Critical Casimir torques and forces acting on needles in two spatial dimensions
aa r X i v : . [ c ond - m a t . s o f t ] A p r Critical Casimir torques and forcesacting on needles in two spatial dimensions
O. A. Vasilyev , , E. Eisenriegler , and S. Dietrich , Max-Planck-Institut f¨ur Intelligente Systeme, Heisenbergstraße 3, D-70569 Stuttgart, Germany IV. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart,Pfaffenwaldring 57, D-70569 Stuttgart, Germany Theoretical Soft Matter and Biophysics, Institute of Complex Systems,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany (Dated: September 27, 2018)We investigate the universal orientation-dependent interactions between non-spherical colloidalparticles immersed in a critical solvent by studying the instructive paradigm of a needle embeddedin bounded two-dimensional Ising models at bulk criticality. For a needle in an Ising strip theinteraction on mesoscopic scales depends on the width of the strip and the length, position, andorientation of the needle. By lattice Monte Carlo simulations we evaluate the free energy differ-ence between needle configurations being parallel and perpendicular to the strip. We concentrateon small but nonetheless mesoscopic needle lengths for which analytic predictions are available forcomparison. All combinations of boundary conditions for the needles and boundaries are consideredwhich belong to either the “normal” or the “ordinary” surface universality class, i.e., which inducelocal order or disorder, respectively. We also derive exact results for needles of arbitrary mesoscopiclength, in particular for needles embedded in a half plane and oriented perpendicular to the cor-responding boundary as well as for needles embedded at the center line of a symmetric strip withparallel orientation.
I. INTRODUCTION
A solvent near its bulk critical point induces long-ranged forces between immersed mesoscopic particleswhich are “universal”, i.e., independent of most micro-scopic details of the system [1–4]. Apart from a few bulkproperties of the solvent and from its local interactionswith the surfaces of the immersed particles, these forcesdepend only on the geometry of the confinement of thecritical fluctuations of the solvent imposed by the parti-cle surfaces, that is on the sizes, shapes, positions, andorientations of the particles. Such forces have been ob-served experimentally in film geometry for He [5, 6]and He/ He mixtures [7] near the superfluid transitionas well as for classical binary liquid mixtures near theirdemixing transition [8] and, directly, between a singlespherical particle immersed in a binary liquid mixturenear the critical demixing point and its confining pla-nar wall [9, 10]. Due to the similarity of these forceswith the Casimir forces arising in quantum electrody-namics [11, 12] they are called critical Casimir forces.
Nonspherical particles near other confinements, suchas a planar wall, experience orientation dependent forcesgiving rise to critical Casimir torques [13, 14].In the following we concentrate on critical solvents be-longing to the Ising universality class such as demixingclassical binary liquid mixtures. In these systems theparticle surfaces generically prefer one of the two com-ponents of the mixture, i.e., in Ising language one of thetwo directions (+ or − ) of the order parameter is pre-ferred. This preference is captured in terms of surfacefields the strength of which under renormalization groupflow attains ±∞ , corresponding to fixed surface spinsand denoted by ± boundary conditions. However, using suitable surface preparation one can suppress this pref-erence, being left with a weakened tendency to demixnear the particle surface [15]. These two types of surfaceuniversality classes [16] are called “normal” (+ / − ) and“ordinary” (O), respectively.Simple universal behavior arises in the scaling regionwhere the ranges of the interactions between the order-ing degrees of freedom other than short-ranged ones andthe lengths characterizing corrections to scaling and thecrossover from less stable bulk and surface universalityclasses, respectively, are much smaller than the increas-ing correlation length of the bulk solvent upon approach-ing criticality, than the distances between the particles,and than the lengths characterizing the sizes and shapesof the particles. In this region of a clear separation of“microscopic” and “mesoscopic” lengths, the forces andtorques are given by universal scaling functions which,apart from the surface universality classes, only dependon ratios of the characteristic mesoscopic lengths.Here, our main interest is in the orientation depen-dent interaction of nonspherical particles with bound-aries of critical systems. For the interaction betweena prolate uniaxial ellipsoid and a planar wall with ++or + − boundary conditions at the corresponding sur-faces the complete universal scaling functions for the crit-ical Casimir forces, both for the disordered and orderedbulk phases, have been obtained within mean field the-ory [14]. Beyond mean field theory such scaling functionshave been determined to a lesser degree of completeness.For nearby particles, i.e., for closest surface to surfacedistances much smaller than their size and radii of cur-vature, the interaction can be expressed in terms of theCasimir force in film geometry by using the Derjaguinapproximation. The scaling function of the latter is avail-able for ++ and + − surfaces in three [17–19] and two [20]spatial dimensions [21]. For a small spherical [22] or non-spherical [13, 23] particle, with a size much smaller thanthe bulk correlation length and the distances to the sur-faces of other particles or to boundaries, the interactionscan be obtained from the so-called “small particle expan-sion” [24]. This is one of the field theoretic operator ex-pansions for small objects the most well known of whichis the product of two nearby operators first consideredaround 1970 by Wilson, Polyakov, and Kadanoff [25, 26].These expansions are to a certain extent reminiscent ofthe multipole expansion in electrostatics.In view of the twofold asymptotic condition, i.e., theparticle being small on mesoscopic scales and large onthe microscopic scale [24], it is a nontrivial issue underwhich circumstances the results of the “small particleexpansion” can be observed for a given actual system.As a first step to address this issue, here we investigatewhether the results of the expansion can be observed inthe two-dimensional Ising model on a square lattice with(ferromagnetic) couplings J > W rows and L columns with an embedded particleresembling a needle. The directions along the rows andcolumns define, respectively, the directions u and v par-allel and perpendicular to the strip of length [27] L andwidth W . While in the u direction we impose periodicboundary conditions by means of couplings J betweenthe first and last spin in each row, we couple all of thespins in the lowest and uppermost row by means of in-teractions 0 or ±J to spins which are fixed in the + di-rection and are located in outside neighboring rows (seeFig. 1). Thus in the v direction, the strip is bounded byfree surfaces or surfaces to which a magnetic field ∝ ±J is applied, i.e., by surfaces of “ordinary” ( O ) or “normal”(+ / − ) character.In order to generate an embedded needle with O or+ / − boundary conditions and orientation parallel to thestrip we remove couplings (break bonds) between twoneighboring lattice rows or we fix spins within a singlelattice row, respectively (see Fig. 1). In the two caseswe define the length D of the needle as the number [27]of broken bonds and of fixed spins, respectively, whichwe choose to be even and odd, respectively, in order thatthe needle centers × coincide with the center of an ele-mentary square and a vertex of the lattice, respectively.This allows us to “turn” the needles abruptly about theircenter by 90 degrees upon rearranging the broken bondsand fixed spins, leading to broken bonds between neigh-boring columns and fixed spins within a single column ofthe lattice, respectively. In order to be able to positionthe centers of O and + / − needles right at the midline ofthe strip (as shown in Fig. 1) we choose W to be evenand odd, respectively.In the simulation we calculate the free energy expense∆ F required to “turn” the needle about its center from PSfrag replacements (a) (b)(c) (d) ( i [ h ] j ) = ( O [ O ] O ) ( i [ h ] j ) = (+[ O ] O )( i [ h ] j ) = ( O [+] O ) ( i [ h ] j ) = (+[+] − ) L = 8, W = 8, D = 4 L = 8, W = 7, D = 5 no couplingno couplingno couplingno coupling no couplingferro couplingferro couplingantiferro coupling vv uuW L v = 0 v = 0 W/ − W/ / / / ( W − / W − / W + 1) / W + 1) / − / − / − / − ( W − / − ( W − / − ( W + 1) / − ( W + 1) / − − − FIG. 1: Needle, with their centers denoted by × , embed-ded in a strip of a ferromagnetic Ising model on a squarelattice comprising W rows and L columns [27] of fluctuatingspins (empty circles) and periodic boundary condition in the u direction parallel to the rows. The two additional rows ofspins fixed in the + direction (full circles) allow one to in-duce positive or negative magnetic fields at the surfaces ofthe strip by coupling them in a ferro- or antiferromagneticway to the bottom and top rows of fluctuating spins (see themain text). (a) and (b): Needle of D = 4 broken bonds ina strip with the number W of rows and the number L ofcolumns equal to W = L = 8. (a) shows the case ( O [ O ] O ) ofa strip with two free surfaces (no coupling to the fixed spins)while (b) shows the case (+[ O ] O ) in which a ferromagneticcoupling to the fixed spins leads to a positive magnetic fieldat the lower surface. (c) and (d): Needle of D = 5 spins fixedin the + direction (full circles) in a strip with W = 7 and L = 8. (c) shows the case ( O [+] O ) while (d) shows the case(+[+] − ) with positive and negative magnetic fields induced atthe lower and upper surfaces by ferro- and antiferromagneticcouplings, respectively. All the needles shown are oriented inthe u direction and have their center × at the midline v = 0 ofthe strip, half way between the strip surfaces, i.e., v N = 0 forthe vertical position of the center of the needle. This requiresto choose W even (odd) for needles of broken bonds (of fixedspins) so that spins reside at half odd integer (integer) valuesof v . Needles oriented in the v direction are shown in Figs. 6and 7 below. an alignment perpendicular to the strip to the parallelalignment. This is a measure for the effective torque act-ing on the needle. In line with the introductory remarksthe free energy cost ∆ F depends on the distances of theneedle center from the strip surfaces.We put the origin of the ( u, v ) coordinate system at themidline of the lattice, i.e., for O and + / − needles at thecenter of an elementary square and at a vertex so thatthe lattice spins are located at half odd integer and in-teger values, respectively, of the coordinates (see Fig. 1).With this choice the mirror symmetry of ∆ F in a stripwith equal boundary conditions i = j is described simplyby its invariance if the coordinate v = v N of the needlecenter changes sign. Here i and j denote the boundaryconditions at the surfaces at v = − W/ v = W/ × of O and + / − needles coincide with the ori-gin, i.e., v N vanishes, while in the general case v N takesinteger values for both types of needles.It is useful to adopt a notation ( i [ h ] j ) which character-izes the boundary conditions i and j of the strip surfacesas well as the boundary condition h of the embedded nee-dle. For example, (+[+] − ) denotes the case of a needleof spins fixed in the + direction which is embedded in astrip with outside couplings J and −J at the lower andupper surfaces inducing strip surfaces i = + and j = − ,respectively, as shown in Fig. 1(d).For completeness we consider also the case of a stripwith periodic boundary conditions in both u and v di-rections by coupling, in both the W rows and in the L columns, the corresponding end spins to each other withstrength J . This is a strip without surfaces and is equiv-alent to a square lattice on a torus.If D, W, L, and the closest distance between the nee-dle and the strip boundaries (corresponding to a N and a < in Figs. 2(a) and 2(b), respectively) are “sufficiently”large on the scale of the lattice constant, the free en-ergy cost ∆ F in the lattice model is expected to dis-play universal scaling behavior and, as mentioned above,∆ F/ ( k B T ) will depend only on the universality classes( i [ h ] j ) of the boundaries and the needle and on three in-dependent ratios of D, W, L , and v N . In this case one PSfrag replacements u uv vL LW W (a) (b) v N v N u N = 0 u N = 0 DD a < a N a N FIG. 2: Continuum [27] description of the geometry of a nee-dle in a strip. In the strip of length L and width W theembedded needle of length D is oriented parallel (a) and per-pendicular (b) to the strip boundaries at v = ± W/
2. Theneedle center × is located at u = u N = 0 and v = v N < a N ≡ v N + W/ > a < ≡ a N − D/ > a > = a < + D ) from the lower boundary. can adopt a mesoscopic continuum description with sharpstrip boundaries at v = ± W/ D − D fixedspins), lead to the same mesoscopic length D . Figure 2shows the various lengths characterizing the geometry ofa needle in a strip within the continuum description.For small mesoscopic needles and a long strip, i.e., for D ≪ W ≪ L , the universal behavior can be predictedby means of the “small particle expansion” and, as men-tioned earlier, it is one of our main goals to quantitativelyinvestigate whether present Monte Carlo simulations canaccess this regime.For a needle perpendicular to the boundary of a halfplane or embedded at the midline of a symmetric ( i = j )strip of infinite length, we derive the analytic form of theuniversal scaling behavior of the critical Casimir forcesin the complete range of mesoscopic needle lengths D ,increasing from small to large. This aspect is interestingin its own right and also allows us to estimate the rangeof validity of the “small needle expansion”.The predictions of the “small particle expansion” for∆ F in critical Ising strips and our analytic results for nee-dles of arbitrary length are presented in Secs. II and III,respectively. In Sec. IV we describe the lattice model inmore detail and explain how one obtains via Monte Carlosimulations the results for ∆ F which are compared withthe analytic scaling predictions in Sec. V . Our resultsare summarized in Sec. VI. In Appendices A and B wepresent the input material and the derivations which arenecessary in order to obtain the predictions and results ofSecs. II and III, respectively. For the convenience of thereader we present a glossary of our symbols and notationsin Appendix C.Besides for critical systems [13, 14] the orientation de-pendent interaction between a wall and a nonsphericalmesoscopic object (such as an ellipsoid, a semi-infiniteplate, or a spherocylinder) has been studied also for quan-tum electrodynamics [28, 29] and for purely entropic sys-tems in which it is induced by spherical so-called deple-tion agents with hard body interaction only [30] or by freenonadsorbing polymer chains [23]. As for correlation-induced forces we mention also the attractive effectiveforce generated by needles (rigid rods) acting as deple-tion agents [31] and the repulsive force generated bya nonadsorbing polymer chain grafted to the tip of a(model-) atomic force microscope near a repulsive wall[32]. Finally we note that the critical Casimir force be-tween inclusions in the two -dimensional Ising model hasbeen suggested as a possible mechanism for the presenceof long-ranged forces between membrane bound proteins[33], which are typically non-circular. II. SMALL NEEDLE EXPANSION
Here we specialize the “small particle expansion” fornon-spherical particles (see Ref. [13]) to the present caseof a needle embedded in the two-dimensional Ising modeland apply it to the geometry of a needle in a strip.We consider a needle of small mesoscopic length D ,centered at r N , and pointing along the unit vector n .Inserting the needle into the d = 2 Ising model at itscritical point [34] changes the Boltzmann weight of thecorresponding field theory by a factor exp( − δ H ) whichcan be represented by the operator [26] expansion [13, 23]exp( − δ H ) ∝ S I + S A (2.1)where S I = X O = φ,ǫ A ( h ) O " D ! x O ++ 116 x O D ! x O ∆ r N O ( r N ) + ... (2.2)with ∆ r N = ∇ r N and S A = X k,l =1 , n k n l − δ kl !" − π D ! T kl ( r N ) ++ X O = φ,ǫ A ( h ) O x O ) D ! x O ∂ ( r N ) k ∂ ( r N ) l O ( r N ) + ... (2.3)are the isotropic and anisotropic contributions, respec-tively. Here O = φ is the order-parameter-density op-erator and O = ǫ is the difference of the energy-density[35] operator e and its average h e i bulk in the unboundedplane (bulk) at bulk criticality (so that their bulk aver-ages h φ i bulk and h ǫ i bulk vanish at the bulk critical point).They are normalized such that the bulk two-point corre-lation functions [34] are [36] hO ( r ) O ( r ′ ) i bulk = | r − r ′ | − x O , (2.4)where (in d = 2) x φ = 1 / x ǫ = 1 are their scaling di-mensions [37]. The affiliation of the surface of the needleto the “ordinary” ( h = O ) or to the “normal” ( h = + or h = − ) surface universality class enters into Eqs. (2.1)-(2.3) via the universal half plane amplitude A ( h ) O . Thelatter is the amplitude of the profile [34] of O in the halfplane, hOi half plane = A ( h ) O a − x O , as function of the meso-scopic distance a from the boundary line of type h andis given by [37] A ( O ) φ = 0 , A (+) φ = −A ( − ) φ = 2 / , A ( O ) ǫ = −A (+) ǫ = −A ( − ) ǫ = 1 / . (2.5) The properties A ( O ) φ = 0 and A ( − ) φ = −A (+) φ of the am-plitudes of the order parameter profile are obvious fromthe up-down symmetry of the Ising spins [34]. The en-ergy density profile increases (decreases) upon approach-ing an “ordinary” (“normal”) boundary where the Isingspins are more disordered (more ordered) than in thebulk [35]. This implies the positive (negative) sign of thecorresponding amplitude A ( O ) ǫ ( A (+) ǫ = A ( − ) ǫ ). The ab-solute values of A ( O ) ǫ and A (+) ǫ = A ( − ) ǫ are the same, dueto duality properties [38, 39]. In Eq. (2.3) the contribu-tion from the stress [37] tensor T kl is the same [40] for allneedle types h = O, + , − .The ellipses in Eqs. (2.2) and (2.3) stand for contribu-tions of fourth order and higher [41] in the small meso-scopic length D . The common factor of proportionalityon the rhs of Eq. (2.1) is given [34] by h exp( − δ H ) i bulk because the bulk averages of S I and S A vanish at thebulk critical point.For a two-dimensional Ising model with boundaries theinsertion free energy of the needle depends on its posi-tion and orientation with respect to these boundaries.Removing the needle from the bulk model and insertingit in the bounded one at a distance from the bound-aries much larger than D changes the free energy by F ( r N , n ) = − k B T ln h (1+ S I + S A ) i BM where BM denotesthe bounded model in the absence of the needle. We shallconcentrate on the geometry of a needle in a strip as de-scribed in the Introduction (see Fig. 2) for which BM ≡ ST is the strip with boundaries ( i, j ) in the absence ofthe needle and correspondingly