Single-Interface Casimir Torque

It is shown that the quantum fluctuations of the electromagnetic field generally induce a torque at the interface of an anisotropic material with another anisotropic or isotropic material. It is proven that this torque depends on an interface zero-point energy determined by the dispersion of the interface (localized and extended) modes. Our theory demonstrates that the single-interface torque is essential to understand the Casimir physics of material systems with anisotropic elements and determines the equilibrium positions of the system.


I. INTRODUCTION
Casimir-Lifshitz interactions [1][2][3]  and its great impact in different areas [4][5], the research of Casimir-Lifshitz interactions has become of great practical importance as well. If, on one hand, Casimir interaction phenomena may lead to potentially undesired effects such as 'stiction' [6][7], on the other hand, they may open new and exciting possibilities in the field of micro and nanomechanics [4,[8][9][10][11].
The study of Casimir-Lifshitz phenomena was pioneered by Casimir for more than 60 years ago [1]. In his seminal work, Casimir showed that as a result of the electromagnetic field quantum fluctuations, two parallel perfectly conducting plates standing in a vacuum may experience an attractive force pushing the plates toward each other. Following Casimir's prediction, Lifshitz, Dzyaloshinskii, and Pitaevskii extended the theory to the more general case of realistic isotropic dielectric plates (including non-ideal metals) [2][3]. Some years later, this theory was further generalized to anisotropic dielectric plates [12][13]. Interestingly, it was shown that the anisotropy may lead to the emergence of qualitatively different phenomena. It was demonstrated that a pair of parallel anisotropic uniaxial plates (with inplane optical anisotropy and misaligned optical axes) separated by an isotropic dielectric, may experience a mechanical torque, designated as Casimir torque, that spontaneously forces the rotation of the plates towards the minimum energy position (with the two optical axes aligned). The Casimir torque in this kind of systems was further investigated in [14][15][16][17][18]. In particular, numerical calculations of the torque were provided in [14][15][16]18], and possible experiments to measure the Casimir torque were proposed in [14,[16][17][18].
With the emergence of metamaterials and their intriguing electromagnetic properties, the study of the Casimir-Lifshitz interactions has been also extended to systems involving complex structural nanoscopic unities [19][20][21][22][23][24][25]. In particular, the Casimir-Lifshitz phenomena have been investigated in systems formed by dense arrays of nanowires [26][27]. It was demonstrated in [26] that interactions mediated by nanowire-based materials result in ultralong-range forces, contrasting with the short-range Casimir forces characteristic of interactions mediated by isotropic dielectrics. The anomalous long-range Casimir force stems from the ultra-large density of photonic states in the nanowire materials, which boosts the quantum fluctuation-induced interactions [26][27].
Furthermore, in a recent work [28], we studied the Casimir interaction torque in nanowire materials, and demonstrated that it is distinctively different from the torques studied hitherto in other systems (e.g., birefringent parallel plates [14]). On one hand, it was proven that the Casimir interaction torque in nanowire structures has an unusual scaling law. Specifically, the torque generated due to the coupling between two interfaces decays as 1/ d at large distances (d is the distance between the two interfaces), which differs markedly from the characteristic 3 1/ d decay in usual configurations wherein the two interfaces are separated by an isotropic background [28]. On the other hand, it was argued that the torque has an additional and dominant contribution, designated by Casimir single-interface torque, which is an interfacial effect and does not vanish even when the two interfaces are infinitely far apart. The study of [28] was however mainly qualitative, and no detailed quantitative analysis of the singleinterface torque was provided. The objective of this work is to study in depth this singleinterface torque and unveil the physical mechanisms associated with this nontrivial Casimirtype interaction.
Even though the analysis in [28] was focused on nanowire materials, the single-interface torque emerges at any interface involving at least an anisotropic material with optical axes out of the interface plane. In these conditions, the zero-point energy of the system depends on the relative orientation of the material optical axes. Thus, rather than considering the particular case of metallic nanowire systems, here we theoretically investigate the Casimir singleinterface torque in general conditions, treating the relevant anisotropic materials as continuous media.

