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.

• Received 10 August 2020
• Accepted 24 December 2020

DOI:https://doi.org/10.1103/PhysRevE.103.012120