use the notation [27] r N = ( u N , v N ) , n = ( n || , n ⊥ ) (2.6)for the center and direction vectors of the needle with thecomponents parallel and perpendicular to the strip. Thedensity averages hOi ST at r N which enter h S I + S A i ST are, within this model, independent of u N and in thescaling region given by hO ( r N ) i ST = W − x O f ( i,j ) O ( v N /W, W/L ) , (2.7)where f are universal scaling functions and f ( j,i ) O followsfrom f ( i,j ) O upon replacing v N /W by − v N /W .Due to the continuity equation of the stress tensor [37]its average h T kl ( r N ) i ( i,j )ST in the strip is independent ofboth u N and v N and follows from the universal, scale-free, and shape-dependent contribution [42] Φ ( i,j )ST ( L/W )to the free energy F ST per k B T of the strip ST withoutthe needle: h [ T ⊥⊥ , T kk ] i ( i,j )ST = − [ L − ∂ W , W − ∂ L ]Φ ( i,j )ST ( L/W )= [1 , − W − ∆ i,j ( W/L ) . (2.8)Like in Eq. (2.6), here k and ⊥ denote directions paralleland perpendicular to the u axis of the strip,∆ i,j ( W/L ) = ( d/dδ )Φ ( i,j )ST ( δ ) , δ = L/W , (2.9)and the off-diagonal components h T ⊥k i ST = h T k⊥ i ST vanish by symmetry. Cardy [43] has obtained an ex-plicit form for all functions Φ ( i,j )ST ( δ ). While an ex-tensive discussion of ∆ i,j ( W/L ) is deferred to Ap-pendix A 2, here we mention a few basic properties.Obviously Φ ( i,j )ST and ∆ i,j are symmetric in i and j .For a long strip the leading behavior Φ ( i,j )ST ( L/W →∞ ) = ∆ i,j (0) × δ of Φ ( i,j )ST is linear in δ with∆ i,j (0) ≡ ∆ i,j given [37] by ∆ i,j = ( π/ − , − , , i, j ) = [( O, O ) , (+ , +) , (+ , − ) , (+ , O )]. While due tothe (+ ↔ − ) symmetry the equalities ∆ + , + (1 /δ ) =∆ − , − (1 /δ ) and ∆ + ,O (1 /δ ) = ∆ − ,O (1 /δ ) hold for arbi-trary 1 /δ , the equality ∆ O,O = ∆ + , + holds only for in-finitely long strips [44], i.e., 1 /δ = 0.For the free energies F k and F ⊥ associated with remov-ing the needle from the bulk system and inserting it inthe strip with its center at r N and with its orientation n parallel and perpendicular, respectively, to the u axis,Eqs. (2.1)-(2.3) and (2.6) yield F k ≡ F ( r N , n = (1 , − k B T ln Z k (2.10)and F ⊥ ≡ F ( r N , n = (0 , − k B T ln Z ⊥ (2.11)with Z k = 1 + ζ I − ζ A , Z ⊥ = 1 + ζ I + ζ A (2.12)where ζ I = X O = φ,ǫ A ( h ) O " D W ! x O + 116 x O D W ! x O ×× ∂ ∂ ( v N /W ) f ( i,j ) O (cid:18) v N W , WL (cid:19) + ... (2.13)and ζ A = − π D W ! ∆ i,j (cid:18) WL (cid:19) + 12 X O = φ,ǫ A ( h ) O x O ) ×× D W ! x O ∂ ∂ ( v N /W ) f ( i,j ) O (cid:18) v N W , WL (cid:19) + ... (2.14)follow from S I and S A in Eqs. (2.2) and (2.3), respec-tively. This implies the expression∆ F = − k B T ln( Z k /Z ⊥ ) (2.15)for the free energy required to turn the needle about itscenter from its orientation perpendicular to the u axis ofthe strip to the parallel orientation.In the expansions in Eqs. (2.13) and (2.14) a term ∝ D X is of the order of ( D/W ) X near the strip cen-ter, where | v N | ≪ W , and of the order of ( D/a N ) X nearthe strip boundaries, where a N ≡ ( W/ − | v N | ≪ W . These expansions for the partition functions Z k and Z ⊥ are useful if D/W and
D/a N are much smaller than 1.However, expanding the free energies F k , F ⊥ , and ∆ F in powers of D is not always useful for the compari-son with our simulation data. While D/W ≪ D/a N ≪ D of manylattice constants, at present it seems to be unrealistic toachieve ( D/W ) / ≪ D/a N ) / ≪
1. Thus weexpand the logarithm in Eq. (2.15) in terms of powers of D only in those cases in which the power D x φ ≡ D / does not appear [45] in Eq. (2.13). These are the caseswith an “ordinary” needle h = O , for which A ( h ) φ van-ishes, and with a “normal” needle in a strip with two“ordinary” boundaries i = j = O , for which the den-sity profile in Eq. (2.7) vanishes for the order parameter O = φ . Expanding the logarithm in these cases yields∆ F = ∆ F l + ∆ F nl + ... (2.16)with the leading contribution∆ F l k B T = − π D W ! ∆ i,j ( W/L ) (2.17)and the next-to-leading contribution∆ F nl k B T = D W ! A ( h ) ǫ " π ∆ i,j ( W/L ) ++ 316 ∂ ∂ ( v N /W ) f ( i,j ) ǫ (cid:18) v N W , WL (cid:19) . (2.18)The ellipses in Eq. (2.16) are of order D .For completeness we also consider a double periodicrectangle or strip with periodic boundary conditions inboth the u and v -directions, i.e., the surface of a torus. Inthis boundary-free case at T c the average of φ vanishes inthe strip, that of ǫ is independent of both u and v , andfor the average of the stress tensor Eq. (2.8) again ap-plies with ∆ i,j ( W/L ) replaced by a function ∆ P ( W/L ) =( d/dδ )Φ ( P )ST ( δ ) with the infinite strip limit ∆ P (0) ≡ ∆ P = − π/
12 (see Subsec. V A and Appendix A 1 for more de-tails). For the double periodic strip Eqs. (2.16)-(2.18)also apply if ∆ i,j ( W/L ) and f ( i,j ) ǫ ( v N /W, W/L ) are re-placed by the corresponding stress amplitude ∆ P ( W/L )and the energy density scaling function f ( P ) ǫ ( W/L ) whichis independent of v N /W .Explicit expressions for the functions ∆( W/L ) areknown [43] for all types of strips considered here. Con-cerning the scaling functions f O the dependence on theaspect ratio W/L is known [46, 47] for f ( P ) ǫ while in thepresence of boundaries the functions f ( i,j ) O ( v N /W, W/L )are, to the best of our knowledge, known only [48] for W/L = 0, i.e., for strips of infinite [49] length L = ∞ .For the convenience of the reader we collect these expres-sions in Appendix A.For the special case in which the distance a N = v N + W/ h from theboundary of type i at v = − W/ D ≪ a N and W/a N , L/a N → ∞ , the aboveexpressions for the needle in the strip with free energies F ≡ F ( i [ h ] j ) reduce to those for the needle in the halfplane v + W/ ≡ a > F ≡ F ( i [ h ]) where Eqs. (2.13) and (2.14) are replaced by ζ I = X O = φ,ǫ A ( h ) O A ( i ) O " ϑ x O + 1 + x O ϑ x O + ... ,ζ A = X O = φ,ǫ A ( h ) O A ( i ) O x O ϑ x O + ... , (2.19)and Eqs. (2.17) and (2.18) by∆ F l k B T = 0 , ∆ F nl k B T = A ( h ) ǫ A ( i ) ǫ ϑ (2.20)where ϑ = D a N . (2.21)As mentioned above these relations only apply if 0 ≤ ϑ ≪ III. NEEDLES OF ARBITRARY LENGTH
The “small needle expansion” is valid if the mesoscopiclength D of the needle is “small” compared to the othermesoscopic lengths of the system, i.e., within the presentmodel much smaller than the width and length of thestrip and the distance of the needle from the two bound-aries. Given the limited set of operators appearing inEqs. (2.2) and (2.3), it is an open issue what in this con-text “small” means quantitatively. In order to get a clue,in this section we study a few situations in which resultsfor an arbitrary mesoscopic needle length D can be ob-tained. These results, which we derive in Appendix B,are interesting also in their own right.(i) Needle in half-plane
We consider a needle of universality class h embeddedin the half plane, oriented perpendicular to the boundaryline of surface universality class i , and with its centerlocated at a distance a N from the boundary. Here wealso introduce the distance a < = a N − ( D/
2) of the closerend of the needle from the boundary (compare Fig. 2(b))so that one has ϑ ≡ D a N = 11 + (2 a < /D ) (3.1)for the length ratio ϑ defined in Eq. (2.21). Notethat ϑ tends to 0 and 1 for a small (or distant) nee-dle with D ≪ a N and a long (or close) needle with a < ≪ D , respectively. Both D and a < are assumed to be mesoscopic lengths, i.e., both are large on the mi-croscopic scale. We are interested in the free energy F ⊥ ≡ F ( i [ h ]) ⊥ = k B T f ⊥ i [ h ] ( ϑ ) required to insert the needlefrom the bulk into the half plane, and in the correspond-ing Casimir force − ( ∂/∂a N ) F ( i [ h ]) ⊥ ≡ − ( ∂/∂a < ) F ( i [ h ]) ⊥ =( k B T /a N ) g ⊥ i [ h ] ( ϑ ) with [50] universal scaling functions f i [ h ] and g i [ h ] . The above partial derivatives are taken atfixed needle length D . The force pushes the needle awayfrom (towards) the boundary if g i [ h ] is positive (negative).Due to symmetries the identities F (+[+]) ⊥ = F ( − [ − ]) ⊥ , F (+[ O ]) ⊥ = F ( − [ O ]) ⊥ , and F ( h [ i ]) ⊥ = F ( i [ h ]) ⊥ hold so that ex-changing the surface universality classes of the needle andof the boundary leaves the free energy unchanged. Thesefollow from the (+ ↔ − ) and the duality [51] symmetriesof the Ising model and are consistent with the symme-tries of the corresponding small needle expression whichfollows from Eqs. (2.11), (2.12), and (2.19).(ia) For i = h = O the effective interaction has theform (see the paragraph containing Eqs. (B2)-(B4)) F ( O [ O ]) ⊥ k B T = f ⊥ O [ O ] ( ϑ ) = − − ln (1 + ϑ ) (1 − ϑ ) (3.2)and is attractive within the entire range 0 ≤ ϑ ≤ ϑ ≪
1, Eq. (3.2) is in agreement with the cor-responding result F ( O [ O ]) ⊥ / ( k B T ) = − ϑ/ ϑ / − ϑ /
12 + O ( ϑ ) of the small needle expansion which fol-lows from Eqs. (2.11), (2.12), and (2.19). For ϑ ր F ( O [ O ]) ⊥ /k B T → − − × / D ) /a < ] shows a loga-rithmic dependence. The logarithmic divergence of thefree energy for D/a < → ∞ is related to the long-rangedbehavior ∝ /a < of the Casimir force for a needle of infi-nite length D = ∞ which is addressed in Eq. (3.9) below;its integral diverges for a < → ∞ (while for a needle of finite length D , the force decays more rapidly than 1 /a < as a < increases beyond D so that the integral is finite).(ib) For i = + , h = O the interaction is always repul-sive and has the form F (+[ O ]) ⊥ k B T = 2 − ln (1 + ϑ ) (1 − ϑ ) , (3.3)i.e., the same form as Eq. (3.2) but with ϑ replaced by − ϑ (see Appendix B 1).(ic) For the Casimir forces, in Appendix B 2 we find ex-pressions which apply to arbitrary combinations ( i, h ) ofthe universality classes. For later convenience we presentthem here in two forms − ∂∂a N F ( i [ h ]) ⊥ k B T = − ∂∂a N F ( O [ O ]) ⊥ k B T − a N − ϑ ˜ τ i,h ( ϑ ) ≡ − a N − ϑ − ϑ − ρ i,h ( ϑ ) ! (3.4)which are equivalent due to Eq. (3.2) and the definition˜ τ i,h ( ϑ ) ≡
112 (1 − ϑ + ϑ ) − ρ i,h ( ϑ ) . (3.5)The dependence of ρ i,h on ϑ is given by ρ i,h ( ϑ ) = 4 π ∆ i,h (1 /δ )[ K ∗ ( ϑ )] = πδ ∆ i,h (1 /δ )4 [ K ( ϑ )] (3.6)with the variable 1 /δ replaced by the function1 /δ = K ∗ ( ϑ ) / [4 K ( ϑ )] (3.7)of ϑ . Here K ∗ ( ϑ ) ≡ K ( ¯ ϑ ) , ¯ ϑ ≡ p − ϑ , (3.8)and K is the complete elliptic integral function (seeEqs. (8.113.1) and (8.113.3) in Ref. [52]). The quantity∆ i,h (1 /δ ) in Eq. (3.6) is the well-studied [43] Casimir (orstress tensor) amplitude for a strip ST without needle,with boundaries ( i, h ), and finite aspect ratio W/L = 1 /δ which has been introduced in Eqs. (2.8) and (2.9) and isgiven explicitly in Appendix A 2.The symmetries of F ( i [ h ]) ⊥ addressed above Eq. (3.2)are reflected in Eqs. (3.4)-(3.8) by the symmetries of∆ i,j (1 /δ ) discussed below Eq. (2.9).For the special cases ( i, h ) = ( O, O ) and (+ , O ) wehave checked that Eqs. (3.4)-(3.8) are consistent with thesimple expressions given in Eqs. (3.2) and (3.3). In par-ticular, ˜ τ O,O ( ϑ ) vanishes for all ϑ .The present results for a needle embedded in a halfplane involve, via ∆ i,h (1 /δ ), knowledge about the stripST without needle but with finite aspect ratio becausethe two geometries are related by a conformal mappingas explained in Appendix B 2.Varying ϑ from 0 to 1, for all combinations ( i, h )Eqs. (3.4)-(3.8) provide the complete crossover of theforce − ( ∂/∂a N ) F ( i [ h ]) ⊥ /k B T as the length D and the posi-tion (determined by the distances a N or a < ) of the needlein the half plane change from small and remote from theboundary to large and close to the boundary. Equation(3.7) tells us that the corresponding change in the ge-ometry of the strip ST is from remote ( δ ց
0) to close( δ ր ∞ ) strip boundaries, as expected.In the small needle limit ϑ ց τ i,h van-ishes (compare Eq. (A13)) and the expansion for small ϑ is provided by Eqs. (A8), (A10), and (A12) and the formof σ in Eq. (B22). We check in the paragraph contain-ing Eq. (B26) that this expansion is consistent with thesmall needle expansion in Eqs. (2.11), (2.12), and (2.19).In the long needle limit ϑ ր τ i,h → i,i (0) − ∆ i,h (0)] /π with ∆ i,h (0) given below Eq. (2.9)so that [53] − ( ∂/∂a N ) F ( i [ h ]) ⊥ /k B T → [ − , − , , / (32 a < ) , ( i [ h ]) = [( O [ O ]) , (+[+]) , ( − [+]) , (+[ O ])] . (3.9)The behavior ∝ a − < of the force per k B T acting on theneedle of infinite length D = ∞ follows easily [53] from comparing its scaling dimension with that of the onlyremaining mesoscopic variable a < . Unlike the expansionof the force for a small needle in which fractional powerssuch as a − N ( D/a N ) / may occur, the expansion aroundthe long needle limit involves only contributions for whichEq. (3.9) is multiplied with positive integer powers of a < /D ≡ α (compare Eqs. (A7) and (A11), the relationbetween κ and ¯ ϑ in Eq. (B22), and the relation ¯ ϑ =4 α (1 + α ) / (1 + 2 α ) which follows from Eq. (3.1)).In the limit ϑ ր k B T tends to F ( i [ h ]) ⊥ / ( k B T ) ≡ f ⊥ i [ h ] → ([ − , − , , /
32) ln( C i,h / (1 − ϑ )) , ( i [ h ]) = [( O [ O ]) , (+[+]) , ( − [+]) , (+[ O ])] , (3.10)where 1 / (1 − ϑ ) → D/ (2 a < ) in terms of a < . Here C i,h are numbers, in particular C O,O = 2 / and C + ,O =2 / (see Eqs. (3.2) and (3.3)). This logarithmic be-havior of F ( i [ h ]) ⊥ / ( k B T ) for the long perpendicular nee-dle should be compared with the power law behavior F ( i [ h ]) k / ( k B T ) → ∆ i,h D/a < of the free energy for a longneedle aligned at a small distance a < parallel to theboundary which is obtained by turning the perpendic-ular needle about that end which is closer to the bound-ary. For a particle of circular shape [22] with a radius R much larger than the distance a < between the clos-est points of the circle and the boundary, the interactionfree energy F ( i [ h ]) / ( k B T ) → π ∆ i,h (2 R/a < ) / exhibits,as expected intuitively, a power law exponent the value ofwhich lies in between those corresponding to perpendic-ular and parallel needles. Unlike the long perpendicularneedle, for the long parallel needle and the large circlethe exact results of the interaction free energies quotedabove are found already within the Derjaguin approxi-mation and exhibit an amplitude proportional to ∆ i,h which depends on ( i, h ). For all three particles (the per-pendicular and parallel needles as well as the circle) theparticle-boundary interaction is attractive (repulsive) for i = h ( i = h ).Figure 3(a) shows a comparison between the exact re-sult for F ( − [+]) ⊥ / ( k B T ) ≡ f ⊥− [+] ( ϑ ) which follows fromEq. (3.4) upon integration [50] and its “small needle ap-proximation” (“sna”) (compare Eqs. (B27) and (B33)).The approximation reproduces the exact result very wellfor ϑ < ϑ with 0 . < ∼ ϑ < ∼ . ϑ > ∼ . strip sit-uation ( − [+] j ) with the center of the perpendicular nee-dle at v N , the question arises down to which proximityof the + needle to the lower − boundary of the strip onecan expect the “small needle approximation” to be rea-sonably valid. Identifying the distance a N = D/ (2 ϑ ) inthe half plane with the distance W/ − | v N | ≡ W/ v N in the strip suggests the approximation for F ( − [+] j ) ⊥ tobe valid if | v N | /W is smaller than [1 − D/ ( W ϑ )] /
2. For0 . < ∼ ϑ < ∼ . D/W = 21 /
101 used in ourMC simulations (see, c.f., Subsec. V C) this rough argu-ment predicts that | v N | /W should be kept smaller than PSfrag replacements (a) ϑ f ⊥− [ + ] ( ϑ ) exactsna Eq.(3.10) - 0.3- 0.2- 0.1 0 0 0.2 0.4 0.6 0.8
PSfrag replacements (b) ϑ f ⊥ O [ O ] ( ϑ ) exactsna Eq.(3.10) -1- 0.8- 0.6- 0.4 0 0.2 0.4 0.6 0.8
PSfrag replacements (c) ϑ f ⊥ + [ + ] ( ϑ ) exactsna Eq.(3.10)
PSfrag replacements (d) ϑ f ⊥ + [ O ] ( ϑ ) exactsna Eq.(3.10)
FIG. 3: Free energy cost per k B T , f ⊥ i [ h ] , to transfer a needle of length D and universality class h from the unbounded plane(bulk) to the half plane, with an orientation which is perpendicular to the boundary of the half plane of universality class i . The dependence on ϑ = D/ (2 a N ), with a N as the distance of the needle center from the boundary, is shown for variouscombinations ( i [ h ]). The exact results for arbitrary D/ ≤ a N are compared with the “small needle approximation” (“sna”,Eq. (B33)), which is valid for small ϑ , and with the long needle behavior for ϑ → i = h ( i = h ) shown in(b) and (c) ((a) and (d)). about 1 / /
4, i.e., v N should be kept rather close tothe midline v = 0. For the anisotropy ∆ F ( − [+] j ) of thefree energy of the needle in the strip there is even moreuncertainty concerning the validity of the “sna” becausethe corresponding comparison for F ( − [+]) k is lacking, i.e.,the counterpart of Fig. 3(a) for a needle parallel to theboundary of the half space is missing.Figures 3(b)-(d) show the corresponding results for thefree energies k B T f ⊥ i [ h ] with ( i, h ) = ( O, O ) , (+ , +), and(+ , O ). For i = h and i = h , f ⊥ i [ h ] is negative and positive,respectively.(ii) Needle in infinitely long symmetric strip
Here we consider a needle of universality class h andwith parallel orientation || at the center line v = 0 ofan infinitely long ( i, i ) strip of width W . We determinethe free energy cost F k ≡ F ( i [ h ] i ) k = k B T f k i [ h ] i ( θ ) and the corresponding disjoining force − ( ∂/∂W ) F ( i [ h ] i ) k =( k B T /W ) g k i [ h ] i ( θ ) between the two boundary lines of thestrip upon inserting the needle from the bulk with par-allel orientation. Here, the partial derivative is takenwith the needle length D kept fixed. The disjoining forceincreases (decreases) if g i [ h ] i is positive (negative). The( i, h ) symmetries of F ( i [ h ] i ) k are those of ∆ i,h as can beinferred from Eq. (3.15) below. The results depend onthe length ratio [54] θ = πDW (3.11)and provide the crossover from the small needle behaviorfor θ ≪
1, consistent with the “small needle expansion”,to the long needle behavior for θ ≫
1. In the latter caseone has F ( i [ h ] i ) k k B T ≡ f k i [ h ] i → D " i,h (0)( W/ − ∆ i,i (0) W = θ [ − , − , , / , ( i, h ) = [( O, O ) , (+ , +) , ( − , +) , (+ , O )] (3.12)because upon inserting a long parallel needle with itscenter at u = v = 0 transforms, within the interval − D/ < u < D/
2, the ( i, i ) strip of width W into twoindependent adjacent ( i, h ) strips of width W/ i = h = O the free energy to insert the needlefrom the bulk reads F ( O [ O ] O ) k k B T = − θ θθ . (3.13)(iib) For i = + , h = O we find F (+[ O ]+) k k B T = θ θθ (3.14)which is Eq. (3.13) with θ replaced by − θ . Equa-tions (3.13) and (3.14) are derived in Appendix B 1.(iic) For arbitrary ( i, h ) we find (see Appendix B 2) − ∂∂W F ( i [ h ] i ) k k B T = − ∂∂W F ( O [ O ] O ) k k B T − W θt ˜ τ i,h ( t ) ≡ W ( −
14 + θ h (cid:16) t + t (cid:17) + 1 t ρ i,h ( t ) i) ,t ≡ tanh( θ/ , (3.15)where ˜ τ i,h ( t ) and ρ i,h ( t ) are the functions ˜ τ i,h ( ϑ ) and ρ i,h ( ϑ ) from Eqs. (3.5) and (3.6), respectively, evaluatedat ϑ = t . Equations (3.13) and (3.14) as well as Eq.(3.15) are consistent with the “small needle expansion”for small θ (see Eqs. (B29)) and with Eq. (3.12) for large θ . Figure 4 shows a comparison between the exact re-sult for F ( − [+] − ) k / ( k B T ) ≡ f k− [+] − ( θ ) which followsfrom Eq. (3.15) via integration [50] and its “small nee-dle approximation” (compare Eqs. (5.4), (B30), (B32),and (B34)). For the value θ = 21 π/
101 = 0 .
