论文标题
具有混合边界条件的二维临界系统：完整的ISING由共形不变性和边界操作员扩展产生
Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions
论文作者
论文摘要
借助保形性方法，伯克哈特，吉姆和Xue研究了关键的伊辛模型，该模型定义在上半平面$ y> 0 $，带有不同边界条件$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $和$ b $。对于$ ab = - +$和$ f+$，他们确定了旋转$σ$和能量$ε$的一个和两点平均值。在这里，$+$，$ - $和$ f $ sant分别用于旋转，旋转和自由旋转边界。 $+ - + - + - +\ dots $，其中边界条件在任意点的$+$和$ - $之间切换，$ζ_1$，$ζ_2$，$ \ dots $也分析了$ x $ axis的$。 在本文中，考虑了三种不同边界条件的交替边界条件$+f+f+f+\ dots $和case $ -f+$。得出了$σ$，$ε$和应力张量$ t $的一个和两点平均值的确切结果。使用$ \ langle t \ rangle $的结果，分析了混合边界条件的临界Casimir与楔形包含边界的关键相互作用。 本文还包括对具有混合边界条件的二维关键系统中边界操作员扩展的全面讨论。考虑了两种类型的膨胀 - 远离边界条件的开关点和在开关点处。两点平均值的渐近行为是根据扩展的帮助表达的。我们还考虑了具有混合边界条件的带状几何形状，并使用边界操纵器的扩展将远处的墙校正得出了一个边缘附近的一点点平均值。边界操作员扩展的预测与Ising系统的确切结果一致。
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 $+-+-+\dots$, where the boundary conditions switch between $+$ and $-$ at arbitrary points, $ζ_1$, $ζ_2$, $\dots$ on the $x$ axis was also analyzed. In this paper the alternating boundary conditions $+f+f+\dots$ and the case $-f+$ of three different boundary conditions are considered. Exact results for the one and two-point averages of $σ$, $ε$, and the stress tensor $T$ are derived. Using the results for $\langle T\rangle$, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is analyzed for mixed boundary conditions. The paper also includes a comprehensive discussion of boundary-operator expansions in two-dimensional critical systems with mixed boundary conditions. Two types of expansions - away from switching points of the boundary condition and at switching points - are considered. The asymptotic behavior of two-point averages is expressed in terms of one-point averages with the help of the expansions. 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 using the boundary-operator expansions. The predictions of the boundary-operator expansions are consistent with exact results for Ising systems.
