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 withthe 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.

FIF in QED rules out the possibility of stable levitation (and consequently force reversals) in most cases [19].
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. [20,21] and references therein). More recently, motivated by the possibility that the lipid mixtures composing biological membranes are poised at criticality [22,23], it has been proposed that inclusions (such as proteins) on such membranes are subject to 2D analogs of critical FIF [24]. A notable advantage is that 2D systems at criticality can be described by conformal field theories (CFT) [25,26]: Casimir forces in a strip are related to the central charge of the CFT [27][28][29], with appropriate modification for boundaries. There are results for interactions between circles [24], needles [30]; Ref. [31] 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 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 T c ) described by a CFT. The boundaries, S 1 and S 2 , impose conformally invariant boundary conditions a and b, 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  are dominated by the tips, we also consider the infinite wedges depicted in Figs. 1(a), (c). Following our approach for compact shapes [31], the strip with deformed boundaries is conformally mapped to a flat strip. Information about the intervening medium enters only via its conformal charge c, and the scaling dimension h ab of the boundary changing operator (BCO) from a to b; with h ab = 0 for like boundaries [32]. 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 ( y) and lateral (x) components of the force into the complex expression F = (F x −i F y )/2. For periodically deformed boundaries with wavelength λ and length W → ∞, where the first contribution is the force on a strip, 1 that is determined by the free energy (per unit length) F strip = −(π /2)(c/12 −η)/ of a flat strip of width = 2π /C cell with C cell the electrostatic capacitance of the deformed strip per unit cell, and η ≡ 2h ab . The second contribution is the geometric force 2 is the Schwarzian derivative of the conformal map w(z) of the deformed to the flat strip [32]. Due to periodicity, it is sufficient to construct w(z) for a unit cell so that integrations in Eqs. (1), (2) are restricted to a path cell that separates S 1 and S 2 within a unit cell [cf. Fig. 1]. Of course, the forces are proportional to the number of unit cells, 1 Throughout the paper we measure energies in units of k B T c ; correspondingly forces are proportional to k B T c divided by an appropriate length scale. W /λ. 2 Whereas the strip force depends on shape simply via the electrostatic capacitance [31], the geometric force has a more intricate dependence on shape.
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 U (x, y) on the strip with the two boundaries held at a fixed potential difference U = 1. The conformal map is then given by w(z) = U + iV where V is the conjugate harmonic function to U . Clearly = U = 1. Since Eqs. (1), (2) involve only derivatives of w(z), we use the Cauchy-Riemann equations to get ∂ z w = ∂ x U − i∂ y U and eliminate V . 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 U . Parameterizing the contour cell by r(s) = [x(s), y(s)] for 0 ≤ s ≤ 1, and splitting into real and imaginary parts, we obtain the force in terms of c, η and U as We note that the strip force F strip 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 c > 0 [15,16] 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 U either analytically (e.g. using the Schwarz-Christoffel (SC) map for polygons [33]), 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 ϑ and L/d.
Due to lack of space, we delegate the full solution to a forthcoming work, and study here short and large distances d only. At small d L, the force is the sum of the contributions from the tips of the wedges, such that the normal force F y = (W /λ)F wedges, y . The FIF between two infinite wedges of opening angle 2ϑ [ Fig. 1(a)] is proportional to 1/d on dimensional grounds, where the first term corresponds to F strip and the second to F geo .
This force is shown in Fig. 2 for different values of η/c. Interestingly, for η/c < 1/8 the force becomes attractive below a critical opening angle ϑ 3 when c > 0. This is different from the asymptotic large distance force between the boundaries, F y = (π /2)(c/12 − η)W /d 2 , which is repulsive for η/c > 1/12. Hence, there is a reversal of the force between triangular corrugations from repulsive to attractive with decreasing separation d if and ϑ is sufficiently small. This is confirmed by our full analytic solution at all distances shown in the inset of Fig. 2. The change of sign corresponds to an unstable point with respect to a normal displacement.
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 H L, as in Fig. 1(d). Indeed, for a single pair of truncated wedges [ Fig. 1(c) which is repulsive for η/c > 12. At distances d H , 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 δ [see Fig. 1(c)]. We also confirm that this configuration is laterally stable.
Combining the above findings, for the truncated triangular corrugations of Fig. 1(d) we expect under the condition (9) and for sufficiently small ϑ 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 δ).
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 3η /c = 1/8 corresponds to a free field theory (c = 1) with mixed Dirichlet and Neumann boundary conditions. ishing lateral shift δ. The results of both methods for the normal force (acting on the lower boundary S 2 ) between truncated triangular corrugations with ϑ = π/10 are shown in Fig. 3(b). 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 δ we used the FES to compute both normal and lateral force. The lateral force at a normal distance close to the stable point (d/H = 3.4) is plotted in the inset of Fig. 3(c), demonstrating mechanical stability also in the lateral direction. The force field and curves of constant Casimir potential are shown in Fig. 3(a), confirming a stable equilibrium point, and a saddle point.
In the following, we identify unitary minimal CFT models with c < 1 that permit boundary conditions consistent with the criteria in Eq. (9). For unitary models the conformal charge is restricted to the discrete values c = 1 − 6/[m(m + 1)] with integer m ≥ 2. The allowed scaling dimensions in Eq. (9) can assume the values [25] with p = 1, 2, · · · , m − 1, and q = 1, 2, · · · , p. Careful inspection of Eq. (11) shows that the condition of Eq.  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. [35] for a review). Possible boundary conditions compatible with a renormalization group fixed point at tricriticality are discussed in Ref. [36]. Fusion rules in CFT [32] provide another route to characterizing conformally invariant boundary conditions. For the TIM with the BCO of scaling dimension h 2,2 = 3/80, the relevant fusion rules are |1/10 × |7/16 = |3/80 , |3/5 × |7/16 = |3/80 , |3/2 × |3/80 = |3/80 , |0 × |3/80 = |3/80 , |1/10 × |3/80 = |3/80 + |7/16 , and |3/5 × |3/80 = |3/80 + |7/16 . The last two rules are relevant since for a semiinfinite 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 [37,38]: (i) A disordered state of free spins, corresponding to |7/16 ; (ii) Maximally ordered (fixed) spins (+ or −), with |0 , |3/2 . The phase transition between the above surface states can occur through (iii) Partially polarized (+ or − with vacancies) at finite surface fields, for |1/10 , |3/5 ; or (iv) Through a so called degenerate point at zero surface field, with |3/80 . The fusion rules show that a stable point with vanishing FIF can occur for the following combinations (a, b) 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 [38]. 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 conformally invariant boundary states are related to surface universality classes leading to a surface phase diagram [39].
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 [19]) on the other. For other examples in 2D, we could look for other realizations of CFT in interface (restricted solid-on-solid) models [40]. 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 d = 3 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 [36] to wedge/cone geometries [41,42].