653 as usedin our MC simulations (see Subsec. V C) there is goodagreement. However, we have no such comparison for F ( i [ h ] i ) ⊥ .All four possible independent free energies k B T f k i [ h ] i ( θ )for needle insertion are collected in Fig. 5. They arenegative (positive) for i = h ( i = h ). In this doublelogarithmic plot the power laws as obtained for small θ from the “small needle expansion” and for large θ fromthe long needle expression (Eq. (3.12)) appear as straightlines.(iii) Periodic strip
Finally we consider an infinitely long strip of width W with periodic boundary condition in v direction. In this PSfrag replacements θ f || − [ + ] − ( θ ) exactsnasnae3 (2 θ ) / Eq.(3.12) − − − FIG. 4: Free energy cost per k B T , f k− [+] − , to remove a needleof universality class + and length D from the bulk and in-sert it along the midline of an infinitely long strip with ( − , − )boundary conditions and width W as a function of θ = πD/W (see Ref. [50] and Eq. (3.15)). The exact result for arbitrarylength ratio θ is compared with the “small needle approxi-mation” (“sna”, Eq. (B34)), valid for small θ , and with thelong needle behavior according to Eq. (3.12). Also shown arethe first term, (2 θ ) / , and the sum of the first three terms(2 θ ) / + θ / / / + (2 θ ) / /
3, denoted as “snae3”, in theexpansion of f k− [+] − in terms of powers of θ for θ ≪ f k− [+] − proves to be lesssuccessful than the “sna” within which exp( − f k− [+] − ) is ex-panded and which agrees with the exact result up to θ ≈ θ → case the free energy cost to insert a needle parallel to thestrip reads (see Appendix B 1) F k k B T = 116 ln sinh θθ . (3.16)This result holds for any needle universality class h = O, + , − and it is valid for an arbitrary ratio θ = πD/W .Equation (3.16) should be compared with Eq. (3.13). For θ ≫ θ/
16 = D × [∆ h,h (0) − ∆ P (0)] /W . This is consistent with the periodic stripof width W being transformed, by inserting a parallelneedle, to a ( h, h ) strip of width W within a u -interval oflength D . Consistent with the “small needle expansion”,for θ ≪ θ /
96 + O ( θ ). IV. FREE ENERGY ANISOTROPY FROMMONTE CARLO SIMULATIONS
The anisotropic shape of a bounded critical systeminduces orientation dependent properties for embeddednon-spherical particles. In Sec. II the operator expan-sion has provided the asymptotic scaling properties for a“small” but “mesoscopic” particle in a “large” system. In0
PSfrag replacements θ | f || ( θ ) | exact Eq.(3.12) sna ( − [+] − )(+[+]+) (+[ O ]+) ( O [ O ] O ) − − − − − − FIG. 5: Same as Fig. 4 for all four independent combina-tions ( i [ h ] i ) of a needle of universality class h placed alongthe midline of an ( i, i ) strip as function of θ = πD/W .Full, dash-dotted, and dotted lines correspond to the ex-act results, the “sna”, and Eq. (3.12), respectively. For h = i ( h = i ) f k i [ h ] i is negative (positive), decreasing (in-creasing) with increasing θ , and implies a negative (posi-tive) contribution − ∂ W F ( i [ h ] i ) k = ( k B T /W ) θ∂ θ f k i [ h ] i to thedisjoining force. Thus the attractive Casimir interaction(see Eq. (2.8)) with the negative universal disjoining force − ∂ W k B T Φ ( i,i )ST = ∆ i,i k B T L/W = − ( π/ k B T L/W be-tween the ( i, i ) boundaries of the long strip without the nee-dle becomes even more attractive (less attractive) due to thepresence of the needle with h = i ( h = i ). the present section we address the issue to which extentthese asymptotic properties capture the actual behaviorin specific critical model systems. Concerning the needlesstudied here, we want to check whether the asymptoticpredictions of Sec. II for ∆ F can already be observedwithin a lattice model with numerous, but not too manyrows and columns so that the model is amenable to sim-ulations. In this section we describe how to set up thecorresponding Monte Carlo simulations and to calculate∆ F . In Sec. V we compare these simulation data withthe corresponding analytic predictions. This allows us tojudge both the achievements and the limitations of the“small particle approximation”. A. Model
For the simulation we use the lattice version of the Isingstrip described in the Introduction and shown in Fig.1. The implementation of the double periodic boundaryconditions is obvious. Beyond that, here we describe inmore detail strips with boundaries ( i, j ). In this case thelattice Hamiltonian H ST for strips without a needle reads(see Sec. I): H S T / J = − X h u,v ; u ′ ,v ′ i J u,v ; u ′ ,v ′ s u,v s u ′ ,v ′ − Λ (1) i X u s u, − ( W +1) / s u, − ( W − / − Λ (1) j X u s u, ( W +1) / s u, ( W − / (4.1)with J > J u,v ; u ′ ,v ′ = 1 for nearest neighbors (de-noted by h u, v ; u ′ , v ′ i ) and zero otherwise. The fluctu-ating Ising spins s = ± W rows and L columns withperiodic boundary conditions in the u direction. Thetwo last terms generate “ordinary” or “normal” (+ / − )boundaries near v = − W/ v = W/ s u, − ( W +1) / = s u, ( W +1) / = 1 for the spinsin the two additional outside rows and by choosing in-dependently Λ (1) i and Λ (1) j equal to 0 (“ordinary”) or ± / − “normal”). This serves as a microscopic realizationof all the pairs ( i, j ) of strip boundary types consideredin Sec. II which exhaust all possible surface universalityclasses in d = 2 (up to the “extraordinary” boundarytype [38, 44] corresponding to infinitely strong couplingsbetween surface spins).For reasons given in the Introduction and shown inFig. 1, for “ordinary” (“normal”) needles the number W of rows is taken to be even 2, 4, 6, ... (odd 1,3, 5, ...) and the components of the lattice vertices u, v are half odd integers ± / , ± / , ± / , ... (integers0 , ± , ± , ... ). The components u = u N and v = v N ofthe needle centers × are integers for both types of nee-dles.In order to be able to compare the Monte Carlo datawith the results of Sec. II we consider the system atits bulk critical point T c with J / ( k B T c ) = ln( √ J is the coupling constant scaled out of H ST (Eq. (4.1)).Inserting a needle of class h = O or h = ± into stripsamounts to appropriately removing bonds or fixing spinsin accordance with Fig. 1 via additional terms in theHamiltonian. For given surface universality classes ( i, j )of the strip, universality class h , needle length D , and thecoordinate v N of the center of the needle we introduce thenotation H = H ≡ H ST + H ( h ) ⊥ (4.2)and H = H ≡ H ST + H ( h ) k (4.3)for the total lattice Hamiltonian H corresponding to theneedle being oriented perpendicular and parallel, respec-tively, to the u axis. The explicit forms of H ( h ) ⊥ and H ( h ) k will be given in Eqs. (4.8), (4.9), and (4.11)-(4.18) below. B. Numerical algorithm
1. Coupling parameter approach
We consider two systems with the same configura-tional space C (i.e., number and spatial connectivity of1spins) and with Hamiltonians H and H as given inEqs. (4.2) and (4.3). The corresponding free energiesare F , = − β ln P C exp( − β H , ); β = 1 / ( k B T ) is theinverse thermal energy. We are interested in the free en-ergy difference ∆ F = F − F .For the computation of this free energy difference weuse the coupling parameter approach [55]. To this endwe introduce the cr ossover Hamiltonian H cr ( λ ) which de-pends on the coupling parameter λ , H cr ( λ ) = H + λ ( H − H ) = H + λ ∆ H , (4.4)with the Hamiltonian difference ∆ H = H − H whichin the present context is H ( h ) k − H ( h ) ⊥ . The deriva-tive of the corresponding cr ossover free energy F cr ( λ ) = − β ln P C exp( − β H cr ( λ )) with respect to the couplingparameter readsd F cr ( λ )d λ ≡ F ′ cr ( λ ) = P C ∆ H e − β H cr ( λ ) P C e − β H cr ( λ ) = h ∆ Hi cr ( λ ) , (4.5)where h ∆ Hi cr ( λ ) is the Hamiltonian difference ∆ H aver-aged with respect to H cr ( λ ). Therefore one can computethe free energy difference by integrating h ∆ Hi cr with re-spect to λ :∆ F = F − F = Z F ′ cr ( λ )d λ = Z h ∆ Hi cr ( λ )d λ. (4.6)For the forms of H and H given by Eqs. (4.2) and (4.3),respectively, one has∆ H = H ( h ) k − H ( h ) ⊥ , H cr ≡ H ( h )cr = H ST + (1 − λ ) H ( h ) ⊥ + λ H ( h ) k . (4.7)
2. Needle of broken bonds
Here we consider the combination ( i [ O ] j ) correspond-ing to an “ordinary” needle which consists of an evennumber D of broken bonds in a strip with an even number W of rows. Figure 6 shows the example of a needle with D = 4 broken bonds in the center of an L × W = 6 × H and H for perpendicularand parallel needle orientation have the same form asthe right hand side of Eq. (4.1) but with reduced inter-action constants J u,v ; u ′ ,v ′ which depend suitably on thecoordinates ( u, v ) and ( u ′ , v ′ ) of nearest neighbor spins.Choosing without restriction u N = 0, the needle withcenter at (0 , v N ) is inserted in perpendicular orientationby “breaking”, i.e., removing the D lattice bonds whichat v = v N ± / , v N ± / , ..., v N ± ( D − / u = − / u = 1 /
2. This is accomplished by adding H ( O ) ⊥ / J = D X k =1 s − / ,v N − ( D +1) / k s / ,v N − ( D +1) / k ≡ X h inc . i (4.8) PSfrag replacements (a) (b) (c) fluctuating spinfixed spin (+1) J u,v ; u ′ ,v ′ = 1 normal bondbroken bonds not shownbonds to boundaries decreasing bond: 1 − λ increasing bond: λ Λ (1) i Λ (1) i Λ (1) j Λ (1) j uuu u = 0 u = 0 u = 0 L − L − L − − − − − − − − L − − L − − L − v v = 0 W/ − W/ / / ( W − / W + 1) / − / − / − ( W − / − ( W + 1) / FIG. 6: Bond arrangements for a needle ( O ) of D = 4 bro-ken bonds with center ( × ) at ( u N , v N ) = (0 ,
0) at the midline v = 0 of a strip with L = W = 6: (a) for the perpendic-ular orientation of the needle with Hamiltonian H ; (b) forthe parallel orientation of the needle with Hamiltonian H ;(c) for the crossover Hamiltonian H ( O )cr ( λ ) which interpolatesbetween (a) for λ = 0 and (b) for λ = 1. Bonds of strength J = 1 are indicated by thin black lines. Broken bonds ( J = 0)are not shown. The bonds indicated by green dashed and bluedot-dashed lines have strengths λ and 1 − λ which increase anddecrease, respectively, as λ increases. As in Fig. 1 the fluc-tuating spins s u,v = ± | v | ≤ ( W − / s u, ± ( W +1) / = 1 in thetwo additional outside rows are indicated by full circles. Thenearest neighbor bonds between fixed and fluctuating spinsare indicated by thick magenta lines. For the lower and up-per boundary i and j they have the strengths Λ (1) i and Λ (1) j ,respectively, where Λ (1) equals 0 for an O boundary and ± / − boundary. to the Hamiltonian H ST / J without needle (seeEq. (4.2)). Similarly for the parallel orientation of theneedle one has to add H ( O ) || / J = D X k =1 s − ( D +1) / k,v N − / s − ( D +1) / k,v N +1 / ≡ X h decr . i (4.9)so that ∆ H = H ( O ) k − H ( O ) ⊥ . Figures 6(a) and (b) illus-trate these configurations for the special case v N = 0.The sums in Eqs. (4.8) and (4.9) have been charac-terized by subscripts h inc . i and h decr . i because in thecrossover Hamiltonian following from Eq. (4.7), H ( O )cr ( λ ) = ˜ H ( O ) − λ J X h inc . i − (1 − λ ) J X h decr . i , (4.10)they appear with a prefactor − λ and − (1 − λ ), respec-tively, representing sums of products of spins coupled bynearest neighbor bonds with strengths λ J and (1 − λ ) J which increase and decrease, respectively, as λ increases.Here ˜ H ( O ) ≡ H ST + J P h inc . i + J P h decr . i equals H ST inEq. (4.1) but with both types of nearest neighbor bondsmissing which are broken in the perpendicular or the2parallel orientation of the needle. This corresponds toFig. 6(c) with both dashed and dash-dotted bonds re-moved (so that in ˜ H ( O ) only those bonds of H ST remainwhich are outside a hole with the shape of a cross). In H ( O )cr ( λ ), however, the two types of missing bonds are re-placed by the bonds of increasing and decreasing strengthas illustrated in Fig. 6(c) by the green dashed and bluedash-dotted lines. This obviously leads to the crossoverfrom the perpendicular to the parallel needle orientationas λ increases from 0 to 1.On this basis, following the steps described by Eqs.(4.4)-(4.7) allows us to calculate F k − F ⊥ ≡ F − F = ∆ F for the combination ( i [ O ] j ).
3. Needle of fixed spins
In this subsection we consider the lattice version of thecase ( i [+] j ) in which a needle consisting of an odd num-ber D of spins fixed in the + direction is embedded in astrip with an odd number W of rows. Figures 7(a) and PSfrag replacements (a) (b) (c) fluctuating spinfixed spin bond to outside boundaryΛ (1) i Λ (1) j vu u u = 0 u = 0 − W +12 − W − W − W +12 − − − − − L + 1 − L + 1 L L xy FIG. 7: Bond arrangements for a needle of D = 5 spins fixed in the + direction with the needle center ( × ) at ( u N , v N ) =(0 ,
0) at the midline v = 0 of a strip with L = 8 and W = 7for the perpendicular direction of the needle (a) and for theparallel direction of the needle (b). The strengths Λ (1) of thebonds near the strip boundaries are as explained in Fig. 6. (b) show the example of a needle with D = 5 in the centerof an L × W = 8 × D +2 fluctuating nearest neighbors are removed andthese 2 D + 2 neighbors are coupled instead with the bulkstrength J to a single exterior spin s = +1 which iskept fixed in the + direction. The coupling to this singlespin or to the D fixed spins has the same effect on thefluctuating spins, namely that of a magnetic field actingon the 2 D + 2 neighboring spins. Once this coupling to s is in place, for the following it is convenient to replaceeach of the D fixed needle spins by a freely fluctuatingspin, i.e., free of any couplings. This changes the freeenergy per k B T only by D ln 2, independent of the orien- PSfrag replacements (a) (b)(c) fixed spin s = +1fluctuating spinfixed spin (+1)needle center J u,v ; u ′ ,v ′ = 1 normal bondbond to boundarybroken bonds not shownincreasing bond: λ decreasing bond: 1 − λ normal bond to s increasing bond to s : λ decreasing bond to s : 1 − λ s s s v uu u u Λ (1) i Λ (1) i Λ (1) j Λ (1) j − W +12 − W +12 − W − − W − W − W − W +12 W +12 − − −
20 00 1 11 − L + 1 − L + 1 − L + 1 L L L − − − xy FIG. 8: (a) and (b) describe how to mimic the configurationsdiscussed in Fig. 7 by means of couplings (north-east arrows)to a fixed external spin s = +1; (c) shows the arrangement ofbonds in the crossover Hamiltonian H (+)cr ( λ ) (see Eq. (4.19))with their strengths indicated in the right margin. The ar-rangement reduces to that of (a) and (b) for λ = 0 and λ = 1,respectively. For further explanations see the main text. tation of the needle and thus drops out of ∆ F . For theabove example discussed in Fig. 7 this alternative modelis illustrated in Figs. 8(a) and (b), with the couplings tothe external spin denoted by north-east arrows.The corresponding additional terms in the Hamilto-nian can be written as H (+) ⊥ = X h zero i + dX h inc . i − (+) X h one i − (+) X h decr . i (4.11)and H (+) || = X h zero i + dX h decr . i − (+) X h one i − (+) X h inc . i (4.12)(so that ∆ H = H (+) k − H (+) ⊥ ) where the sums X h zero i = s ,v N ( s − ,v N + s ,v N + s ,v N +1 + s ,v N − ) , (4.13)3 dX h inc . i = ( D − / X k =1 h s ,v N + k ( s − ,v N + k + s ,v N + k ) (4.14)+ s ,v N − k ( s − ,v N − k + s ,v N − k )+ s ,v N + k s ,v N + k +1 + s ,v N − k s ,v N − k − i , and dX h decr . i = ( D − / X k =1 h s k,v N ( s k,v N +1 + s k,v N − ) (4.15)+ s − k,v N ( s − k,v N +1 + s − k,v N − )+ s k,v N s k +1 ,v N + s − k,v N s − k − ,v N i contain products of nearest neighbor lattice spins whilethe sums (+) X h one i = s − ,v N +1 + s ,v N +1 + s − ,v N − + s ,v N − , (4.16) (+) X h decr . i = s ,v N +( D − / + s ,v N − ( D − / − + s − ,v N + s ,v N (4.17)+ ( D − / X k =2 h s − ,v N + k + s ,v N + k + s − ,v N − k + s ,v N − k i , and (+) X h inc . i = s ( D − / ,v N + s − ( D − / − ,v N + s ,v N +1 + s ,v N − (4.18)+ ( D − / X k =2 h s k,v N +1 + s k,v N − + s − k,v N +1 + s − k,v N − i contain products of a lattice spin and the fixed exter-nal spin s (which has the value s = +1 and thus doesnot appear in Eqs. (4.16)-(4.18)). Here P h zero i contains theproducts of the center ( × ) spin of the needle with its fournearest neighbor spins which correspond to lattice bondsbroken in both the perpendicular and parallel needle ori-entation (compare Figs. 8(a) and (b)). The productsin dP h inc . i and dP h decr . i correspond to the remaining nearestneighbor bonds which are broken in the perpendicularand parallel needle orientation, respectively. The fourterms in (+) P h one i correspond to the bonds between the ex-ternal spin and those four lattice spins which are coupledto it in both the perpendicular and the parallel needleconfiguration (compare Figs. 8(a) and (b)). The sums (+) P h decr . i and (+) P h inc . i contain the terms which correspond tothe rest of the bonds to the external spin in the perpen-dicular and the parallel needle orientation, respectively.As in the previous subsection the notation for the vari-ous sums reflects the modulus of their corresponding pref-actors in the crossover Hamiltonian H (+)cr ( λ ) = ˜ H (+) − λ J dX h inc . i − (1 − λ ) J dX h decr . i −J (+) X h one i − λ J (+) X h inc . i − (1 − λ ) J (+) X h decr . i . (4.19)The first term ˜ H (+) ≡ H ST + J P h zero i + J dP h inc . i + J dP h decr . i corresponds to a strip with a cross-shaped hole wherethe bonds belonging to P h zero i , dP h inc . i , and dP h decr . i are miss-ing (i.e., in Fig. 8(c) this means that the dashed anddash-dotted bonds and all the arrows are removed). In H (+)cr ( λ ) the contributions due to the two last types ofbonds missing in ˜ H (+) , i.e., dP h inc . i and dP h decr . i carry the pref-actors − λ J and − (1 − λ ) J of increasing and decreasingstrengths , respectively. Moreover, bond contributionsare added which couple the lattice spins contained in (+) P h one i , (+) P h inc . i , and (+) P h decr . i with strengths 1, λ , and 1 − λ ,respectively, to the external spin s = 1. For L = 8, W = 7, D = 5, and the needle center × at v N = 0 atthe midline of the strip, the various bond strengths in H (+)cr ( λ ) are shown in Fig. 8(c) which clearly illustratesthe crossover from the perpendicular to the parallel nee-dle orientation considered in Figs. 8(a) and 8(b) as λ increases from 0 to 1.
