平均值的渐近行为。我们还考虑了具有混合边界条件的条带几何结构，并推导了由于另一条边导致的靠近一条边的单点平均值的远壁修正。最后，我们在本文的前半部分和后半部分中确认了用共形不变性方法和边界算子展开法得到的预测结果的一致性。