A. Zero-point energy
We are interested in the Casimir-type interactions between different anisotropic materials at zero-temperature. Even though at a later stage the relevant media will be modeled as continuous anisotropic uniaxial dielectrics, in a first step it is convenient to visualize each material as a periodic arrangement of inclusions embedded in a vacuum ( The zero-point energy C ε of the system can be calculated with the help of the argument principle [29][30][31][32]. In this section, we consider a generic double-interface configuration ( Fig.   1(b)) and revisit the usual derivation of the zero-temperature Casimir energy [29][30][31][32]  α has no meaning and can be ignored. Yet, we will keep it in the formulas because at a later stage we will consider the general case where the middle region is an anisotropic material. Generalizing the approach of Ref. [32] to three-dimensional geometries, it follows that for a periodic system the characteristic function D may be chosen of the form where 1 is a unit matrix, L,R R are the reflection matrices for the left and right interfaces, and  . .
After integration by parts, the right-hand side of this formula reduces to the familiar Casimir interaction energy defined as: where ( ) One important aspect is that the spatial domain is required to be electromagnetically closed.
Hence, the cavity should be terminated with some type of opaque boundary, for example with periodic boundary conditions or a perfectly electric conducting wall placed at z = ±∞ . Thus, strictly speaking the poles of L R and R R do not need to be associated with waves localized at the interfaces, and may be associated with spatially extended modes.
In summary, we formally demonstrated that when the materials response ceases for ω → ∞ the zero-point energy of the double-interface configuration ( Fig. 1(b)) can be written as (apart from an irrelevant constant independent of the system configuration):  8 The first term C,int dε corresponds to the usual Casimir interaction energy that appears due to the coupling between the two interfaces, whereas the other two terms are associated with the Casimir single-interface energies determined by the orientation of the optical axes. These single-interface components are due to the anisotropy of the materials because the energy of the system depends on the angles ( ) that dictate the orientation of the inclusions.
Even though the single-interface terms C,12 ε and C,23 ε are distance independent, and therefore do not contribute to the usual Casimir force, they can contribute to the Casimir torque. This will be discussed in detail in the next subsection.

B. Casimir torque
Next, we derive the Casimir torque in the considered material structure, and highlight the differences compared to the torques induced in conventional systems with in-plane anisotropy.
The total Casimir torque acting on the i-th body (i=1,2,3) in the double-interface configuration is ( ) and hence from Eq. (8) it is given by: In systems where the interaction is mediated by an isotropic material and when the optical axes of the materials 1 and 3 are parallel to the interface, C,12 ε and C,23 ε are evidently independent of i α , and hence it is possible to assume that the Casimir zero-point energy of the system C,tot ε can be replaced by the interaction energy C,int dε . Thus, in such a scenario the Casimir torque is simply given by A single-interface torque also occurs in the single-interface configuration ( Fig. 1(a)). For such a configuration it is physically evident that there must be a preferred orientation for the optical axis of the medium, and hence some associated zero-point energy. To determine the single-interface energy C,s.i.
ε and torque C,s.i. M , we adapt the ideas of our previous work [28], and consider the scenario where the gap d between the two interfaces in the double-interface configuration ( Fig. 1 From here, we see that ( ) . But for a twin- . Therefore, it follows that the single-interface Casimir torque is such that: dε is a short-hand notation for

M
represents the torque acting on half of the crystalline structure. Thus, ( ) corresponds to the additional stress due to the asymmetry created by the interface, and consistent with this it is proportional to the area of the interface.
Even though the described theory is completely rigorous, the granularity of the crystal does not allow for a simple analytical treatment. To circumvent this issue, in the next section we consider the continuous medium approximation.

