Reversing the Critical Casimir force by shape deformation
The exact critical Casimir force between periodically deformed boundaries of a 2D semi-infinite strip is obtained for conformally invariant classical systems. Only two parameters (conformal charge and scaling dimension of a boundary changing operator), along with the solution of an electrostatic problem, determine the Casimir force, rendering the theory practically applicable to any shape and arrangement. The attraction between any two mirror symmetric objects follows directly from our general result. The possibility of purely shape induced reversal of the force, as well as occurrence of stable equilibrium points, is demonstrated for certain conformally invariant models, including the tricritical Ising model.
pacs:11.25.Hf, 05.40.-a, 68.35.Rh
Fluctuation-induced forces (FIF) are ubiquitous in nature Kardar and Golestanian (1999); prominent examples include van der Waals Parsegian (2005), and closely related Casimir forces Casimir (1948); Bordag et al. (2009), originating from quantum fluctuations of the electromagnetic field. Thermal fluctuations in soft matter also lead to FIF, most pronounced near a critical point where correlation lengths are large de Gennes and Fisher (1978); Krech (1994). Controlling the sign of FIF (attractive or repulsive) is important to myriad applications in design and manipulation of micron scale devices. This has been achieved with judicious choice of materials in case of quantum electrodynamic (QED) Casimir forces Munday et al. (2009), and with appropriate boundary conditions for critical FIF Soyka et al. (2008); Hertlein et al. (2008).
The non-additive character of FIF has also prompted a quest for reversing the sign of Casimir forces solely by manipulation of shapes. The original impetus comes from the intriguing result by Boyer Boyer (1968) for the modification of QED zero point energy by a spherical metal shell. The suggestion that this result may imply repulsion between two hemispheres was later ruled out by a general theorem for attraction between mirror symmetric shapes Kenneth and Klich (2006); Bachas (2007). There are indeed specific geometrical arrangements in which the normally attractive QED force in vacuum appears repulsive when constrained along a specific axis (e.g. Levin et al. (2010)), but is unstable when moved off such axis. Indeed, a generalized Earnshaw’s theorem for FIF in QED rules out the possibility of stable levitation (and consequently force reversals) in most cases Rahi et al. (2010).
Two dimensional (2D) membranes have provided yet another arena for investigation of FIF, mostly focused on interactions arising due to modifications of capillary fluctuations (see, e.g. Yolcu et al. (2012); Noruzifar et al. (2013) and references therein). More recently, motivated by the possibility that the lipid mixtures composing biological membranes are poised at criticality Veatch et al. (2007); Baumgart et al. (2007), it has been proposed that inclusions (such as proteins) on such membranes are subject to 2D analogs of critical FIF Machta et al. (2012). A notable advantage is that 2D systems at criticality can be described by conformal field theories (CFT) Friedan et al. (1984); Cardy (1989): Casimir forces in a strip are related to the central charge of the CFT Cardy (1986); Kleban and Vassileva (1991); Kleban and Peschel (1996), with appropriate modification for boundaries. There are results for interactions between circles Machta et al. (2012), needles Vasilyev et al. (2013); Ref. Bimonte et al. (2013) describes any compact shapes. Here, we consider the interaction between two wedges, or an array of wedges, as depicted in Fig. 1. We show that (with appropriate choice of CFT and boundary conditions) the FIF can be attractive or repulsive depending on the angle of the wedge; and that arrangements of stable equilibrium can be obtained with truncated wedges and arrays of them.
Consider two identically corrugated, infinite boundaries that enclose
a critical classical medium (e.g., a fluid or magnetic system at its
critical temperature ) described by a CFT. The boundaries,
and , impose conformally invariant boundary conditions
and , respectively, on the
medium. While our method is applicable to any shape, as specific
examples we study the periodic, wedge-like shapes in
Figs. 1(b,d). As interactions at proximity are
dominated by the tips, we also consider the infinite wedges
depicted in Figs. 1(a,c). Following our approach for
compact shapes Bimonte et al. (2013),
the strip with deformed boundaries is conformally mapped to a flat strip.
Information about the intervening medium enters only
via its conformal charge , and the scaling dimension
of the boundary changing operator (BCO)
from to ;
with for like boundaries di Francesco et al. (1997).
All information about the shape of the deformed strip is encoded in the conformal map
to the flat strip. This map, and hence the FIF, can be obtained from the
solution to an electrostatic problem.
In the following, we combine the normal () and lateral ()
components of the force into the complex expression .