4. Details of the numerical implementation
For the sequential generation of system configurationswe have used the hybrid Monte Carlo method [56]. Onestep consists of updating a Wolff cluster [57] followed by L × W/ D + 3) updates of ran-domly chosen spins in the square of size ( D + 3) × ( D + 3)with the center at position (0 , v N ).In order to determine the dependence of the free en-ergy on v N we have used system sizes 1000 ×
100 and1000 ×
101 for ( i [ O ] j ) and ( i [+] j ) needles, respectively.For thermalization we have used 1 . × MC steps, fol-lowed by the computation of the thermal average using8 × MC steps. These latter MC steps have been splitinto 16 intervals which facilitates to estimate the numer-ical inaccuracy.In order to determine the aspect ratio dependence ofthe free energies we have used various numbers of MC4steps (split into 8 intervals for estimating again the nu-merical inaccuracy) for various system sizes, varying from6 × MC steps for L = 4000 to 2 . × for L = 200.We have used one fifth of these MC steps in order toachieve thermalization.Concerning the numerical integration over thecrossover variable λ , for every selected set of parame-ters (i.e., type of needle and boundary conditions ( i [ h ] j ), L, W, D , and v N ) we have performed computations for 32points λ k = k , k = 0 , , . . . ,
31 and then we have car-ried out the numerical integration by using the extendedversion of Simpson’s rule.
V. COMPARISON OF ANALYTIC RESULTSWITH SIMULATION DATA
Here we compare the “small needle” predictions fromSec. II for the quasi-torque ∆ F with corresponding re-sults obtained by the Monte Carlo simulations describedin Sec. IV. On one hand this allows us to assess theperformance of the truncated form of the “small needleapproximation”, i.e., to determine the smallness of themesoscopic needle length D needed in order to be ableto neglect higher order terms in this expansion. On theother hand, good agreement signals that all the meso-scopic distances and lengths, including D , chosen in thesimulations turn out to be large enough for the latticesystem to lie within the universal scaling region. A. Aspect ratio dependence for the doubleperiodic strip
For a needle in a double periodic strip, one can studythe full dependence of ∆ F (Eq. (2.15)) on the aspectratio W/L . Due to the symmetry of these boundary con-ditions, ∆ F keeps its modulus but changes its sign as thevalues of W and L are exchanged. This implies that ∆ F vanishes for W = L . Within the small needle approxima-tion, its quantitative behavior follows from the remarksbelow Eq. (2.18) yielding (see Eqs. (2.16)-(2.18))∆ Fk B T ≃ ∆ F l + ∆ F nl k B T = − π D W ! ∆ P ( W/L ) ×× " (cid:26) − , (cid:27) D W f ( P ) ǫ ( W/L ) (5.1)for an { “ordinary”, “normal” } needle with A ( h ) ǫ = { / , − / } (see Eq. (2.5)). Equation (5.1) is con-sistent with the sign change mentioned above becauseboth Φ ( P )ST (see Eqs. (2.8) and (2.9)) and h ǫ i ( P )ST = W − f ( P ) ǫ ( W/L ) (see Eq. (2.7) with x ǫ = 1) re-main unchanged upon exchanging the values of W and L , implying ∆ P ( W/L ) /W = − ∆ P ( L/W ) /L and PSfrag replacements
W/LW/LW/LW/LW/LW/LW/LW/LW/LW/L [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) π / π / π / π / π / π / π / π / π / π / [∆ F l / ( k B T )] / ( D/W ) ( P ) , [ O ] , W = 100 , D = 10( P ) , [ O ] , W = 100 , D = 20( P ) , [+] , W = 101 , D = 11( P ) , [+] , W = 101 , D = 21 FIG. 9: Normalized quasi-torque ∆ F = F || − F ⊥ (Eq. (2.15))acting on a small mesoscopic needle of length [27] D in a dou-ble periodic strip P as a function of the aspect ratio W/L of the strip. ∆
F > F vanishes at W/L = 1 andis antisymmetric around this point. The black dotted linedenotes its limit for
W/L = 0. For “ordinary” [ O ] and “nor-mal” [+] needles with D = 20 and D = 21, respectively, thesimulation data (green squares and magenta diamonds) agreequite well with the analytic prediction (red dash-dotted line)according to the first term on the rhs of Eq. (5.1). The factthat the (green) squares lie above the (magenta) diamondsagrees with the tendency arising from the second term ∆ F nl in Eq. (5.1) due to f ( P ) ǫ <
0. However, the splitting between[ O ] and [+] due to ∆ F nl is quite small. (Considering D = 20and W = 100, the explicit form of f ( P ) ǫ given in Eqs. (A2)-(A4) leads at, e.g., W/L = 0 . W/L = 0 . ± .
003 and 1 ± . D, W , and L which are closer to the asymp-totic scaling limit than the ones presently available. For moredetails see the main text. f ( P ) ǫ ( W/L ) /W = f ( P ) ǫ ( L/W ) /L . Since, due to Ap-pendix A 1, − sgn( L − W ) × ∆ P ( W/L ) is positive and f ( P ) ǫ is negative, Eq. (5.1) implies the following:(i) Both “ordinary” and “normal” needles prefer toalign perpendicular to the longer axis of the dou-ble periodic strip. (Concerning related effects seeRef. [59].)(ii) While in leading order ∝ D the strength of thispreference is independent of the needle type h , it isstrengthened (weakened) for an “ordinary” (“nor-mal”) needle by the correction of order D .A quantitative comparison of the aspect ratio depen-dence predicted in Eq. (5.1) with our simulation data isprovided in Fig. 9. This figure shows data [60] for the nor-malized quantity [∆ F/ ( k B T )] / ( D/W ) within the range0 < D/W < /
2. For the “ordinary” needle of “bro-ken bonds” the data were obtained from systems with5 W = 100 and D = 10 and 20. The data for the “normal”needle of “fixed spins” stem from systems with W = 101and D = 11 as well as D = 21. The data for D = 20and D = 21 are in rather good agreement with the ex-pression [∆ F l / ( k B T )] / ( D/W ) , as predicted accordingto Eq. (5.1) and Appendix A 1 for the leading order con-tribution ∝ D , shown as the red dash-dotted line. Thedata for the “ordinary” needle are indeed slightly largerthan those for the “normal” needle, as predicted by thenext-to-leading order contribution ∝ D in Eq. (5.1).The stronger deviations of the data for D = 10 and D = 11 from the analytic approximation presumably in-dicate that these smaller needle lengths lie outside themesoscopic scaling region required [24] for the validity ofEq. (5.1). B. Dependence of the free energy anisotropy onthe spatial position of needles in strips.
In strips with boundaries ∆ F depends on the position v N of the needle in the strip. Here we consider strips ofinfinite length L , either with arbitrary boundaries ( i, j )and containing an “ordinary” needle or with boundaries( O, O ) and containing a “normal” needle. In all thesecases ∆ F ≃ ∆ F l + ∆ F nl is predicted to have the formgiven by Eqs. (2.16)-(2.18). Ordinary needles
While ∆ F l does not depend on v N and is given byEq. (2.17) with ∆ i,j given below Eq. (2.9), in the cases( i [ O ] j ) one has∆ F nl k B T = 1256 πDW ! g i,j (5.2)which depends on v N via the following simple expressionsfor g i,j : g O,O = 3(cos V ) − − (5 / V ) − g + , + = − g O,O g + ,O = − g O, + = (tan V )[3(cos V ) − + (1 / g + , − = g − , + = − [3(cos V ) − ++(7 / V ) − − V )] (5.3)with V = πv N /W . They follow upon inserting f ( i,j ) ǫ fromEq. (A14) and A ( h ) ǫ ≡ A ( O ) ǫ = 1 / i [ O ] j ) with the corresponding analytic predictions forvarious positions v N of the needle and for various bound-ary conditions ( i, j ) of the strip. The plots show the datafor [(∆ F − ∆ F l ) / ( k B T )] / ( D/W ) with ∆ F obtained froma system with L = 1000 , W = 100, and D = 20. Thecomparison with [∆ F nl / ( k B T )] / ( D/W ) ≡ c i,j followingfrom Eqs. (5.2) and (5.3) and shown by lines is very fa-vorable (i.e., ∆ F is captured well by ∆ F l +∆ F nl ), except -0.15-0.1-0.05 0 0.05 0.1 -0.2 -0.1 0 0.1 0.2 PSfrag replacements v N /W [ ( ∆ F − ∆ F l ) / ( k B T ) ] / ( D / W ) MC (+[ O ]+)MC ( O [ O ] O )MC ( − [ O ]+)MC ( O [ O ]+) (+[ O ]+) ( O [ O ] O )( − [ O ]+)( O [ O ]+) L = 1000 , W = 100 , D = 20 W = 100 , D = 20 FIG. 10: Quasi-torque ∆ F acting on a small mesoscopic “or-dinary” needle O in a long strip with boundaries ( i, j ) cor-responding to the cases ( i [ O ] j ). This plot shows its depen-dence on the position v N of the needle in the strip and on theboundary conditions ( i, j ). The lines are suitably normalizedexpressions c ij (see Eqs. (5.2) and (5.3) and the text belowthe latter one) for ∆ F nl which, according to Eq. (5.3), aresymmetric around v N /W = 0 for ( i, j ) = ( O, O ), (+ , +), and( − , +) but antisymmetric for ( i, j ) = ( O, +). In order to ob-tain the numerical data we have computed ∆ F by means ofthe coupling parameter approach (Sec. IV) and then have sub-tracted the analytic expression (2.17) for ∆ F l . There is veryfavorable agreement of the simulation data, which correspondto
W/L = 0 .
1, with the analytic predictions for
W/L = 0.The case ( − [ O ]+) is exceptional in that for it the agreementis only fair, i.e., in this case there are sizeable corrections to∆ F = ∆ F l + ∆ F nl beyond ∆ F nl . The statistical error barsare comparable with the symbol sizes and therefore they areomitted. for the case ( − [ O ]+) in which it is only fair. According toEqs. (5.2) and (5.3) one has c i,j = ( π / D/W ) g i,j .In order to visualize the approach of the limit of in-finite strip length L = ∞ , Figs. 11 and 12 show [60]the dependence of the quasi-torque on the aspect ra-tio W/L for needles of length D = 10 or D = 20 withtheir center v N = 0 at the midpoint of a strip of width W = 100. The cases ( O [ O ]+) and ( O [ O ] − ) should leadto the same ∆ F , due to the (+ ↔ − ) symmetry [34] ofthe Ising model, and ∆ F nl should vanish for v N = 0, dueto Eq. (2.18) and Ref. [48]. These properties are reflectedrather well by the data in Fig. 11 for D = 20 (shown assquares and diamonds) which are close to the values of[∆ F l / ( k B T )] / ( D/W ) , equal to − π /
96 for
W/L = 0(shown as the horizontal straight line) and about 5%smaller for
W/L = 1 /
2. Here we discard the data for
W/L ≤ . O [ O ] O ) discussed in Fig. 12where in a strip of width W = 100 the data for the nee-dle of length D = 20 (shown by squares) attain muchcloser the asymptotic value ( π / π/ F l + ∆ F nl ) / ( k B T )] / ( D/W ) predicted for v N = 0and L = ∞ (shown as the uppermost straight line) as L -0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0 0.1 0.2 0.3 0.4 0.5 PSfrag replacements
W/LW/LW/LW/LW/LW/LW/LW/LW/L [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / − π ∆ O, + ( W/L ) / eq.(5.1) ( O [ O ]+) , D = 10( O [ O ]+) , D = 20 ( O [ O ] − ) , D = 10( O [ O ] − ) , D = 20 W = 100 W = 100 W = 100 W = 100 W = 100 W = 100 W = 100 W = 100 W = 100 FIG. 11: Dependence of the quasi-torque ∆ F on the aspectratio W/L of the strip for ( O [ O ]+) and ( O [ O ] − ) bound-ary conditions with the needle at the strip center v N = 0.In these cases the next-to-leading contribution ∆ F nl van-ishes [48] for arbitrary W/L and the simulation data for D = 20 (squares and diamonds) are indeed close to the val-ues given by − π ∆ O, + ( W/L ) / F l which are − π /
96 for
W/L = 0 and about 5 percent smaller for
W/L = 1 / ↔ − ) symmetry, one has ∆ F ( O [ O ]+) = ∆ F ( O [ O ] − ) for arbitrary values of D , v N , W , and L . This exact identity isembodied in the form of the corresponding lattice Hamiltoni-ans ∆ H and H cr ( λ ) in Sec. IV. However, the results for thethermal average h ∆ Hi cr ( λ ) and its integral ∆ F in Eq. (4.6),calculated by means of the statistical Monte Carlo method,violate this identity within the numerical inaccuracy. Notethat the ensuing deviations in the above data are of tolerablesize, at least for the larger value of D . increases, i.e., as W/L becomes smaller.
Normal needles in strips without broken symmetry
Now we consider the (equivalent) cases ( O [+] O ) and( O [ − ] O ) of a “normal” needle with h = + or − in an ( O, O ) strip. Figure 13 shows data [60] for[∆ F/ ( k B T )] / ( D/W ) for such a needle of length D = 21at various positions v N in a strip with L = 1000 and W =101. They compare favorably with the corresponding an-alytic expression − ( π/ O,O + ( π / D/W )( − g O,O )for [(∆ F l + ∆ F nl ) / ( k B T )] / ( D/W ) with L = ∞ , whichis shown as dashed line. This expression follows fromEqs. (2.16)-(2.18) by repeating analogously the line ofarguments leading to Eqs. (5.2) and (5.3). Here the signin front of g O,O differs from that in Eq. (5.2) for the “ordi-nary” needle because A (+) ǫ = A ( − ) ǫ = −A ( O ) ǫ (Eq. (2.5)).In strips of infinite length L the pseudo-torques ∆ F ( v N )for the present case ( O [+] O ) and for the case (+[ O ]+)considered above are identical . This follows from a dual-ity argument similar to the one in Ref. [44], is in agree-ment with the expression ∆ F l + ∆ F nl in Eqs. (2.16)-(2.18), and is quite well reflected by the simulation datain Figs. 10 and 13. Concerning the aspect ratio depen- -0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 PSfrag replacements
W/LW/LW/LW/LW/LW/LW/LW/LW/L [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) π / π / π / π / π / π / π / π / π / ( O [ O ] O ) , D = 10( O [ O ] O ) , D = 20 ( O [+] O ) , D = 11( O [+] O ) , D = 21 D = 10 D = 10 D = 10 D = 10 D = 10 D = 10 D = 10 D = 10 D = 10 D = 11 D = 11 D = 11 D = 11 D = 11 D = 11 D = 11 D = 11 D = 11 D = 20 D = 20 D = 20 D = 20 D = 20 D = 20 D = 20 D = 20 D = 20 D = 21 D = 21 D = 21 D = 21 D = 21 D = 21 D = 21 D = 21 D = 21 ( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [+] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O )( O [ O ] O ) FIG. 12: Similar as Fig. 11 but for the cases ( O [ O ] O ) and( O [+] O ) in which, for L = ∞ , [(∆ F l +∆ F nl ) / ( k B T )] / ( D/W ) equals ( π / πD/W ] and ( π / − πD/W ], respec-tively, so that the leading contributions (dashed line) are thesame. For small W/L the simulation data for D = 20 , W =100 and D = 21 , W = 101 (green squares and magenta dia-monds) are indeed close to the corresponding uppermost andlowest horizontal line, respectively. -0.1-0.05 0 0.05 0.1 0 0.05 0.1 0.15 0.2 0.25-0.1-0.05 0 0.05 0.1 0 0.05 0.1 0.15 0.2 0.25 PSfrag replacements v N /Wv N /W [ ∆ F / ( k B T ) ] / ( D / W ) [ ∆ F / ( k B T ) ] / ( D / W ) ( O [+] O )from Eqs.(2.17) and (2.18) L = 1000 , W = 101 , D = 21 L = 1000 , W = 101 , D = 21 FIG. 13: Dependence of the quasi-torque ∆ F on the position v N of the needle in the ( O [+] O ) case. The simulation data(triangles) agree well with the analytic prediction (dashedline) given by ∆ F l + ∆ F nl (see the main text). dence, for W/L ≤ . O [+] O ) ofa “normal” needle with v N = 0 and D = 21 (shown bydiamonds in Fig. 12) have the tendency to approach thepredicted limiting value ( π / − π/ C. “Normal” needles in strips with brokensymmetry
In this subsection we consider a “normal” needle with h = + embedded in strips with at least one “normal”7 -0.45-0.4-0.35-0.3-0.25-0.2-0.15 0 0.1 0.2 0.3 0.4 0.5 PSfrag replacements
W/L [ ∆ F / ( k B T ) ] / ( D / W ) D = 11 D = 21( − [+] − ) W = 101 FIG. 14: Quasi-torque ∆ F acting on a + needle in the cen-ter of a ( − , − ) strip. We show simulation data for needlesof length D = 11 and 21 in strips of width W = 101 and ofvarious lengths L . The data for D = 21 (diamonds) agreequite well with the “sna” prediction − . D = 11 there is a significant discrepancy between the simula-tion data and the “sna” prediction -0.3072 (dashed line) fora strip of infinite length. boundary. These are the boundary conditions ( i, j ) =[(+ , +) , ( − , − ) , (+ , − ) , (+ , O ) , ( − , O )] for which the orderparameter profiles h φ i ST = W − / f ( i,j ) φ in Eq. (2.7) arenonvanishing and ζ I in Eq. (2.13) contains a contribution ∝ D / . In these cases we do not compare our simulationdata with the expanded analytic form of ∆ F , becauseconfining this expansion to the two leading powers of D ,as in Eq. (2.16), is expected to be insufficient for reachingagreement with presently accessible simulation data. Asexplained there, we rather set out to compare the simu-lation data with the full expression “sna” of the “smallneedle approximation” following from Eqs. (2.10)-(2.15).While we disregard the terms of order D denoted bythe ellipses [41] in Eqs. (2.13) and (2.14), we leave thelogarithm in Eq. (2.15) unexpanded.We illustrate this point for the case ( − [+] − ), i.e., a +needle embedded in a ( − , − ) strip. Figure 14 shows[60]that [∆ F/ ( k B T )] / ( D/W ) as obtained from simulationsfor a needle (with D = 21 and located in the center ofa strip with W = 101) agrees, for smaller aspect ratios W/L , quite well with the prediction − . L = ∞ (see the diamonds and the dash-dotted bottom line inFig. 14). The prediction follows by inserting the presentvalue θ = 21 π/
101 = 0 .