A. Continuum approximation
It is possible to considerably simplify the problem using an effective medium approximation wherein each material region is seen as a uniaxial anisotropic dielectric with permittivity: where ˆˆĉos sin In the continuum limit, for each fixed || k the electromagnetic fields in the vacuum region can be expanded simply in terms of the usual plane wave modes, similar to Ref. [28]. Hence, in this case the matrices L,R R and (2) become 2×2 matrices and can be determined using standard analytical methods [28] (see also Appendix A). Indeed, within the effective medium framework the wave propagation is described by an ordinary wave (transverse electric (TE) mode) and an extraordinary wave (transverse magnetic (TM) mode [28].
At this point, it is important to discuss the validity of the continuous medium approximation. Typically, effective medium methods are valid for interactions such that || 1 k a < and / 1 a c ω < . In the microscopic formulation C,int dε must be calculated in the limit for which the structure becomes periodic (a crystal). In this limit the distance between adjacent layers of inclusions is nonzero, but is as small as d a ≈  , i.e., on the order of the lattice constant. Thus, it is possible to estimate that the modes relevant for the Casimir interaction satisfy || 1 k a < and / 1 a c ω < , which is precisely the limit of validity of the effective medium approximation. Due to this reason, it follows that the effective medium framework is only approximately satisfactory, and in particular it may not yield quantitatively precise results. Yet, the effective medium theory has the advantage that it enables a simple analysis of the problem, and we expect it to provide at least a qualitatively correct description of the physics of the single-interface Casimir torque.
Another important aspect is that in the continuum limit the torque in a bulk material must vanish ( bulk 0 M = ) because any orientation of the optical axis is energetically equivalent when there is no underlying granularity. Hence, in the continuum limit Eq. (11) becomes: so that the single-interface torque is only due to surface effects. We used the fact that in the vanish when x → ∞ , and it can be checked that this implies that C,int dε is finite.
In summary, the single-interface torque is originated by interactions between bodies that are nearly in contact ( d a ≈  ) and hence an effective medium description of the problem depends critically on the high-frequency (both spatial and temporal) response of the materials.
The precise knowledge of the effective dielectric function for ω → ∞ and the precise wave vector cut-off max k are critical to make quantitative predictions.

B. Generalization
So far it was assumed that the middle layer (region 2 in Fig. 1) is a vacuum, so that ( ) 1 C,s.i.

M
corresponds to the single-interface torque when the material 1 is adjacent to a vacuum.
However, within the effective medium description there is no difficulty in generalizing the theory to the case wherein the middle layer is an arbitrary anisotropic dielectric (Fig. 2)  A straightforward analysis analogous to that reported in Sect. II, but using as a starting point the macroscopic framework with the physical cut-off max~/ k a π , shows that Eq. (13) remains valid when the middle region is an arbitrary dielectric. As before, ( ) is understood as the single-interface torque acting on medium 1 for an interface between medium 1 and medium 2. However, when the second material is not isotropic the torque on medium 2 is typically nonzero, and can be calculated using ( ) It is important to prove that the theory is self-consistent. Indeed, C,s.i.
The indices "123" and "121" identify the configuration used to evaluate the interaction energy and the single-interface energy, respectively. Does the above formula give the same result as ∂ ? We will not attempt to give a direct proof of this property but in the next section it is shown with numerical simulations that the answer is affirmative. This result demonstrates that the theory is fully self-consistent, and that the calculated torque is, indeed, independent of the considered limit process.

IV. NUMERICAL EXAMPLES
In order to characterize the single-interface energy and torque in the considered systems ( Fig. 1), next we carry out extensive numerical simulations based on Eqs. (9) and (13)

A. Single-interface configurations
To begin with, we study the single-interface Casimir interactions at the junction of an anisotropic and an isotropic material ( Fig. 1(a)).
In the first example (Fig. 3 (2 ) α , somewhat analogous to the typical angle-dependence of the interaction torque but here the optical axis is not parallel to the interface [14].    Fig. 4(b)), and hence the system has two equilibrium positions. The single-interface torque induced in the region 2 acts to rotate the "inclusions" towards the closest equilibrium point. In order to further characterize the considered system, next it is supposed that the two particle sets are free to rotate around the x-axis. Figure 5  A similar trend is observed when the two anisotropic materials are different. Figure 6 shows the single-interface Casimir energy for a system formed by an anisotropic material with anisotropy ratio Now the global energy minima occur for ( ) ( ) To conclude this subsection, we note that for any system with the generic geometry considered in this section the single-interface Casimir energy has the following symmetries: The first property is trivial and is a simple consequence that the system is unchanged if the optical axis of the uniaxial dielectrics is rotated by 180º. The second property is a consequence of the fact that the zero-point energy is unaffected by a transformation of the type y y → − . These properties imply that the torques must vanish ( ( ) ( )