For periodically deformed boundaries with wavelength and length
, , where the first
contribution is the force on a strip,
that is determined by the free energy (per unit length) of a flat strip of width , and . The second contribution is the geometric force
where is the Schwarzian derivative of the conformal map of the deformed to the flat strip di Francesco et al. (1997). Due to periodicity, it is sufficient to construct for a unit cell so that integrations in Eqs. (1,2) are restricted to a path that separates and within a unit cell [cf. Fig. 1]. Of course, the forces are proportional to the number of unit cells, . Whereas the strip force depends on shape simply via the electrostatic capacitance Bimonte et al. (2013), the geometric force has a more intricate dependence on the boundary shapes. (The Schwarzian derivative vanishes if and only if is a global conformal map.)
Conformal maps are physically realized as equipotential curves and stream lines in electrostatics. We employ this analogy to derive a general result for the Casimir force in terms of the electrostatic potential on the strip with the two boundaries held at a fixed potential difference . The conformal map is then given by where is the conjugate harmonic function to . Clealry . Since Eqs. (1,2) involve only derivatives of , we use the Cauchy-Riemann equations to get and eliminate . For practical computations (e.g. using finite element solvers) it is useful to express the Casimir force in terms of line integrals of real valued vector fields that are fully determined by derivatives of . Parametrizing the contour by for , and splitting into real and imaginary parts, we obtain the force in terms of , and as
We note that the strip force is proportional to the usual electrostatic force. This result also implies that the critical Casimir force between any pair of mirror symmetric boundaries is attractive for Kenneth and Klich (2006); Bachas (2007): In this case the electrostatic potential must be constant along the -axis of mirror symmetry. Choosing this axis as gives and hence shows that both and have a vanishing real part and a negative imaginary part for , which includes like boundaries (). This implies a vanishing lateral force and positive normal force that corresponds to attraction in our notation.
Due to the simplicity of the related electrostatic problem, virtually any boundary shape can be studied by computing either analytically (e.g. using the Schwarz-Christoffel (SC) map for polygons Smythe (1950)), or numerically (using finite element solvers). Here we consider a simple profile composed of a periodic array of (truncated) wedges as in Figs. 1(b,d). For the triangular corrugations in Fig. 1(b), the SC map yields an analytic result for the force in terms of a single parameter implicitly determined in terms of and . Due to lack of space, we delegate the full solution to a forthcoming work, and study here short and large distances only. At small , the force is the sum of the contributions from the tips of the wedges, such that the normal force . The FIF between two infinite wedges of opening angle [Fig. 1(a)] is proportional to on dimensional grounds, and given by
where the first term corresponds to and the second to
. The amplitude of this force is shown in
Fig. 2 for different values of
, corresponding to unlike boundary conditions ().
Interestingly, for the
force becomes attractive below a critical opening angle
and is sufficiently small. This is confirmed by our full analytic solution at all distances which is shown in the inset of Fig. 2 for different opening angles and a BCO of the tricritical Ising model that obeys Eq. (9). The change of sign corresponds to an unstable point.
However, these results together with the expected validity of the proximity force approximation (PFA) at very short separations suggest the possibility of a stable point if the tips of the wedges are truncated and replaced by plateaus of width , as in Fig. 1(d). Indeed, for a single pair of truncated wegdes [Fig. 1(c)], PFA at short distances suggests
which is repulsive for . At distances , the plateaus become irrelevant, and the force approaches the result of Eq. (8). Hence two truncated wedges must have a stable point at intermediate distance if Eq. (9) holds and the opening angle is sufficiently small. This expectation is confirmed by an exact computation (using a SC map) of the normal and lateral force between two truncated wedges with lateral shift [see Fig. 1(c)]. We additionally confirm that this configuration is stable with respect to displacements in the lateral direction.
Combining the above findings, for the truncated triangular corrugations of Fig. 1(d) we expect under the condition (9) and for sufficiently small a stable equilibrium point (in both directions), and a saddle point at larger separations. For this geometry, a SC transformation can be performed in principle, but has to be evaluated numerically (which is cumbersome for a finite lateral shift ). Hence, we employ the analogy to electrostatics as described by Eqs. (3,4). The electrostatic potential is computed by a finite element solver (FES) and subsequently the resulting vector fields [see Eqs. (5-7)] are integrated along the contour . To be specific, we chose the unitary CFT with conformal charge that describes the tricritical Ising model (TIM) and chose boundary conditions that are connected by the BCO with scaling dimension so that condition in Eq. (9) is fulfilled. (The reason for these particular choices shall become clear below when we discuss possible CFT’s.) The accuracy of the FES can be established by comparing its results to those from a SC transformation for vanishing lateral shift . The results of both methods for the normal force (acting on the lower boundary ) between truncated triangular corrugations with are shown in Fig. 3. The agreement is excellent and confirms sufficient accuracy of the FES. The normal force shows the expected sign reversals from repulsive to attractive and back to repulsive. For a finite lateral shift we used the FES to compute both normal and lateral force. The lateral force at a normal distance close to the stable point () is plotted in the inset of Fig. 3, demonstraing mechanical stability also in the lateral direction. The global force field and curves of constant Casimir potential are shown in Fig. 4. The presence of a stable equilibrium point, and a saddle point, (both at ) is clearly confirmed.
