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 $\sigma$ and energy $\epsilon$. Here $+,-$, and $f$ stand for spin-up, spin-down, and free-spin boundaries, respectively. The case $+-+-+\cdots$, where the boundary condition switches between $+$ and $-$ at arbitrary points, ${\zeta }_1,{\zeta }_2,\cdots$ 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 $+f+f+\cdots$ and the case $-f+$ of three different boundary conditions. Exact results for the one- and two-point averages of $\sigma ,\epsilon$, and the stress tensor $T$ are derived with conformal-invariance methods. From the results for $\langle T\rangle$, 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.

中文翻译：

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具有混合边界条件的二维临界系统：共形不变性和边界算子展开的精确伊辛结果

通过保形不变方法，Burkhardt，Guim和Xue研究了在上半平面上定义的临界伊辛模型 $ÿ>0$ 具有不同的边界条件 $\mathrm{一种}$ 和 $b$ 在负面和正面 $X$轴。对于$\mathrm{一种}b=-+$ 和 $F+$，他们确定了旋转的一点和两点平均值 $\sigma$ 和能量 $\epsilon$。这里$+，-$和 $F$分别代表向上旋转，向下旋转和自由旋转的边界。案子$+-+-+\cdots$，边界条件在 $+$ 和 $-$ 在任意点， ${\zeta }_{\mathrm{1个}}，{\zeta }_2，\cdots$ 在 $X$轴也进行了分析。在本文的上半部分，对交替边界条件进行了类似的研究$+F+F+\cdots$ 和案件 $-F+$三个不同的边界条件。一点和两点平均值的精确结果$\sigma ，\epsilon$和应力张量 $Ť$由保形不变方法得出。从结果$〈Ť〉$，对于混合边界条件，得出了与楔形夹杂物边界临界的卡西米尔相互作用。在本文的后半部分，使用边界算子展开对具有混合边界条件的任意二维临界系统进行了分析。考虑了两种不同类型的展开-远离边界条件的切换点和在切换点处。使用扩展，我们用一点平均值表示边界附近的两点平均值的渐近行为。我们还考虑了带边界条件混合的条形几何形状，并由于一个边缘而将远壁校正导出到一个边缘附近的单点平均。最后，我们确认了采用保形不变方法和边界算子展开式获得的预测的一致性，