Casimir Interactions in Ising Strips with Boundary Fields: Exact Results

Abstract

An exact statistical mechanical derivation is given of the critical Casimir forces for Ising strips with arbitrary surface fields applied to edges. Our results show that the strength as well as the sign of the force can be controled by varying the temperature or the fields. An interpretation of the results is given in terms of a linked cluster expansion. This suggests a systematic approach for deriving the critical Casimir force which can be used in more general models.

Critical Casimir forces between homogeneous and chemically striped surfaces

“…In order to complement this spatial dependence and to further probe the relevance of fluctuations, it would be interesting to investigate the corresponding system in d = 2, where some exact results are available [45] and conformal invariance allows one to determine exactly certain critical properties.…”

“…This case has been studied so far only for laterally homogeneous BC in the presence of variable boundary fields. This case already gives rise to interesting crossover phenomena, which have been studied within mean-field theory [44], by exact calculations in two spatial dimensions [45,46], and with MC simulations [33,34].…”

Reversing the critical Casimir force by shape deformation

“…Controlling the sign of FIF (attractive or repulsive) is important to myriad applications in design and manipulation of micron scale devices. Theoretical results for FIF in various critical systems [7] have shown that sign changes of the force can be achieved by varying boundary fields [8,9]. Sign control has been achieved experimentally with judicious choice of materials in case of QED Casimir forces [10], and with appropriate boundary conditions for critical FIF [11][12][13].…”

Generalised Wick theorem and the Fredholm alternative

“…Taking symmetrized products in (13), we see that to evaluate the large N limit, the maximal eigenvalue of say V 1/2 2 V 1 V 1/2 2 is needed; this entails a competition between x-ordering in V 2 and having spins point in the z-direction in V 1 . The quantum character of the problem and its relationship to one-dimensional spin chains is now rather transparent.…”