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# Boundary conditions and the critical Casimir force on an Ising model film: exact results in one and two dimensions 

Joseph Rudnick, 1 Roya Zandi, 2 Aviva Shackell, 1 and Douglas Abraham 3 

$^{1}$Department of Physics and Astronomy, UCLA, Box 951547 Los Angeles California 90095-1547 

$^{2}$Department of Physics and Astronomy, University of California, Riverside 

$^{3}$Theoretical Physics, Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom 

(Dated: November 17, 2018) 

Finite size effect in critical systems can be interpreted as universal Casimir forces. Here we compare the Casimir force for free, periodic and antiperiodic boundary conditions in the exactly calculable case of the Ising model in one and two dimensions. We employ a new procedure which allows us to calculate the Casimir force with periodic and antiperiodic boundary conditions analyt- ically in a transparent manner. Quite interestingly, we find an attractive Casimir force for the case of periodic boundary condition but a repulsive Casimir force for the anti-periodic one. 

Since the original work by Fisher and de Gennes [1], it has been well accepted that in the vicinity of a critical point the developing long range fluctuations give rise to a Casimir force in a system with a film-like geometry. As a consequence of finite size scaling [2], the generic form of the force is 

where T$_{C}$ is the critical temperature, L is the thickness of the film, d is the bulk dimensionality of the system, ξ is the correlation length which diverges at the critical point in the bulk, and ϑ is a universal function of its one argument. 

The notion of universality requires some qualification in the context of Casimir forces. Generally, for a sys- tem with a critical point, universality implies dependence only on the universality class to which that system be- longs. In the case of an Ising model, that universality class is characterized as O (1), corresponding to the dis- crete symmetry that is broken in the ordered phase. Sys- tems belonging to this class include uniaxial ferromag- nets, simple liquid-vapor systems and systems exhibiting de-mixing [3]. 

The universality class, however, is not the only char- acteristic of the system that dictates the behavior of the function ϑ ( x ); one must also consider the boundary con- ditions at the two extremities of the film. This effect is dramatically exhibited in the results of measurements performed in Ref. [4, 5]. Those experiments monitor the thickness of a film of $^{4}$He as the temperature passes through the lambda point at which bulk Helium acquires superfluid characteristics. It was found that the thickness of the film, which directly tracks the Casimir force ap- pearing in the vicinity of this critical point, changes sub- stantially when the temperature is close to and slightly below T$_{λ}$ . Quite interestingly, deep in the superfluid 

PACS numbers: 05.50.+q, 64.60.an, 64.60.De, 64.60.fd, 68.35.Rh 

## I. INTRODUCTION 

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phase, the film partially recovers its thickness but still remains thinner than in the normal phase. 

The thinning of the film well below the transition point was successfully explained in Ref [6], where it is shown that the confinement of bulk Goldstone modes combined with the effect of finite film thickness on surface fluc- tuations gives rise to a Casimir force that reduces the thickness of the film in the superfluid phase as compared to that of the normal film. However, the much more sub- stantial thinning of the film right below the critical point remained unexplained for many years. 

Earlier predictions of the Casimir force at the vicinity of λ transition based on calculations emphasizing the cor- rect universality class, O (2) for the superfluid transition in $^{4}$He [7, 8], differ markedly from the experimentally de- termined force, which is greater in magnitude by a factor of about thirty than those theoretical results. 

In Refs. [9–11] it was shown that the discrepancy arises from the boundary conditions assumed in the previous calculations. In particular, prior field theoretical calcu- lations [7, 12] as well as the simulations in [7] assume periodic boundary conditions, while physical reality re- quires the superfluid order parameter to vanish immedi- ately outside of the Helium film, thus mandating Dirich- let boundary conditions. It has been known for a long time that the boundary conditions operative in a sys- tem play an important role in determining the behavior of Casimir forces [13, 14]. However, there is no a priori reason to assume that the strength of Casimir force with Dirichlet boundary conditions will be significantly larger than that for the case of periodic boundary conditions. In fact, the absence of this expectation undoubtedly hin- dered our understanding of the dip in Helium films for many years. 

The strong effect of boundary conditions on the Casimir force in the particular instance of an O (2) system in three dimensions leads one to ask if there are any gen- eral inferences one can arrive at regarding the interplay of boundary conditions, dimensionality, and universality class on the Casimir force acting on a system at and 

