Geodesic
Casimir effect for critical fluctuations of the 2d Ising model with mixed boundary conditions: The results of exact solution
Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 2, pp. 270-286

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The auxiliary Grassmanian field method is used in the derivation of the partition function of the 2D Ising model with periodic boundary conditions with respect to one of the cycles of the torus and mixed (periodic–antiperiodic) conditions with respect to the other cycle. The Casimir energy for critical fluctuations, and also for fluctuations in the neighborhood of the critical point is calculated. 
@article{TMF_1994_100_2_a9,
     author = {A. I. Bugrij and V. N. Shadura},
     title = {Casimir effect for critical fluctuations of the 2d {Ising} model with mixed boundary conditions: {The} results of exact solution},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {270--286},
     publisher = {mathdoc},
     volume = {100},
     number = {2},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1994_100_2_a9/}
}
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A. I. Bugrij; V. N. Shadura. Casimir effect for critical fluctuations of the 2d Ising model with mixed boundary conditions: The results of exact solution. Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 2, pp. 270-286. http://geodesic.mathdoc.fr/item/TMF_1994_100_2_a9/