653 of θ ≡ πD/W into the “sna”expression ∆ F/ ( k B T ) = − ln( Z k /Z ⊥ ) where Z k , ⊥ = 1 − (2 θ ) / − − / θ / + 2 − θ + 2 − θ + (1 , − × [ − − (1 / θ +2 − / (1 / θ / − − θ ] , (5.4)see Eqs. (2.5) and (2.12)-(2.14) with the value ∆ − , − = − π/
48 taken from below Eq. (2.9) as well as Eqs. (A14)and (A15). For this value of θ the validity of the “sna”is confirmed by Fig. 4, as far as the contribution F k to∆ F is concerned (compare the discussion in the para-graph below Eq. (3.15)). This should be contrasted withthe prediction − . − . F being truncated af-ter the leading and the next-to-leading order in D , i.e.,from (∆ F l + ∆ F nl ) /k B T = [(2 θ ) − (2 θ ) / ] / (3 × ) (byusing again Eq. (5.4)) and which would disagree with thesimulation data by a factor of about 250.The reasonable agreement between the simulation datafor D = 21 and the “sna” persists for the dependence onthe needle position v N shown as the squares and the greendashed line in Fig. 15, provided | v N | /W remains small.However, for | v N | /W > ∼ .
15 an even qualitative de-viation develops in that the analytic approximation pre-dicts a point of inflection and a minimum which is notsupported by the simulation data. We attribute this fail-ure to inadequacies of this approximation near the stripboundary which have been discussed in detail for F ⊥ inthe half plane (see the paragraph in Sec. III addressingFig. 3(a)). In particular the unphysical minimum of thegreen dashed line in Fig. 15 is related to the maximumof the “sna” result shown in Fig. 3(a).This dependence on v N /W is shown in Fig. 15 also forthe other cases ( − [+] O ) , (+[+] O ) , (+[+] − ), and (+[+]+)of a + needle embedded in strips with at least one “nor-mal” boundary. Also in these cases the dependences ofthe data on v N /W are reproduced reasonably well by the“sna”. VI. SUMMARY AND CONCLUDINGREMARKS
A critical solvent such as a binary liquid mixture atits continuous demixing transition induces a long-ranged,so-called critical Casimir interaction between immersedparticles which is universal on mesoscopic length scales.For nonspherical particles the interaction depends notonly on their sizes and distances among each other butalso on their shapes and mutual orientations. We havestudied a critical system belonging to the Ising bulk uni-versality class in two spatial dimensions and particles ofneedle shape.As described in the Introduction we have consideredIsing strips at the bulk critical point with an embeddedneedle which is oriented either parallel or perpendicularto the symmetry axis of the strips. For such systemsour analysis benefits from the wealth of knowledge ac-cumulated for the two-dimensional Ising model at criti-cality and the comparative ease to implement a needlealong a row or column of the square lattice. The cor-responding effective interaction between the needle andthe two strip boundaries is probably the simplest exam-ple to check by Monte Carlo simulations the range of va-lidity of asymptotic analytic predictions for the orienta-8tion dependence of critical Casimir interactions betweennonspherical mesoscopic particles which go beyond theDerjaguin approximation.In Sec. IV and Figs. 1, 6, 7, and 8 we explain how toimplement in the lattice model boundary properties atthe two confining surfaces of the strip and for the needle,which locally induce one of the two demixing bulk phases(“normal” surface universality classes + (or − )) or in-duce disorder and suppress demixing (“ordinary” surfaceclass O ). The geometrical features of the correspondingcontinuum description are explained in Fig. 2.Primarily, we analyze the free energy ∆ F = F k − F ⊥ required at bulk criticality to turn the needle from analignment perpendicular to the strip (with free energy F ⊥ ) to parallel alignment (with free energy F k ). Thusone has ∆ F >
F < F depends on the length D of the needle, the width W andlength L of the strip, the distance v N of the needle centerfrom the midline of the strip, and the surface universalityclasses h of the needle and ( i, j ) of the two boundaries ofthe strip. Here we consider needles of small mesoscopiclength D for which predictions are available from the so-called “small needle expansion” explained in Sec. II.Before presenting below an itemized summary of thequantitative comparison with the simulations we pointout a few qualitative observations. For the needle it isadvantageous to reside in a spatial region and take anorientation which suits its boundary condition.(i) First, consider a needle of universality class h in a half plane with the boundary belonging to surface univer-sality class i . The needle will prefer the vicinity of thisboundary if i = h . Since in the case i = + ( i = O )the boundary-induced order (disorder) increases uponapproaching the boundary - as described, according toEq. (2.5), by the concomitant increasing density profilesof the order parameter (of the energy) in the half planewithout needle - this leads to an attractive force betweenneedle and boundary. Since both increases are strongerthan linear, for a fixed needle center in both cases the nee-dle will adopt an orientation perpendicular to the bound-ary of the same class. Likewise, for different universalityclasses i = h of the needle and the boundary the forcewill be repulsive and for a fixed center the needle willorient parallel to the boundary [13]. These orientationalpreferences depend on the needle position, increasing anddecreasing, respectively, with decreasing and increasingdistance a N of the needle center from the boundary.(ii) In a strip there is an additional qualitative effectin that there is a contribution to the orientational pref-erence of the needle which is independent of its position v N in the strip but depends on the combination ( i, j )of surface universality classes of the two strip bound-aries. In a long strip this contribution favors an align-ment of the needle perpendicular (parallel) to the bound-aries if they belong to the same (different) universalityclass i = j ( i = j ). For the double periodic strip itfavors an alignment perpendicular to the longer axis. This is reminiscent of - and related to - a well knowncorresponding anisotropy [59] of the two-point averages hO ( r − s / O ( r + s / i ST in the strip of the densities O = φ of the order parameter and O = ǫ of the energy.For small mesoscopic distances | s | the anisotropy causesthe two-point averages to be larger for s perpendicular(parallel) to the two strip boundaries if i = j ( i = j )and for s perpendicular to the longer axis of the doubleperiodic strip.All of these expectations are confirmed and further re-fined by the quantitative predictions of the “small needleexpansion”. It predicts, in particular, that the anisotropy∆ F due to [61] the aforementioned effect (ii) depends,apart from the boundary classes ( i, j ) of the strip, onlyon the length and not [40] on the universality class( h = + , − , or O ) of the small mesoscopic needle.In the following three blocks A, B, and C we list ourmain results.(A) We have demonstrated that the quantitative pre-dictions of Sec. II for the rich structure of the orientation-dependent interaction of non-spherical particles with a“small mesoscopic size” can actually be observed in a lat-tice model. This is a nontrivial result in view of a twofoldsize condition: the particle size being small comparedwith other geometric features and being large on the scaleof the lattice constant. Our Monte Carlo simulationswith needle lengths of about 20 lattice constants [27] havethe potential to closely approach the asymptotic regimeof the “small mesoscopic needle”, leading to results for∆ F which agree quite well with the analytic predictions,without adjusting any parameter. At the same time, theparameter range in which there is good agreement pro-vides a (conservative) estimate for the range of validityof the “small needle approximation” (“sna”), which is atruncated form of the systematic expansion for the uni-versal quasi-torque ∆ F in terms of the needle size.A1. For a strip with double periodic boundary condi-tions, for which the aforementioned effect (i) is ab-sent, our data for ∆ F reproduce, for the larger nee-dle lengths D = 20 ,
21, quite well the predicteddependence of ∆ F on the aspect ratio W/L of thestrip and the predicted independence of ∆ F of theuniversality classes h = O and h = + of the needle,in leading order (see Fig. 9 and the discussion inSubsec. V A).A2. In strips with actual boundaries, effects of bothtype (i) and (ii) are present and the former oneslead to a dependence of F k , F ⊥ , and ∆ F on theposition v N of the needle center within the strip.Here the analytic predictions hold for strips of in-finite length L = ∞ and Figs. 11, 12, and 14 showthe dependence of the simulation data on the as-pect ratio W/L of the strip.A2a. For the five possible and relevant combinations( i [ h ] j ) = ( O [ O ] O ) , (+[ O ]+), ( − [ O ]+) , ( O [ O ]+),and ( O [+] O ) of universality classes of the two9 -2.5-2-1.5-1-0.5 0 0.5 -0.2 -0.1 0 0.1 0.2 PSfrag replacements v N /W [ ∆ F / ( k B T ) ] / ( D / W ) (+[+]+) ( − [+] − )(+[+] − )(+[+] O )( − [+] O ) L = 1000 , W = 101 , D = 21 FIG. 15: Quasi-torque ∆ F acting on a small mesoscopic “nor-mal” needle + in a long strip in which one or both bound-aries break the Ising symmetry. We show the dependencesof ∆ F on the position v N of the needle in the strip and onthe boundary conditions ( i, j ) of the strip. There is reason-able agreement of the simulation data for W/L = 0 . W/L = 0 which are discussed in the first paragraph ofSubsec. V C. For the case ( − [+] − ) the “sna” can be trustedonly for | v N | /W smaller than ≈ .
15 (see the discussion in thelast but one paragraph of Subsec. V C). The statistical errorof the simulation data is comparable with the symbol sizes. boundaries of the strip and of the embedded nee-dle, for which the expansion in Eqs. (2.16)-(2.18)of ∆ F in terms of powers of D is appropriate, thedependence on v N of our data for ∆ F is discussedin Subsec. V B. In Figs. 10 and 13 this dependenceis compared with the approximate analytic predic-tions. There is fair agreement for the case ( − [ O ]+)while in the four other cases the agreement is verygood.In the case ( O [ O ]+), due to effect (i) one has∆ F ( v N ) > F ( v N ) < O and +boundary, respectively, so that the needle prefersthe perpendicular and parallel orientation, respec-tively. Since effect (ii) for unequal strip boundariesturns the needle parallel to the strip, the range of v N values allowing for parallel orientation is thelarger part of the accessible values of v N , i.e., thepoint v N where ∆ F changes sign is closer to the O boundary.In the strip of infinite length L = ∞ duality ar-guments [38, 44] predict for the cases (+[ O ]+) and( O [+] O ) the same expression for ∆ F ( v N ) whichwithin the “sna” (Eqs. (2.16)-(2.18)) is visualizedas the full line in Fig. 10 and as the dashed linein Fig. 13. It predicts that in this case the nee-dle prefers the orientation perpendicular to thestrip for small | v N | where effect (ii) prevails sothat ∆ F >
0, while for larger | v N | where effect (i) prevails it prefers the parallel orientation with∆ F <
0. The good agreement with the data inFigs. 10 and 13 tells again that within our latticemodel one can access the scaling region and thuscapture the corresponding universal small particlebehavior.A2b. Unlike the combinations considered above(in A2a.) the five combinations ( i [ h ] j ) =(+[+]+) , ( − [+] − ) , (+[+] − ) , (+[+] O ) , ( − [+] O )discussed in Subsec. V C involve a needle which at T c is subject to the order parameter profile inducedby the strip boundaries ( i, j ) via its nonvanishingamplitude A ( h ) φ (see Eqs. (2.10)-(2.14)). This leadsto a contribution to the partition function ∝ D / .For the comparison with the simulation data for∆ F , in these five cases the corresponding “smallneedle approximation” should not be implementedby directly expanding ∆ F in terms of powers of D but by expanding the corresponding partition func-tions (see the remarks above Eq. (2.16) and in thefirst paragraph of Subsec. V C). The comparison ofthe analytical predictions for these cases with thecorresponding MC data is shown in Fig. 15. Theorientation preferred by the needle (i.e., the signof ∆ F ) is in conformance with [62] the qualitativeobservation (i); also quantitatively the MC dataagree quite well with the predictions. Accordingto Fig. 15, in the case ( − [+] − ), however, the goodagreement for small values of | v N | /W does notextend beyond | v N | /W ≈ .
15. In this case theexpected breakdown of the “small needle approxi-mation” near strip boundaries starts already ratherclose to the strip center producing an unphysicalminimum of ∆ F ( v N ) at | v N | /W ≈ . arbitrary mesoscopic length D , i.e., beyond the regime ofthe “small needle expansion”.B1. These exact results have been derived in Sec. IIIand Appendix B for the free energy of a needle inthe half plane with an orientation perpendicular toits boundary i and of a needle embedded along themidline of a strip of infinite length and with thetwo boundaries being members of the same sur-face universality class ( i = j ). These geometriesare related to the W × L strip without a needle(Appendix A 2) via conformal transformations ofthe Schwarz-Christoffel type [63]. For various com-binations ( h, i ) of the surface universality classesthese effective interactions are shown in Figs. 3-5and display the crossover between the small needleregime and the limiting behavior for which the nee-dle approaches and nearly touches the boundary orbecomes much longer than the width of the strip.B2. Apart from the importance in their own right theseuniversal results allow us to better understand the0limitations of the “small needle approximation”.For example, in Fig. 3(a), for ( i [ h ]) = ( − [+])this approximation reproduces the exact result for F ( − [+]) ⊥ very well down to a distance of the closerend of the perpendicular needle from the boundarycorresponding to half its length ( a < = D/ a N / ϑ = D/ (2 a N ) = 0 . F in thecase ( − [+] − ) (see Fig. 15). In contrast, the exactresult crosses over to a logarithmic increase whichdiverges when the needle “touches” the bound-ary. We recall that the validity of the “small nee-dle approximation” requires that the mesoscopiclength D of the needle is small compared withthe distance a < between the closer end of the nee-dle and the boundary, whereas the exact result isvalid if only the microscopic lengths are sufficientlysmall compared with a < . Concerning the free en-ergy F ( − [+] − ) k for embedding a + needle extendingalong the midline of a ( − , − ) strip, Fig. 4 explicitlydemonstrates the ensuing improvement if one ex-pands the corresponding partition function ratherthan the free energy in terms of the needle length(see the remarks above Eq. (2.16) and in the firstand second paragraph of Subsec. V C).(C) Our results allow us to conclude which of the fea-tures studied here are of a more general character andthus can be expected to show up also in spatial dimen-sion d = 3 and which ones are specific for d = 2.C1. In dimensions d > i [ h ]). Both for particlesizes small [13, 64] and comparable [14] with re-spect to the distance from the wall, these particlesprefer, at fixed particle center , orientations perpen-dicular (parallel) to the boundary of the half spaceif i = h ( i = h ). These are the same preferences asdescribed in paragraph (i) above for our small nee-dles in the half plane. In the mean-field treatmentof Ref. [14] it was pointed out that at fixed closestsurface-to-surface distance of particle and wall thepreferred orientations display the opposite trend,i.e., being parallel (perpendicular) to the bound-ary of the half space if i = h ( i = h ). These lat-ter preferences are in agreement with our findingin d = 2 that the free energy F ( i [ h ]) k − F ( i [ h ]) ⊥ , re-quired to turn a long needle in the half plane aboutits closer end from the perpendicular to the par-allel orientation, is dominated by the first term, F ( i [ h ]) k = k B T ∆ i,h D/a < (note the signs of ∆ i,h given below Eq. (2.9)). The reason is that the second term, given by Eq. (3.10), depends only log-arithmically on the large ratio D/a < of the needlelength D and the distance a < of the closer end ofthe needle from the boundary and thus can be ne-glected relative to the linear increase exhibited by F ( i [ h ]) k .C2. However, there are also effects which are specific totwo dimensions, due to the symmetries based on theduality transformation and the much wider class ofconformal mappings. For example, the aforemen-tioned equality of the particle insertion free ener-gies as function of v N for the cases ( O [+] O ) and(+[ O ]+) with infinite strip length - based on theduality symmetry [38, 44] of the d = 2 Ising modelat the bulk critical point - has no correspondencefor a particle between parallel walls in d >
2. Like-wise, the independence of h of the leading smallparticle contribution ∆ F l to the quasi-torque ∆ F in the case ( O [ h ] O ) (see Eq. (2.17)) is valid in d = 2but not in d >
2. The derivation of this result in d = 2 hinges on using a conformal transformationas described in Ref. [40] which is not available fora non-spherical particle in d > d = 2 and d = 3 which arisesfrom the dependence on their width W . Here wehave considered needles with a mesoscopic width W which is much smaller than the needle length D . For both “ordinary” and “normal” needles in d = 2 the effects they induce in the embeddingcritical system (such as the density profiles), andthus their effective interactions with the bound-aries leading to force and torque, only depend ontheir length D but not [65] on their width W . Thisapplies also [66] to a “normal” needle in d = 3.However, for an “ordinary” needle in d = 3 withfixed length D the strength of these effects de-creases [67] upon decreasing the width W . For ex-ample, within the small needle expansion the pref-actor of the energy density O = ǫ is proportionalto D W x ǫ − = D W . in d = 3 while in d = 2 itis proportional to D x ǫ = D and independent of W (see Eq. (2.2) and Ref. [23]).Extending our detailed investigations to non-sphericalparticles in three spatial dimensions and for the wholeneighborhood [68] of the critical point is desirable butbeyond the scope of the present study. For Monte Carlosimulations in d = 3, studies of (square shaped) disksand of “normal” needles with their properties being in-dependent of their width look most promising [65]. Cor-responding quantitative analytical predictions remain achallenge. The “small particle expansion” can in prin-ciple be extended to, e.g., a circular disk. However, theamplitudes and averages of the corresponding operatorscan be obtained only approximately. A mean field treat-ment corresponding to d = 4 is given in Ref. [13] but al-1ready one-loop field theoretic calculations, correspondingto the first order contribution in an expansion in termsof 4 − d , look quite demanding. Acknowledgments
We are grateful to T.W. Burkhardt for communicat-ing his unpublished results [69] concerning Eqs. (B1) and(B8) and for useful discussions.
Appendix A: STRIP WITHOUT NEEDLE1. Double periodic boundary conditions
The behavior near bulk criticality of finite Ising stripswith periodic boundary conditions in both Cartesian di-rections has found a long lasting theoretical interest,starting to the best of our knowledge in 1969 with theseminal paper [46] by Ferdinand and Fisher. Thereinthe aspect ratio dependence of the free energy is givenexplicitly and was re-derived by other methods in laterstudies. The quantity − Φ ( P )ST addressed below Eqs. (2.18)and (5.1) is given by the scale free part of ln Z nm inEq. (3.37) of Ref. [46]. It is equal to ln Z I in Ref. [47]and to ln Z P P for the Ising model in Ref. [43]. Adopt-ing Cardy’s notation (see Eqs. (2.4)-(2.6) and Table 1 inRef. [43]) the relation for the Casimir (or stress tensor)amplitude ∆ P given below our Eq. (2.18) yields the result∆ P ( W/L ) ≡ ∆ P (1 /δ ) = − π/ − ( d/dδ ) ln { [ χ ( δ )] +[ χ ( δ )] + [ χ ( δ )] } (A1)where for the Ising model one has the conformal charge c = 1 / m = 3 in the functions χ pq introduced by Cardy.The energy density h ǫ i ST = f ( P ) ǫ ( W/L ) /W defined be-low Eqs. (2.3) and (5.1) is of course independent of u and v and turns out to vanish [46, 70] for strips of infinitelength L , i.e., f ( P ) ǫ (0) = 0. The aspect ratio dependencehas been determined in Refs. [46] and [47] and in ournotation reads f ( P ) ǫ ( W/L ) ≡ f ( P ) ǫ (1 /δ ) = − πE ( δ ) /Z ( δ ) (A2)where E ( δ ) = [ U / Π ∞ n =1 (1 − U n )] (A3)with U = exp( − πδ ) and Z ( δ ) = (1 / { E (2 δ ) /E ( δ ) + [ E ( δ )] / [ E (2 δ ) E ( δ/ E ( δ/ /E ( δ ) } . (A4)Our quantities f ( P ) ǫ , δ , E , and Z correspond to thequantities −h ǫ i I , − iτ , | η | , and Z I , respectively, inRef. [47] (see the introductory remarks in Sec. III andEqs. (3.3), (3.4), and (3.14) therein).