B. Double-interface configurations
It is interesting to extend the analysis of the previous section to the case of double-interface configurations ( Fig. 1(b)). In the first example, we consider an anisotropic dielectric-airisotropic dielectric system. Figure 7 shows  In the second example, we consider an anisotropic dielectric-anisotropic dielectric-air configuration. We assume that the two anisotropic materials, apart from the orientation of the optical axes, are identical and have the anisotropy ratio  As previously seen (Fig. 5), for a single-interface configuration formed solely by the two anisotropic materials, i.e., in the absence of the air layer, the global energy minimum is not attained when the two dipole sets are aligned. Indeed, the simulations of Fig. 5 suggest that if all the individual particles are free to rotate (a case which can be studied using our analytical framework) then the most energetically favorable configuration is not reached when the particles are all aligned along the same direction, but likely when they are "randomly" oriented. In other words, in a bulk material there is no "anchor" to fix a preferred alignment direction and hence unconstrained particles tend to be oriented in a "random" fashion. In contrast, the presence of the air region promotes the direction parallel to the interface as the most favorable from an energetic point of view.

V. CONCLUSION
We studied the zero-temperature Casimir single-interface torque at the junction between different isotropic and anisotropic materials using both microscopic and macroscopic formulations. The single-interface torque arises due to the quantum fluctuations associated with interface-type (both localized and extended) modes. These quantum fluctuations originate internal material stresses that act to change the internal configuration of the materials, i.e. to rotate the particles. The single-interface torque is quite different from the more familiar Casimir interaction torque, which is determined by the interaction of two rigid bodies separated by an isotropic material. Relying on a microscopic theory, it was proven that, in general, the single-interface torque may have a "bulk" (volumetric) contribution and a surface contribution. The torque surface component can be written in terms of the interaction energy of the system for a twin-material configuration. It was shown that the single-interface torque can be as well computed using the effective medium approximation. However, since the single-interface torque is determined by interactions of bodies that are almost in contact the use of effective medium methods is only approximately satisfactory and requires the use of a physical wave vector cut-off.
Our numerical results obtained with the continuum approximation demonstrate that in isotropic-anisotropic material systems the isotropic region acts as an anchor, imposing a preferential orientation for the particles of the anisotropic material. In particular, when the isotropic region is the vacuum the global energy minimum is reached when the dipoles are parallel to the interface. For conventional dielectrics the energy minimum moves towards the normal direction. On the other hand, in anisotropic-anisotropic material systems the global energy minimum does not correspond to a configuration with aligned dipoles, and in some cases -most remarkably when the two materials are identical -it is reached when they are approximately perpendicular. This property suggests that if all the dipoles were unconstrained and free to rotate then the configuration associated with the global energy minimum would correspond to some amorphous (non-periodic) structure with the dipoles oriented in a "random" fashion.
In future work, it will be relevant to calculate the single-interface torque with the rigorous microscopic model to assess the accuracy of the effective medium approximation. To conclude, we point out that perhaps some of the ideas discussed in this article can be experimentally validated by characterizing interfaces of liquid crystals and conventional dielectrics.

Appendix A: Properties of the reflection matrices
In this Appendix, we derive some useful properties of the reflection matrices L R and R R for two anisotropic dielectric semi-spaces modeled as a continuum. Without loss of generality, it is supposed that the interface is normal to the z-direction.
To begin with, we define the transverse fields as: Let us introduce admittance matrices such that for plane waves propagating along the +z and -z directions one has: T T In general, ± Y depend on the considered material (and in particular on the orientation of the optical axis), on the frequency ω , and on the transverse wave vector || k .
Let us consider a twin-interface configuration of the type 1-2-1. Imposing the continuity of The second property follows from reciprocity of the materials and establishes that: The last identity is a consequence of Eqs. (A3) and (A9 This result proves the desired identity [Eq.(B3)].