It is interesting to explore which critical systems allow for a stable equilibrium point in the above geometry. In the following, we identify unitary minimal CFT models with that permit boundary conditions consistent with the criteria in Eq. (9). For unitary models the conformal charge is restricted to the discrete values with integer . The allowed scaling dimensions of the primary operators can assume the values Friedan et al. (1984)
with , and . Cardy has shown that all possible highest weight states with scaling dimension may be realized by a BCO for an appropriate choice of boundary conditions on the flat strip Cardy (1984). The two conformally invariant boundary conditions, or states that are connected by a BCO, are determined by the fusion rules for the two (bulk) primary fields that correspond to the boundary states. The condition of Eq. (9) can only be fulfilled for . It turns out that for (Ising model), (3-state Potts model), and no primary operator obeys the condition. For all other models with there is exactly one operator whose dimension obeys the condition, while for two or more operators with suitable dimensions may exist. The simplest models with suitable BCO’s are the TIM (, , ) and the tricritical 3-state Potts model (, , ). These considerations underlie our choice of the TIM for Figs. 3, 4. There are certainly other minimal models that also allow for conformally invariant boundary conditions that lead to a stable point.
Tricritical points are a common feature of many phase diagrams, corresponding to the point where a continuous transition becomes first order, as can be observed by addition of vacancies or other impurities to an Ising magnet, or helium 3 to superfluid helium 4 (see, e.g. Ref. Lawrie and Sarbach (1984) for a review). Possible boundary conditions compatible with a renormalization group fixed point at tricriticality are discussed in Ref. Krech and Dietrich (1992). Fusion rules in CFT di Francesco et al. (1997) provide another route to characterizing conformally invariant boundary conditions. For the TIM with the BCO of scaling dimension , the relevant fusion rules are , , , , , and . The last two rules are relevant since for a semi-infinite strip the lowest dimension determines the free energy. Through appropriate choice of surface couplings and magnetic fields, the TIM admits the following conformally invariant boundary states Chim (1996); Affleck (2000): (i) A disordered state of free spins, corresponding to ; (ii) Maximally ordered (fixed) spins ( or ), with , . The phase transition between the above surface states can occur through (iii) Partially polarized ( or with vacancies) at finite surface fields, for , ; or (iv) Through a so called degenerate point at zero surface field, with . The fusion rules show that a stable point with vanishing FIF can occur for the following combinations of boundary conditions:
(fixed spin, degenerate),
(partially polarized, free spin),
(partially polarized, degenerate).
The stability of these boundary states (fixed points) with respect to a boundary magnetic field and spin couplings is determined by the boundary phase diagram of the TIM Affleck (2000). The free and fixed boundary conditions can be achieved relatively easily (at least in simulations), while the degenerate and partially polarized states require tuning one parameter (the surface coupling, or surface field). We expect that the combination of partially polarized and free spin conditions is the most promising candidate.
The conditions obtained here for the observation of a stable equilibrium point with FIF are rather restrictive. This demonstrates on the one hand the difficulty of achieving stability solely by FIF, on the one hand, and absence of its strict impossibility (ala Earnshaw Rahi et al. (2010)) on the other. For other examples in 2D, we could look for other realizations of CFT in interface (restricted solid-on-solid) models Pasquier (1987). It would be quite interesting to explore the possibility of stability with critical FIF in three dimensions. It is not a priori clear if the necessary conditions in higher dimensions will be less or more restrictive. Since is the upper critical dimension for the TIM, at least in this case the question could in principle be resolved by generalizing standard field theory methods Krech and Dietrich (1992) to wedge/cone geometries Maghrebi et al. (2011a, b).
We thank R. L. Jaffe for valuable discussions. This research was supported by Labex PALM AO 2013 grant CASIMIR, and the NSF through grant No. DMR-12-06323.
- Throughout the paper we measure energies in units of .
- The threshold corresponds to a free field theory with mixed Dirichlet and Neumann boundary conditions.
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