2. Strip with boundaries (i, j)
For our strip (ST) with the aspect ratio 1 /δ = W/L and boundaries( i, j ) = [(
O, O ) , (+ , +) , (+ , − ) , (+ , O )] (A5)Cardy’s results [43] for the partition functions and ourEqs. (2.8) and (2.9) yield∆ i,j (1 /δ ) = − π/ − ( d/dδ ) ln[ χ ( δ/
2) + χ ( δ/ ,χ ( δ/ , χ ( δ/ , χ ( δ/ χ pq introduced in Ref. [43]. For later referencewe rewrite Cardy’s expressions in the form∆ i,j (1 /δ ) = ( π/ − κ ( d/dκ ) ln n [Σ + Σ , Σ , Σ , Σ ]Π ∞ n =1 (1 − κ n ) − o! (A7)and alternatively as∆ i,j (1 /δ ) = ( πδ − / − σ ( d/dσ ) ln n(cid:16) S + S ) / , S , S , S ] (cid:17) ∞ Y n =1 (1 − σ n ) − o! (A8)which converge rapidly for long strips with large δ (cor-responding to extended, closely spaced boundaries) andfor short strips with small δ (i.e., short, widely separatedboundaries), respectively. In Eqs. (A7) and (A8) one has κ = e − πδ/ , Σ pq ≡ Σ pq ( κ ) == ∞ X l = −∞ κ [(24 l +4 p − q ) − / − ( q → − q ) ! (A9)and σ = e − π/δ , S pq ≡ S pq ( σ ) = ∞ X r =2 σ ( r − / ×× sin πpr πqr .(cid:16) sin πp πq (cid:17) , (A10)respectively. The three functions Σ pq ( κ ) in Eq. (A7) canbe written asΣ pq ( κ ) = κ [(4 p − q ) − / (cid:16) ∞ X n =1 a n κ n (cid:17) (A11)where the coefficients a n take the integer values 0, ± σ the three functions S pq ( σ ) have the following explicit forms: S = 2 / σ / + σ − σ − / σ / − σ + ... S = − / σ / + σ − σ + 2 / σ / − σ + ... S = − σ − σ + σ + ... . (A12)2For closely spaced boundaries with i = j the deriva-tive with respect to κ in Eq. (A7) contributes due to theprefactor in Eq. (A11) even in leading order W/L ≪ i,j (0) depends on ( i, j ) as given below Eq. (2.9).For widely spaced boundaries ( i, j ) the derivative with re-spect to σ in Eq. (A8) does not contribute to the leadingbehavior and∆ i,j (1 /δ ) → πδ − /
12 = − ∆ P (0) /δ , /δ → ∞ , (A13)is independent of ( i, j ). In this case of W/L ≫ u direction and deter-mined by h T k k i ST ≡ h T u u i ST → ∆ P (0) /L with ∆ P (0)given below Eq. (2.18).The above expressions in Eqs. (A5)-(A12) do not onlyserve to provide the aspect ratio dependence of the lead-ing contribution ∆ F l in Eq. (2.17) of the free energy re-quired to rotate the small needle in the strip, but alsoto calculate the effective interactions for certain parti-cles of arbitrary size: in the first entry of Ref. [22] fortwo particles of circular shape as well as in Appendix B 2below for certain configurations of two needles in the un-bounded plane, of one needle in the half plane, and ofone needle in a strip.Now we present the explicit forms of the scaling func-tions f ( i,j ) O ( v N /W,
0) of the density profiles in Eq. (2.7)in a strip of infinite length L = ∞ . These can be inferredfrom, e.g., Ref. [49]. In our notation they are given by f ( O,O ) φ = 0 , f ( O,O ) ǫ = C/ f (+ , +) φ = (2 C ) / , f (+ , +) ǫ = − C/ f (+ , − ) φ = − (2 C ) / s , f (+ , − ) ǫ = ( C/ − s ] f (+ ,O ) φ = ( C/ / [1 − s ] / , f (+ ,O ) ǫ = ( C/ s (A14)where C ≡ π/ cos( πv N /W ) , s ≡ sin( πv N /W ) . (A15)One can easily check that near the boundaries the cor-responding profiles hO ( r N ) i reduce to the half planelimits hO ( r N ) i half plane with the amplitudes provided inEq. (2.5). In particular, in the strip with i = + the threeprofiles f (+ ,j ) φ given in Eq. (A14) exhibit the behavior f (+ ,j ) φ ( v N /W → − / → / (( v N /W ) + (1 / − / =(2 W/a N ) / corresponding to the half plane with bound-ary +. Appendix B: NEEDLES OF ARBITRARYLENGTH1. Symmetry preserving cases
Needle and both boundaries belonging to O Here we establish Eqs. (3.2) and (3.13) for the freeenergy associated with the insertion of a needle in thecase that both the needle and the boundaries are of thesymmetry preserving “ordinary” type. We start withBurkhardt’s result [69] for the thermal average of thestress tensor induced by n nonoverlapping “ordinary”needles embedded in the x axis of the unbounded ( x, y )plane. If the n needles extend from x < to x > , from x < to x > , ... , and from x n< to x n> , respectively, witharbitrary real numbers x < ≤ x > ≤ x < ≤ x > ≤ ... ≤ x n< ≤ x n> , the stress tensor averages h T kl ( x, y ) i at apoint ( x, y ) follow from the analytic function h T ( z ) i ([ O ][ O ] ... [ O ]) = 2 − z − x < − z − x > + 1 z − x < − z − x > + ... + 1 z − x n< − z − x n> ! (B1)of the complex variable z = x + iy via [71] the rela-tions h T xx ( x, y ) i = −h T yy ( x, y ) i = − Re h T ( z ) i /π and h T xy ( x, y ) i = h T yx ( x, y ) i = Im h T ( z ) i /π where Re andIm denote real and imaginary parts, respectively. Wepoint out the consistency of Eq. (B1) in the special cases x m< → x m> , in which the needle m disappears, and for x m> → x m +1 < , in which two consecutive needles m and m + 1 merge into a single one. For n = 1 this reproducesthe expression h T ( z ) i ([ h ]) which is independent [40] of h and follows from mapping the half plane onto the entireplane outside a single needle (cp. Eqs. (A8) and (A9) inRef. [23].In order to derive Eq. (3.2) we use Eq. (B1) for n = 2,put ( x < , x > ) = ( −∞ , x -axis, and denote ( x < , x > ) by ( x < , x > ). Theconformal transformation z = c /l with c = a + ib mapsthe ( x, y ) plane with the two “ordinary” needles onto thehalf plane ( a > , b ) with the “ordinary” boundary line a = 0 and a single embedded “ordinary” needle extendingfrom ( a = √ lx < ≡ a < , b = 0) to ( a = √ lx > ≡ a > , b = 0).This leads to a stress tensor function h T ( c ) i = − A/ c A /
16 (B2)where A = 1 c − a < − c − a > . (B3)In order to obtain this result one uses the transforma-tion formula for the stress tensor which includes theSchwartz derivative [71]. The arbitrary length l , intro-duced for dimensional reasons, does not appear in therelation between h T ( c ) i and a < , a > . Shifting the nee-dle away from the boundary line, i.e., increasing the dis-tance a N = ( a > + a < ) / a > − a < and orientationfixed, leads to a change in the free energy F ⊥ determined3by [22, 37] ∂∂a N F ⊥ k B T = − Z ∞−∞ db h T ⊥⊥ ( a, b ) i (B4)where h T ⊥⊥ ( a, b ) i = − Re h T ( c ) i /π is the diagonal stresstensor component perpendicular to the boundary line a = 0. The integration path must extend between theboundary and the needle, i.e., 0 ≤ a < a < in Eq. (B4).In this region the integral over b is independent of a and is carried out most easily for a = 0. Integratingthe result with respect to a N and denoting the needlelength a > − a < by D leads to the result in Eq. (3.2) for F ⊥ ≡ F ( O [ O ]) ⊥ .In order to establish Eq. (3.13) one uses the transfor-mation c/l = exp( πw/W ), w = u + iv , in order to mapthe half plane ( a > , b ) with its embedded needle ontoa needle in strip geometry as described in the context ofEq. (3.13). With the ends of the needle at u = ± D/ h T ( w ) i = ( π/W ) ( − / − B/ B /
16) (B5)where B = sinh θ cosh θ − cosh(2 πw/W ) (B6)with θ defined in Eq. (3.11). The free energy change( d/dW ) F ( O [ O ] O ) k / ( k B T ) upon widening the strip followsfrom the right hand side of Eq. (B4) by replacing ( a, b )by ( u, v ) and identifying h T ⊥⊥ ( u, v ) i with Re h T ( w ) i /π .Here and below Eq. (B4) the real parts of h T ( w ) i and h T ( c ) i enter with a plus and minus sign, respectively, be-cause in the complex w and c planes the directions ⊥ per-pendicular to the boundaries point along the imaginaryand the real axis, respectively. Performing the integraland integrating with respect to W leads to the result for F ( O [ O ] O ) k given in Eq. (3.13). Needle in a strip with periodic boundary condition
Now we consider the geometry corresponding to Eq.(3.16). For the infinitely long strip with periodic bound-ary condition containing a needle in parallel direction thestress tensor function h T ( w ) i follows from Eq. (B1) with n = 1 and the transformation z/l = exp(2 πw/W ) yield-ing h T ( w ) i = ( π/W ) ( − /
12 + B /
16) (B7)with B given in Eq. (B6). Proceeding as for the ( O, O )strip above, Eq. (B7) leads to the result for F k given inEq. (3.16). Ordinary needle and + boundaries Finally, we derive Eqs. (3.3) and (3.14) from the ex-pression [69] h T ( z ) i ([ E ][ O ]) = 2 − ×× z − x < − z − x > − z − x < + 1 z − x > ! (B8) for the stress tensor function induced by an “extraordi-nary” needle E extending from x < to x > and an “or-dinary” needle from x < to x > . An “extraordinary”needle preserves the (+ ↔ − ) symmetry and can berealized in a lattice model as a line of spins with in-finitely strong nearest neighbor ferromagnetic couplingsbetween them so that they all point either in the + orall in the − direction [38, 44]. The partition function Z ([ E ][ O ]) ≡ Z ([+][ O ]) + Z ([ − ][ O ]) in the presence of thetwo needles E and O differs by only a factor of 2 fromthe two identical partition functions Z ([+][ O ]) = Z ([ − ][ O ]) in the presence of two needles + and O or − and O .(This argument holds if the partition functions are finite.This can be achieved by enclosing the whole system ina large box with “ordinary” boundaries.) Thus the cor-responding free energies differ by an additive constant − k B T ln 2 which drops out from the free energy differ-ence upon changing the needle geometry as well as fromthe stress tensor so that h T ( z ) i ([ E ][ O ]) = h T ( z ) i ([+][ O ]) = h T ( z ) i ([ − ][ O ]) . (B9)Note the different sign sequences in h T ( z ) i ([ E ][ O ]) on therhs of Eq. (B8) and in h T ( z ) i ([ O ][ O ]) on the rhs of Eq. (B1)with n = 2. This implies that for obtaining the (+[ O ])case Eqs. (B2) and (3.2) have to be modified by replac-ing A → − A and ϑ → − ϑ , respectively, which leadsto Eq. (3.3). Similarly, for the (+[ O ]+) case Eqs. (B5)and (3.13) have to be modified by replacing B → − B and θ → − θ , respectively, in order to obtain Eq. (3.14).
2. Cases with genuinely broken symmetry
In this subsection we derive the general expressions inEqs. (3.4) and (3.15) for the Casimir forces which encom-pass also the cases (+[+]), ( − [+]) and (+[+]+), ( − [+] − )which cannot be reduced to cases with symmetry preserv-ing boundaries. We start by considering two needles 1and 2 on the x axis, similar as in Appendix B 1, but with arbitrary universality classes [ i ] and [ j ], respectively. Thecorresponding stress tensor h T ( z ) i ([ i ][ j ]) can be inferredfrom the difference h T ( z ) i ([ i ][ j ]) − h T ( z ) i ([ O ][ O ]) == 1( z − x < )( z − x > )( z − x < )( z − x > ) ×× ( x > − x < )( x > − x < )(1 − k ) τ i,j ( k ) (B10)where τ i,j ( k ) = 148 ( − k − k ) − π ∆ i,j (1 /δ )(2 K ( k )) (B11)and h T ( z ) i ([ O ][ O ]) is the stress tensor for two “ordinary”needles given by Eq. (B1) with n = 2. The quantity0 ≤ k ≤ x > − x < )( x > − x < )( x < − x > )( x > − x < ) = (1 − k ) k (B12)and ( x < − x < )( x > − x > )( x < − x > )( x > − x < ) = (1 + k ) k . (B13)The amplitude functions ∆ i,j (1 /δ ) are given byEqs. (A6) - (A10) with the argument 1 /δ related to k via 1 /δ = K ( k ) /K ∗ ( k ) (B14)with the complete elliptic integral [52] functions K ( k )and K ∗ ( k ) = K ( √ − k ) (see Eq. (3.8)).Note that the pole of second order at z equal to aneedle end, present with equal residues 2 − in the stresstensor averages in Eqs. (B1) and (B8), is absent in thedifference of averages in Eq. (B10). This implies that,e.g., near the needle end at x < the leading contribu-tion to the corresponding average of the stress tensor el-ements T kl (see below Eq. (B1)) is independent not only[40] of the length and universality class of needle 1 butalso of the presence (i.e., distance, length, and universal-ity class) of another needle 2. Moreover - as shown byEqs. (B2), (B5), and (B7) above as well as by Eqs. (B17)and (B28) below - this contribution is also independentof the presence of the concomitant boundaries of the halfspace and the strip.Equations (B10) - (B14) follow from the Schwarz-Christoffel transformation [63, 72] which conformallymaps the z = x + iy plane with the two needles [ i ] and[ j ] embedded in the x axis onto the rectangle or strip STwith boundaries ( i, j ) at v = ± W/
2, periodic boundarycondition in u direction, and Casimir amplitudes ∆ i,j , asintroduced in the paragraph containing Eqs. (2.7)-(2.9).The Schwartz derivative in the corresponding transfor-mation law of stress tensors[37, 71] drops out from thestress tensor difference given by Eq. (B10).The vanishing of τ i,j in Eq. (B11) for ( i, j ) = ( O, O )provides, together with Eq. (B14), another expressionfor ∆
O,O (1 /δ ) besides (but equivalent to) the ones inEqs. (A6), (A7), and (A8). For ∆ + ,O (1 /δ ) the cor-responding other expression follows from Eqs. (B10)and (B11) with ( i, j ) = (+ , O ) when combined withEqs. (B1), (B8), and (B9).The difference of the stress tensors in Eq. (B10) mustvanish in the limit of distant needles with no correlationbetween them because the stress tensor for a single nee-dle in unbounded space is independent of its universalityclass [40]. For the same reason it must also vanish iftwo needles of the same universality class i = j = + (or − ) come close so that they merge, i.e., x > = x < , andform a single + (or − ) needle. These expectations are inagreement with the behaviorslim k → τ i,j ( k ) = 0 (B15) and lim k → τ i,j ( k ) = [∆ i,i (0) − ∆ i,j (0)] /π (B16)of τ i,j for distant and close needles with δ ց k ր δ ր ∞ , k ց
0, respectively. Equation (B15) followsfrom Eqs. (A13) and (B14).Proceeding as described above Eq. (B2), the differenceof the stress tensors for the needle in the half plane isobtained as h T ( c ) i ( i [ j ]) − h T ( c ) i ( O [ O ]) = − A (1 − k ) τ i,j ( k ) (B17)with h T ( c ) i ( O [ O ]) ≡ h T ( c ) i from Eq. (B2) and the func-tion A from Eq. (B3). Via a < /a > = 2 k / / (1 + k ) ,k = (1 − ϑ / ) / (1 + ϑ / ) (B18) k is related to the needle parameters a < /a > and ϑ =( a > − a < ) / ( a > + a < ) introduced above Eq. (B2) andin Eq. (2.21). Using the relation between k and ϑ inEq. (B18) together with the suitable functional relations K ( k ) = (1 + ϑ / ) K ∗ ( ϑ ) / ,K ∗ ( k ) = (1 + ϑ / ) K ( ϑ ) (B19)between elliptic integrals (see Eqs. 8.126.1 and 8.126.3in Ref. [52]) yields together with Eq. (B11) the moreconvenient expression4(1 − k ) τ i,j ( k ) = 14 ϑ ˜ τ i,j ( ϑ ) (B20)for the amplitude in Eq. (B17) where ˜ τ i,j is taken fromEqs. (3.5)-(3.7). Equation (B4) finally yields the expres-sion − ∂∂a N F ( i [ j ]) ⊥ − F ( O [ O ]) ⊥ k B T = − ϑ ˜ τ i,j ( ϑ ) a < − a > ! (B21)for the difference of the Casimir forces acting on the nee-dle in the cases ( i, j ) and ( O, O ) which implies Eq. (3.4)upon renaming the dummy index j as h .For the special combinations (+ , O ), ( O, +), ( − , O ),and ( O, − ) the expression in Eq. (B20) becomes indepen-dent of k and ϑ and equals − /
4. This follows from com-paring Eq. (B10) with the simple expressions in Eqs. (B1)and (B8) and by using Eq. (B9). For combinations ( i, j )with both i and j being “normal” universality classes theexpression in Eq. (B20) displays a nontrivial dependenceon k and ϑ .In Eq. (3.6) the dependence on ϑ of ˜ τ i,h and ρ i,h , with( i, h ) being arbitrary, can be readily calculated for a nee-dle nearly touching the boundary ( ¯ ϑ = √ − ϑ ց , δ ր ∞ ) as well as for a distant or small needle5( ϑ ց , δ ց
0) by combining the first and second ex-pression in Eq. (3.6) with Eqs. (A7) and (A11) and withEqs. (A8), (A12), and (A13), respectively. In obtainingthese limiting behaviors one uses the fact that Eqs. (A9)and (A10) and Eqs. (3.7) and (3.8) imply the simple de-pendences κ = [( ¯ ϑ/ P ( ¯ ϑ )] , σ = ( ϑ/ P ( ϑ ) (B22)of κ and σ in Eqs. (A7) and (A8) on ¯ ϑ and ϑ , where P ( x ) can be expressed in terms of a power series in x given by P ( x ) = exp {− Q ( x ) /R ( x ) } = 1 + x / O ( x ) , (B23)with R ( x ) = (2 /π ) K ( x ) = 1 + x / O ( x ) (B24)and (see Ref. [52]) Q ( x ) = K ( p − x ) − R ( x ) ln(4 /x )= − ( ! × x ++ × × ! " × × x + ... ) . (B25)We use these relations to show that for small ϑ theexact expressions in Eq. (3.4) for the Casimir forces areconsistent with the “small needle expansion”. To this endwe first note that expanding the expressions in Eq. (3.4)yields − a N ∂∂a N F ( i [ h ]) ⊥ k B T = − ϑ ddϑ ln {} + O ( ϑ ) (B26)with the curly bracket being identical to the curly bracketin the expression for ∆ i,h in Eq. (A8). This follows frominserting Eqs. (3.6) and (A8) into the second expressionin Eq. (3.4) and by using Eq. (B24) for K ( ϑ ) as well asthe relation σ d/dσ = (1 − ϑ / O ( ϑ )) ϑ d/dϑ due toEq. (B22). Furthermore, apart from terms of order ϑ thecurly bracket in Eq. (B26) is equal to 1 + ζ I + ζ A with theexpressions ζ I and ζ A from Eq. (2.19) for a small needlein a half plane. For example, in the case ( i [ h ]) = ( − [+]),with the small needle approximation [41] − a N ∂∂a N F ( − [+]) ⊥ k B T = − ϑ ddϑ ln(1 + ζ I + ζ A )= − ϑ ddϑ ln[1 − / ϑ / + ϑ/ − / (3 / ϑ / + (5 / ) ϑ ] , (B27)the aforementioned relation between the curly bracketand 1 + ζ I + ζ A follows from the second part of Eq. (A12) and from Eqs. (B22) and (B23). Accordingly, the differ-ence between the exact expression for the critical Casimirforces in Eq. (3.4) and their small needle approximationsis of the order ϑ .Proceeding as in the paragraph containing Eqs. (B5)and (B6), for the geometry of a needle of class j extend-ing along the midline of an ( i, i ) strip, as discussed inparagraph (ii) in Sec. III, one obtains h T ( w ) i ( i [ j ] i ) − h T ( w ) i ( O [ O ] O ) = − B × ( π/W ) ˜ τ i,j ( t ) / (4 t )(B28)with the function B given by Eq. (B6), h T ( w ) i ( O [ O ] O ) ≡h T ( w ) i from Eq. (B5), ˜ τ i,j as in Eq. (3.5), and t relatedto θ ≡ πD/W as stated in Eq. (3.15). The force inEq. (3.15) follows from the stress tensor difference inEq. (B28) upon replacing [ j ] by [ h ] and by using a so-called shift equation as in Eq. (B4).Similar as in the paragraph containing Eq. (B26), theexpanded expression − W ∂∂W F ( i [ h ] i ) k k B T = − eθ − θ ddθ ln { aθ / + ˆ bθ ++ˆ cθ / + ˆ dθ } + O ( θ ) , ˆ e = − (1 / − , (B29)of the exact result in Eq. (3.15) with coefficients ˆ a − ˆ e follows from Eqs. (3.6), (A8), (A12), (B22), and (B23)and is related to the “small needle approximation” [41] − W ∂∂W F ( i [ h ] i ) k k B T = − θ ddθ ln(1 + ζ I − ζ A )= − θ ddθ ln(1 + ˆ aθ / + ˆ bθ + ˆ eθ + ˆ f θ / + ˆ gθ )(B30)where ζ I and ζ A are given by Eqs. (2.13) and (2.14)with needle center v N = 0 at the midline of the ( i, i )strip. Note that the term ˆ eθ with ˆ e given in Eq. (B29)equals the contribution ( π/ D/ W ) ∆ i,i with the crit-ical Casimir amplitudes ∆ i,i = − π/
48 for ( i, i ) stripswhich enter into ζ A in Eq. (2.14) and Eq. (B30). Unlikeˆ e , the other coefficients ˆ a, ˆ b, ˆ c, ˆ d, ˆ f , and ˆ g depend on theboundary and needle universality classes ( i, h ). The devi-ation of the exact expression − W ( d/dW ) F ( i [ h ] i ) k / ( k B T )from its “small needle approximation” is given by thedifference of Eq. (B29) and Eq. (B30) and is of the order θ because ˆ a ˆ e + ˆ c − ˆ f = ˆ b ˆ e + ˆ d − ˆ g = 0 . (B31)For example, in the case ( i [ h ] i ) = ( − [+] − ), in which theexpression for the argument of the logarithm in Eq. (B30)equals Z k as given by Eq. (5.4), one hasˆ a = − / , ˆ b = 1 / , ˆ c = − (5 / − / , ˆ d = − (1 / − , ˆ f = − (1 / − / , ˆ g = − − , (B32)6which satisfy Eq. (B31).For the convenience of the reader we provide the ex-plicit expressions (cid:16) f ⊥ O [ O ] ( ϑ ) (cid:17) sna = − ln { ϑ/ / ϑ ] } , (cid:16) f ⊥ +[+] ( ϑ ) (cid:17) sna = − ln { / ϑ / [1 + (3 / ϑ ]+( ϑ/ / ϑ ] } , (cid:16) f ⊥− [+] ( ϑ ) (cid:17) sna = − ln { − / ϑ / [1 + (3 / ϑ ]+( ϑ/ / ϑ ] } (B33)and (cid:16) f k O [ O ] O ( θ ) (cid:17) sna = − ln n − (1 / − θ +( θ/ − − θ ] o , (cid:16) f k +[+]+ ( θ ) (cid:17) sna = − ln n − (1 / − θ +(2 θ ) / [1 + (1 / − θ ] + ( θ/ − − θ ] o , (cid:16) f k− [+] − ( θ ) (cid:17) sna = − ln n − (1 / − θ − (2 θ ) / [1 + (1 / − θ ] + ( θ/ − − θ ] o (B34)for the [41] “small needle approximations” ( f ⊥ i [ h ] ) sna and ( f k i [ h ] i ) sna of f ⊥ i [ h ] ≡ F ( i [ h ]) ⊥ / ( k B T ) and f k i [ h ] i ≡ F ( i [ h ] i ) k / ( k B T ). For the cases (+[ O ]) and (+[ O ]+) theexpressions follow from those for ( O [ O ]) and ( O [ O ] O ) bythe replacements ϑ → − ϑ and θ → − θ , respectively. Appendix C: GLOSSARY
In order to ease the reading of the text, here we com-pile the symbols and notations used and explain theirmeanings. A : Expression given by Eq. (B3) which serves todisplay the dependence on c of the analytic functions h T ( c ) i ( i [ h ]) (see Eqs. (B2) and (B17) as well as the textbelow Eq. (B9)) and thus to display the position depen-dence of the stress tensor averages for the geometries of aneedle of boundary universality class h in the half planewith orientation perpendicular to the boundary of bound-ary universality class i . A ( h ) O : Universal amplitudes of the profiles hO ( r ) i half plane in the half plane with boundary univer-sality class h where O = φ and O = ǫ are the normalizedoperators (see Eq. (2.4)) of the order parameter densityand of the deviation of the energy density from its bulkvalue, respectively (see Eq. (2.5)). a and b : Half plane coordinates perpendicular andparallel to the boundary a = 0 (see above Eqs. (2.5) and(B2)). a N : Distance of the needle center from the boundaryof the half plane (or from the lower boundary of the strip)(see Fig. 2). a < and a > : Distance of the closer and farther needleend from the boundary for a needle in the half planewith perpendicular orientation (see Fig. 2(b) and the textabove Eqs. (3.1) and (B2)); a > = a < + D . B : Expression given by Eq. (B6) which serves todisplay the dependence on w of the analytic functions h T ( w ) i ( i [ h ] i ) (see Eqs. (B5) and (B28) as well as the textbelow Eq. (B9)) and thus to display the position depen-dence of the stress tensor averages for the geometries of aneedle of class h embedded in the midline of a strip with( i, i ) boundaries. c = a + ib : Complex variable specifying the positionvector ( a, b ) in the half plane (see above Eq. (B2)). c i,j ≡ [∆ F nl / ( k B T )] / ( D/W ) : Normalized next-to-leading contribution to the quasi-torque acting on a small“ordinary” needle O in an ( i, j ) strip (see the paragraphfollowing Eqs. (5.2) and (5.3) and the caption of Fig. 10). D : The number of missing bonds (fixed spins) in thelattice description of the “ordinary” (“normal”) needle inSecs. I and IV (see Fig. 1 as well as Figs. 6 and 7) or [27]the length of the needle in the continuum descriptionsused in Secs. I, II, III, and Appendix B (see Fig. 2). E : “Extraordinary” boundary with infinitely strongferromagnetic nearest neighbor couplings between sur-face spins (see Ref. [38]). F k and F ⊥ : Free energy cost to transfer the needlefrom the bulk into the strip (or into the half plane) withits orientation parallel and perpendicular to the bound-aries, respectively (see Eqs. (2.10)-(2.14) and the textabove Eq. (3.11) or below Eq. (3.1)). In particular, F k ≡ F ( i [ h ] i ) k = k B T f k i [ h ] i ( θ ) and F ⊥ ≡ F ( i [ h ]) ⊥ = k B T f ⊥ i [ h ] ( ϑ )for needles h of arbitrary length embedded in the mid-line of an ( i, i ) strip and perpendicular to the boundary i of the half plane, respectively.∆ F ≡ F k − F ⊥ ≡ F − F : Free energy required toturn the needle about its center from the perpendicularto the parallel orientation (see Eq. (2.15)).∆ F l and ∆ F nl : Leading and next-to-leading contri-butions, respectively, to ∆ F for the cases ( i [ O ] j ) and( O [+ / − ] O ) of a small needle (see Eqs. (2.16)-(2.18)as well as the corresponding half plane relations in7Eq. (2.20)). F ( i,j )ST : Free energy of the strip ST without needle andwith boundaries ( i, j ) (see Ref. [42]). F cr ( λ ) : The free energy belonging to the latticecrossover Hamiltonian (see the paragraph containingEqs. (4.4)-(4.6)). f k i [ h ] i ( θ ) ≡ F ( i [ h ] i ) k / ( k B T ) : See Eq. (3.12) as well asFigs. 4 and 5. f ⊥ i [ h ] ( ϑ ) ≡ F ( i [ h ]) ⊥ / ( k B T ) : See Fig. 3 and belowEq. (3.1). f b and f s : Bulk free energy per area and surface freeenergy per length (see Ref. [42]). f ( i,j ) O : Universal scaling functions of the profiles hO ( r ) i ST for O = φ , ǫ in the needle-free strip ST withboundaries ( i, j ) (see Eqs. (2.7) and (A14)). f ( P ) ǫ : Position-independent universal scaling functionof h ǫ i ST in the needle-free strip ST with double peri-odic boundary conditions (see below Eq. (2.18) and Ap-pendix A 1). g ⊥ i [ h ] ( ϑ ) and g k i [ h ] i ( θ ) : Scaling functions for the effectiveforce acting on a needle h perpendicular to the boundary i of the half plane and for the disjoining force induced inan ( i, i ) strip by a needle h embedded in its midline (seethe text below Eq. (3.1) or above Eq. (3.11)). H ST : Lattice Hamiltonian for strips without a needle(see Eq. (4.1)). H ST + H ( h ) ⊥ ≡ H and H ST + H ( h ) k ≡ H : LatticeHamiltonians for strips containing an embedded needlewith orientation perpendicular and parallel to the bound-aries, respectively (see Eqs. (4.2) and (4.3)).∆ H ≡ H − H ≡ H ( h ) k − H ( h ) ⊥ : Difference of latticeHamiltonians for parallel and perpendicular orientationof the needle h with a fixed center (see Eqs. (4.4)-(4.7)). H ( O )cr ( λ ) and H (+)cr ( λ ) : Crossover Hamiltonian for theneedle of broken bonds and of fixed spins, respectively(see Eqs. (4.10) and (4.19) as well as Figs. 6(c) and 8(c)).˜ H ( O ) and ˜ H (+) : Lattice Hamiltonian for a strip con-taining a cross-shaped hole with the two bars correspond-ing to the two orientations of the needle of broken bondsand of fixed spins, respectively (see below Eqs. (4.10)and (4.19)). h : Characterizes the universality class of the needlesurface.( i, j ) : Characterizes the surface universality classes ofthe (lower, upper) boundary of the strip.( i [ h ]) : Needle of class h in the half plane with bound-ary of class i .( i [ h ] j ) : Needle of class h in the strip with boundariesof classes i and j . J > T c = [2 / ln( √ J /k B ) (see Eq. (4.1) and the paragraph above Eq.(4.2)). J u,v ; u ′ ,v ′ J : Coupling between nearest neighbors ( u, v )and ( u ′ , v ′ ) in a strip containing a needle of broken bonds(see Fig. 6(a) and 6(b)). J equals 0 and 1 for broken andunbroken bonds, respectively. k : Parameter characterizing the configuration of twoneedles embedded in the x axis of the unbounded ( x, y )plane via the cross ratio of their endpoints (see Eqs. (B12)and (B13)). L : The number of columns in the lattice model forthe strips considered in Secs. I and IV (see Fig. 1 andFigs. 6-8) or [27] the length of the strip in the continuumdescriptions used in Secs. I, II, III, and Appendix A (seeFig. 2). l : Arbitrary length in the conformal transformationsgiven in the text above Eqs. (B2), (B5), and (B7) whichrelate different geometries. It is introduced for dimen-sional reasons only and, due to dilatation invariance,drops out from equations relating quantities which be-long to the same geometry. n : Unit vector describing the orientation of the needle(see Eq. (2.3)). In the strip, n = ( n k , n ⊥ ) (see Eq. (2.6)). O : The “ordinary” surface universality class. Cor-responding boundaries or needles induce disorder in thesystem of Ising spins, i.e., in their vicinity the probabilityto find parallel nearest neighbor spins is smaller than inthe bulk.+ and − : The two “normal” surface universalityclasses with the tendency to order the Ising spins in the+ and − directions, respectively. r N : Position vector of the center of the needle (seeEqs. (2.2) and (2.3)). In the strip, r N = ( u N , v N ) (seeEq. (2.6)).8 S I and S A : Operator contribution to the normalizedBoltzmann weight of the small needle which is isotropicand anisotropic, respectively, with respect to the orien-tation of the needle (see Eqs. (2.1)-(2.3)).ST : Denotes the strip in the absence of the needle.“sna” (“small needle approximation”): Truncatedform of the “small needle expansion” (see the paragraphcontaining Eqs. (2.16)-(2.18) and the first paragraph ofSubsec. V C). For explicit expressions see Eqs. (5.4),(B33), and (B34). s : Exterior spin fixed to the value +1 (see the firstparagraph in Sec. IV B 3 as well as Fig. 8(c)). T kl : Stress-tensor operator (see Eq. (2.3)) with its el-ements in the strip denoted by T k k , T k ⊥ , T ⊥ k , and T ⊥ ⊥ (see Eq. (2.8)). h T ( z ) i , h T ( c ) i , and h T ( w ) i : Analytic functions in theunbounded plane with needles, in the half plane witha needle, and in the strip with a needle, respectively,which determine the corresponding stress tensor averagesas explained in Appendix B 1 (see Eqs. (B1), (B2), (B5),(B7)-(B10), (B17), and (B28) as well as Ref. [71]). t ≡ tanh( θ/
2) : Useful short notation according toEqs. (3.15) and (B28). u and v : Strip coordinates [27] parallel and perpen-dicular, respectively, to the boundaries of the strip. Inthe continuum description the boundaries of the strip areat v = ± W/ u and v ofthe lattice vertices have integer (half odd integer) values(see Fig. 1 and Figs. 6-8). u N and v N : Coordinates parallel and perpendic-ular, respectively, to the strip of the position vector r N = ( u N , v N ) of the center of the needle. As explainedin Sec. I and Ref. [27], u N and v N are lengths in thecontinuum description used in Sec. II and in Eqs. (A14)and (A15) of Appendix A, while in the lattice descriptionused in Sec. IV they are measured in units of the latticeconstant and have integer values for both “ordinary” and“normal” needles. W : The number of rows in the lattice model for thestrip or [27] the width of the strip in the continuum de-scription. w = u + iv : Complex variable specifying the positionvector ( u, v ) in the strip (see above Eq. (B5)). x φ = 1 / x ǫ = 1 : Scaling dimensions of the orderparameter and energy densities, respectively (see belowEq. (2.4)). Z k and Z ⊥ : Partition functions corresponding to F k and F ⊥ (see Eqs. (2.10) and (2.11)). Z ( i,j )ST : Partition function of a W × L strip ST withoutneedle and with boundaries ( i, j ) (see Refs. [42] and [44]). Z ([ h ][ h ]) : Partition function of a large system con-taining two needles h and h (see below Eq. (B8)). z = x + iy : Complex variable specifying the po-sition vector ( x, y ) in the unbounded plane (see belowEq. (B1)).∆ i,j (1 /δ ) : Casimir amplitude describing the univer-sal contribution − L − ∂ Φ ( i,j )ST ( δ ) /∂W = ∆ i,j (1 /δ ) /W tothe disjoining pressure per k B T of an ( i, j ) strip withoutneedle (see Eqs. (2.8), (2.9) and Appendix A 2).∆ i,j ≡ ∆ i,j (0) : Casimir amplitude for an ( i, j ) stripof infinite length L = ∞ and without needle (see belowEq. (2.9)).∆ P (1 /δ ) : Casimir amplitude for the double periodicstrip without needle (see below Eq. (2.18) and AppendixA 1).∆ P ≡ ∆ P (0) = − π/ δ ≡ L/W : Characterizes the shape of the strip (seeEq. (2.9)). We call
W/L ≡ /δ the aspect ratio of thestrip. ǫ ( r ) : Energy-density operator with its average in theunbounded plane (bulk) at bulk criticality subtracted(see below Eq. (2.3)) and normalized according to Eq.(2.4). ζ I and ζ A : Contributions to Z k and Z ⊥ which arisevia the “small needle expansion” from the operatorcontributions S I and S A to the normalized Boltzmannweight of the needle; they are isotropic and anisotropicwith respect to the needle orientation, respectively (seeEqs. (2.1)-(2.3), (2.13), and (2.14)). ϑ ≡ D/ (2 a N ) : Characterizes the size versus the dis-tance to the boundary for a needle in the half plane withits orientation perpendicular to the boundary line (seeEq. (3.1)).¯ ϑ ≡ √ − ϑ = √ a < a > /a N : Approaches zero if thecloser end of the needle approaches the boundary. ϑ : Threshold value of ϑ above which the interac-tion f ⊥− [+] ( ϑ ) between a − boundary of the half planeand a + needle perpendicular to it deviates significantlyfrom the corresponding “small needle approximation”,9i.e., from the last equation in Eq. (B33) (see the discus-sion of Fig. 3(a) in the paragraph between Eqs. (3.10)and (3.11)). θ ≡ πD/W : Characterizes the size versus the dis-tance to the boundaries for a needle embedded withinthe midline of the strip (see Eq. (3.11)). κ ≡ exp( − πδ/
4) : Variable used for the aspect ratiodependence of ∆ i,j in the case of a long strip with
L/W ≡ δ ≫ (1) i and Λ (1) j : Strengths of the coupling to the lowerand upper additional outside row, respectively, of fixedspins generating strip boundaries i and j of “ordinary”or “normal” character in the lattice model (see Eq. (4.1)and Figs. 1 and 6-8). λ : Parameter within the lattice model describing thecrossover of the needle orientation from perpendicular( λ = 0) to parallel ( λ = 1) orientation with respect to theboundaries of the strip (see Eq. (4.7) as well as Figs. 6(c)and 8(c)). ρ i,h ( ϑ ) : Auxiliary function determining the Casimirforce on a needle h in the half plane with boundary i anddetermining the disjoining force induced in an ( i, i ) stripof infinite length upon inserting a needle h (see Eqs. (3.4)and (3.15), respectively). Via Eqs. (3.6) and (3.7) thefunction ρ i,h ( ϑ ) is related to the dependence ∆ i,h (1 /δ ) onthe aspect ratio of the Casimir amplitude of the needle-free strip with boundaries ( i, h ). P h inc . i and P h decr . i : Sum of products of those nearestneighbor spins in the crossover Hamiltonian H ( O )cr ( λ ) theferromagnetic coupling strength of which increases anddecreases, respectively, upon increasing λ (see Eqs. (4.8)-(4.10) and Fig. 6(c)). (+) P h one i : Sum of those four spins which are coupled toan external spin (and carry a northeast arrow) for both needle orientations shown in Figs. 8(a) and 8(b) (seeEq. (4.16)). P h zero i : Sum of products of the center spin of the “nor-mal” (+) needle and of its four nearest neighbor spins(see Eq. (4.13)). These four products are missing in thecrossover Hamiltonian H (+)cr (see Eq. (4.19)) and the fourcorresponding couplings are absent in Figs. 8(a), 8(b),and 8(c). dP h inc . i and dP h decr . i : Sum of products of those near-est neighbor spins in the crossover Hamiltonian H (+)cr ( λ ) the ferromagnetic coupling strength of which increasesand decreases, respectively, upon increasing λ (seeEqs. (4.14), (4.15), and (4.19) as well as Fig. 8(c)). (+) P h inc . i and (+) P h decr . i : Sum of those spins which in thecrossover Hamiltonian H (+)cr ( λ ) are coupled with increas-ing and decreasing strength, respectively, to the externalspin s = 1 (see Eqs. (4.17)-(4.19) and Fig. 8(c)). σ ≡ exp( − π/δ ) : Variable used for the aspect ra-tio dependence of ∆ i,j in the case of a short strip with L/W ≡ δ ≪ τ i,h ( ϑ ) : Auxiliary function with the same character as(and simply related to) ρ i,h ( ϑ ) (see Eq. (3.5)).Φ ( i,j )ST ( δ = L/W ) : Universal shape dependent andscale free contribution to the free energy per k B T of L × W strips with boundaries ( i, j ) but without needle(see Eq. (2.8) and Ref. [42]). Φ ( P )ST ( δ ) denotes the corre-sponding contribution for the double periodic strip. φ ( r ) : Order-parameter-density operator, normalizedaccording to Eq. (2.4). χ pq ( δ ) or χ pq ( δ/
2) : Auxiliary functions for theCasimir amplitudes of W × L strips without a needle andwith double periodic boundary conditions or with ( i, j )boundaries (see Eqs. (A1) or (A6) as well as Ref. [43]). h ... i : Thermal average which may be specified bymeans of subscripts and superscripts such as h ... i bulk : for the unbounded plane without embeddedparticles (see Eq. (2.4)). h ... i half plane : for the half plane without embedded par-ticles (see above Eq. (2.5)). h ... i ST and h ... i ( i,j )ST : for a strip without embedded par-ticles and with boundaries ( i, j ) (see Eqs. (2.7) and (2.8)). h ... i ( i [ h ]) : for the half space with boundary i and anembedded needle h (see Eq. (B17)). h ... i ( i [ h ] j ) : for the strip with boundaries ( i, j ) and anembedded needle h (see Eq. (B28)). h ... i ([ h ][ h ] .. [ h n ]) : for the unbounded plane with n em-bedded needles h , h , .., h n (see Eqs. (B1) and (B8)-(B10)). h ... i cr : Thermal average based on the lattice crossoverHamiltonian (see Eqs. (4.5) and (4.6)).0 [1] M.E. Fisher and P.G. de Gennes, C. R. Acad. Sci. Paris, , B-207 (1978).[2] M. Krech, The Casimir Effect in Critical Systems (WorldScientific, Singapore, 1994).[3] I. Brankov, D.M. Danchev, and N.S. Tonchev,
Theory ofCritical Phenomena in Finite-Size Systems: Scaling andQuantum Effects (World Scientific, Singapore, 2000).[4] A. Gambassi, J. Phys.: Conf. Ser. , 012037 (2009).[5] R. Garcia and M.H.W. Chan, Phys. Rev. Lett. , 1187(1999).[6] A. Ganshin, S. Scheidemantel, R. Garcia, and M.H.W.Chan, Phys. Rev. Lett. , 075301 (2006).[7] R. Garcia and M.H.W. Chan, Phys. Rev. Lett. ,086101 (2002).[8] M. Fukuto, Y.F. Yano, and P.S. Pershan, Phys. Rev.Lett. , 135702 (2005).[9] C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C.Bechinger, Nature , 172 (2008).[10] A. Gambassi, A. Maciolek, C. Hertlein, U. Nellen, L.Helden, C. Bechinger, and S. Dietrich, Phys. Rev. E ,061143 (2009).[11] M. Kardar and R. Golestanian, Rev. Mod. Phys. , 1233(1999).[12] M. Bordag, U. Mohideen, and V. M. Mostepanenko,Phys. Rep. , 1205 (2001).[13] E. Eisenriegler, J. Chem. Phys. , 3299 (2004).[14] S. Kondrat, L. Harnau, and S. Dietrich, J. Chem.Phys. , 204902 (2009).[15] U. Nellen, L. Helden, and C. Bechinger, EPL , 26001(2009).[16] (a) K. Binder, in Phase Transitions and Critical Phenom-ena , edited by C. Domb and J.L. Lebowitz (Academic,London, 1983), Vol. , p. 1; (b) H.W. Diehl, in PhaseTransitions and Critical Phenomena , edited by C. Domband J.L. Lebowitz (Academic, London, 1986), Vol. ,p. 76; (c) H.W. Diehl, Int. J. Mod. Phys. B , 3503(1997).[17] O. Vasilyev, A. Gambassi, A. Maciolek, and S. Dietrich,EPL , 60009 (2007).[18] O. Vasilyev, A. Gambassi, A. Maciolek, and S. Dietrich,Phys. Rev. E , 041142 (2009).[19] M. Hasenbusch, Phys. Rev. B , 165412 (2010).[20] R. Evans and J. Stecki, Phys. Rev. B , 8842 (1994).[21] The neighborhood of the actual second order phase tran-sition in an Ising film system (which, e.g., for ( i, j ) =( O, O ) boundaries appears at a temperature below thebulk transition temperature and at zero bulk field) ischaracterized by a correlation length ξ k parallel to thefilm which is much larger than its width and is governedby long-ranged fluctuations of d − ε = 4 − d with themean-field theory as the starting point ( d = 4) cannot beexpected to work in this region of the spatial dimensionalcrossover d → d −
1. This has been substantiated for therelated crossover phenomena d → d → ε -expansions in E. Brezin and J.Zinn-Justin, Nucl. Phys. B , 867 (1985)).[22] T. W. Burkhardt and E. Eisenriegler, Phys. Rev. Lett. , 3189 (1995); , 2867 (1997); E. Eisenriegler and U.Rietschel, Phys. Rev. B , 13717 (1995). [23] E. Eisenriegler, J. Chem. Phys. , 144912 (2006).[24] In order to avoid confusion we stress that the “small”mesoscopic particle has to be sufficiently large on the mi-croscopic scale for displaying universal scaling properties.This is the analogue of the “small” distance between thetwo operators in the operator product expansion to bestill sufficiently large.[25] K.G Wilson and J.B Kogut, Phys. Rep. C , 75 (1974);A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation The-ory of Phase Transitions (International Series in NaturalPhilosophy, Vol. , Pergamon, New York, 1979); A. A.Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nu-clear Phys. B , 333 (1984).[26] The “operators” in the present systems of classical sta-tistical mechanics are classical objects fluctuating withthe basic statistical variables (which in the Ising latticemodel are the Ising spins). In the operator expansionsone uses local operators in coarse grained models (suchas the densities of the order parameter and of the en-ergy), see Refs. [13], [23], and [35] as well as the entriesin Refs. [22] and [25].[27] The symbols D, W, L, u, u N , v, v N are used with anobvious double meaning. While they are introduced as numbers in the lattice model (see the Introduction andSec. IV) we consider them in the mesoscopic contin-uum descriptions of Secs. II and III as lengths which fol-low from the corresponding numbers upon multiplicationwith the lattice constant.[28] T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys.Rev. A , 054901 (2009).[29] H. Gies and K. Klingmueller, Phys. Rev. Lett. , 220405(2006); A. Weber and H. Gies, Phys. Rev. D , 065033(2009).[30] R. Roth, R. van Roij, D. Andrienko, K.R. Mecke, and S.Dietrich, Phys. Rev. Lett. , 088301 (2002).[31] L. Helden, R. Roth, G.H. Koenderink, P. Leiderer, andC. Bechinger, Phys. Rev. Lett. , 048301 (2003).[32] M. F. Maghrebi, Y. Kantor, and M. Kardar, EPL ,66002 (2011); Phys. Rev. E , 061801 (2012).[33] B.B. Machta, S.L. Veatch, and J.P. Sethna, Phys. Rev.Lett. , 138101 (2012). While the Monte Carlo resultsin Figs. 1 and 3 of this paper are new, the correspondingconformal field theory (CFT) results for circular inclu-sions in the d = 2 Ising model at criticality have al-ready been obtained earlier in the paper cited first in thepresent Ref. [22] (see Fig. 1 therein) by using Cardy’s ex-act partition functions of the finite critical Ising model ona cylinder (see Ref. [43] below) and a conformal mappingonto the geometry with two circles.[34] Our study is exclusively about properties for which thebulk of the d = 2 Ising model is at the critical point. Thusfor all expressions shown we correspondingly assume thatthe temperature is at bulk T c and that there is no bulkfield coupled to the order parameter.[35] The density operators φ ( r = ( u, v )) and e ( r = ( u, v )) in-troduced below Eq. (2.3) are the coarse-grained (and nor-malized) counterparts of the fluctuating values of the spin s u,v and of the nearest neighbor sum −J s u,v P u ′ ,v ′ ′ s u ′ ,v ′ ofthe energy contributions, respectively, for the Ising lat-tice model introduced in Sec. IV. [36] With the normalization used in Eq. (2.4) the inverselength dimensions of φ and ǫ are equal to their scalingdimensions and the half plane amplitudes in Eq. (2.5) areuniversal.[37] J.L. Cardy, in Phase Transitions and Critical Phenom-ena , edited by C. Domb and J.L. Lebowitz (Academic,New York, 1986), Vol. 11, p. 55; J.L. Cardy, Nucl. Phys.B , 514 (1984).[38] Under the duality transformation the bulk of the Isingmodel remains at the critical point, the energy densityoperator ǫ turns into − ǫ , and an O boundary of the halfplane or strip with a free surface becomes an “extraor-dinary” ( E ) boundary [16] with infinitely strong edgecouplings (see T. W. Burkhardt, J. Phys. A , L307(1985)). Likewise, a needle of D missing bonds (as inFigs. 1(a) and 1(b)) becomes a line of D + 1 spins withinfinite couplings between nearest neighbors which canbe called an “extraordinary” needle if its length is meso-scopic. For each of these “extraordinary” objects the in-finitely strongly coupled spins are in the same type ofstate (all + or all − ). Since ǫ is invariant against a si-multaneous change of sign of all spins, h ǫ i O = −h ǫ i E = −h ǫ i + = −h ǫ i − for energy density profiles in the halfplane or around a single needle embedded in the bulk.[39] For the Ising model in spatial dimensions d larger than2 the universal half-space amplitudes A (+) ǫ ≡ A ( − ) ǫ and A ( O ) ǫ have the same signs as their half-plane counterpartsin Eq. (2.5), but their absolute values are not equal anymore. Using well-known field theoretic techniques (see,e.g., H.W. Diehl and M. Smock, Phys. Rev. B , 5841(1993) and B , 6740 (1993) as well as E. Eisenriegler,M. Krech, and S. Dietrich, Phys. Rev. B , 14377(1996), and references therein) explicit expressions forall amplitudes A ( h ) O , including the energy-density ampli-tude for a surface belonging to the multicritical “special”surface universality class [16], can be obtained within the ε ≡ − d expansion about d = 4. These results are pre-sented in Ref. [13] and in the first two entries of Ref.[22].[40] This property is special for two-dimensional conformalmodels in which the space outside a (simply connected)particle and its surface, embedded in the unboundedplane, can be conformally mapped onto the half plane a > a = 0, respectively.Since the thermal average of the stress tensor in the halfplane vanishes for all surface universality classes h , itsaverage outside the particle is solely determined by theSchwartz derivative of the mapping [37] and thus dependsonly on the shape of the particle and not on its surfaceclass h . This should be compared with the density pro-files hOi half plane = A ( h ) O a − x O in the half plane (see Eq.(2.5)) and the ones outside the particle, which both dodepend on h .[41] For small needles we shall neglect the contributions offourth or higher order in D which are indicated by theellipses in Eqs. (2.2), (2.3), (2.13), (2.14), and (2.19). Wefrequently call this truncated expansion the “small needle approximation ”.[42] For the strip ST with boundaries ( i, j ), free energy F ST ≡ F ( i,j )ST = − k B T ln Z ( i,j )ST , and partition function Z ( i,j )ST the universal contribution Φ ( i,j )ST ( L/W ) introducedabove Eq. (2.8) is the limit of [ F ( i,j )ST − W Lf b − L ( f ( i )s + f ( j )s )] /k B T as L and W tend to infinity with L/W fixed.Here f b is the bulk free energy per area while f ( i )s and f ( j )s are the surface free energies per length correspondingto the surface classes i and j , respectively.[43] J. L. Cardy, Nucl. Phys. B [FS17], 200 (1986).[44] While duality implies ∆ O,O = ∆
E,E for all values of δ = L/W , the identity ∆
E,E = ∆ + , + and its conse-quence ∆ O,O = ∆ + , + only hold for long strips with δ ≫ Z (+ , − )ST in the actualexpression Z ( E,E )ST ≡ Z (+ , +)ST + Z (+ , − )ST ] for the parti-tion function of a strip with two “extraordinary” bound-aries. This is expected intuitively and follows formallyfrom Z (+ , − )ST /Z (+ , +)ST ∼ = exp[ − Φ (+ , − )ST ] / exp[ − Φ (+ , +)ST ] = χ /χ → exp[ − ( π/ L/W ]. Here, in the first step, wehave used Ref. [42] and the equality f (+)s = f ( − )s and,in the second step, Cardy’s results [43] exp[ − Φ (+ , +)ST ] = χ exp[ πδ/
48] and exp[ − Φ (+ , − )ST ] = χ exp[ πδ/
48] with χ pq ≡ χ pq ( δ/
2) as introduced in Ref. [43] for Ising stripswith boundaries. (Via Eq. (2.9) these results are, ofcourse, equivalent to our corresponding Eqs. (A6).) For arbitrary δ , the above expression for Z ( E,E )ST and Ref.[42] for ( i, j ) = (+ , +) and (+ , − ) tell that Z ( E,E )ST × exp( W Lf b / ( k B T )) equals the product of exp[ − Φ (+ , +)ST ] +exp[ − Φ (+ , − )ST ] and a factor which depends on L but noton W . Thus by using the above relations for Φ ST and therelation ∆ E,E = ( W /L ) ∂ W [ln Z ( E,E )ST + W Lf b / ( k B T )],which follows from Eq. (2.8) and Ref. [42] with ( i, j ) =( E, E ), one confirms that the duality relation ∆
E,E =∆ O,O is in accordance with Cardy’s results in the formof our Eq. (A6).[45] Unlike Z k , Z ⊥ an expansion of F k , F ⊥ would involve ar-bitrary integer powers of ( D/W ) / for a “normal” needlein a strip with at least one “normal” boundary.[46] A. E. Ferdinand and M. E. Fisher, Phys. Rev. , 832(1969).[47] P. di Francesco, H. Saleur, and J. B. Zuber, Nucl. Phys.B [FS20], 527 (1987).[48] There are two consequences of duality for the energy den-sity scaling functions. The property, that f (+ ,O ) ǫ is an oddfunction of v N /W , is not only valid for infinite strips W/L = 0 (see Eq. (A14)) but even for arbitrary values of
W/L . However, the identity f ( O,O ) ǫ = − f (+ , +) ǫ only holdsfor infinite strips (compare the corresponding argumentsfor ∆ O,O = ∆ + , + in the present Ref. [44]).[49] T. W. Burkhardt and T. Xue, Nucl. Phys. B , 653(1991).[50] For the needle perpendicular to the boundary ofthe half plane the dimensionless and universal scal-ing function f ⊥ i [ h ] ( ϑ ) = F ( i [ h ]) ⊥ / ( k B T ) of the free en-ergy is related to the scaling function g ⊥ i [ h ] ( ϑ ) = − a N ( ∂/∂a N ) F ( i [ h ]) ⊥ / ( k B T ) of the force by f ⊥ i [ h ] ( ϑ ) = R ϑ d ˆ ϑg ⊥ i [ h ] ( ˆ ϑ ) / ˆ ϑ because f ⊥ i [ h ] ( ϑ = 0) = 0. The corre-sponding relations for the needle with parallel orientationand at the center line of a symmetric strip follow upon thereplacements F ( i [ h ]) ⊥ → F ( i [ h ] i ) k , a N → W , ( ϑ, ˆ ϑ ) → ( θ, ˆ θ ),and f ⊥ i [ h ] → f k i [ h ] i , g ⊥ i [ h ] → g k i [ h ] i , where θ = πD/W .[51] Arguing along the lines of Ref. [38] leads to F ( O [+]) ⊥ = F (+[ O ]) ⊥ .[52] I. S. Gradsteyn and I. M. Ryshik, Table of Integrals, Se- ries, and Products (Academic, New York, 1965). In thisreference the function K ∗ appearing in Eqs. (3.7), (3.8),(B14), and (B19) is denoted as K ′ .[53] The geometry underlying Eq. (3.9) of an infinitely longneedle perpendicular to a boundary line is reminiscentof an (idealized) AFM geometry of a cone near a wall(mimicking the surface of a colloidal particle), albeit intwo spatial dimensions and for a vanishing opening an-gle of the cone. It would be interesting to use AFM inorder to investigate critical Casimir forces for such a sys-tem immersed in a critical binary liquid mixture. Thesomewhat related critical behavior of the cone-plate in-teraction induced by a long flexible polymer chain at-tached to the cone tip has been discussed in Ref. [32].Still another example for two interacting scale-free ob-jects is provided in Ref. [29] analyzing the quantum-electrodynamical Casimir effect between a plane and asemi-infinite plate. In all these cases only a single lengthappears and the force between such objects is related, viasimple dimensional considerations, to a negative integerpower of this length times a universal constant, like inthe original geometry of two infinitely extended parallelplates.[54] The factor π in the definition in Eq. (3.11) simplifies laterformulae.[55] C. H. Bennett, J. Comp. Phys. , 245 (1976).[56] D. P. Landau and K. Binder, A Guide to Monte CarloSimulations in Statistical Physics (Cambridge UniversityPress, London, 2005), p. 155.[57] U. Wolff, Phys. Rev. Lett. , 361 (1989).[58] N. Metropolis and S. Ulam, J. Amer. Stat. Assoc. ,335 (1949).[59] For any W × L strip, the leading anisotropic contribu-tion of the average of O ( r − s / O ( r + s /
2) with O = φ or ǫ (normalized in accordance with Eq. (2.4)) is givenby hO ( r − s / O ( r + s / i (aniso)ST = 4 πx O ∆ s − x O ( s u − s v ) /W . This result is independent of r and holds if | s = ( s u , s v ) | ≡ s is of mesoscopic size but much smallerthan W , L , and the distance of r from the boundaries.In this expression one has ∆ ≡ ∆ i,j ( W/L ) and ∆ ≡ ∆ P ( W/L ) for strips with boundaries ( i, j ) and double pe-riodic boundary conditions, respectively (see J. L. Cardy,Nucl. Phys. B , 355 (1987) and Ref. [13]). The reasonfor this position-independent anisotropy of both the two-point average of close operators ( ∝ s u − s v , see above)and of the free energy F ( r N , n ) of a small particle (seeEqs. (2.3), (2.14), and (2.17)) is the appearance of thestress tensor in the form P kl s k s l T kl and P kl n k n l T kl ,respectively, in the operator expansion of these small ob-jects in the strip. A particularly direct connection be-tween these two types of anisotropies can be establishedfor an anisotropic quasi-particle consisting of two weakpoint defects a small mesoscopic distance s apart, repre-sented by the perturbation δ H ∝ O ( r − s /
2) + O ( r + s / O = φ or ǫ , and embedded in a double periodic strip.In this case the anisotropic part of the embedding freeenergy is, apart from a minus sign, proportional to theaforementioned anisotropy hO ( r − s / O ( r + s / i (aniso)ST of the two-point average in the strip ST. The sign of ∆ P implies that, like the small needles, this quasi-particlealso prefers to align perpendicular to the longer axis ofthe double periodic strip.[60] The error bars in Fig. 9 and in Figs. 11-14 depict the standard deviations of numerical simulation data, calcu-lated via averaging over ten series of Monte Carlo steps.[61] The effect (ii) described in the Summary prevails for casesin which ∆ F can be expanded in terms of D and is dom-inated by its leading order ∝ D . This applies to needlepositions near the strip center for the cases ( i [ h ] j ) con-sidered in Subsec. V B. However, for the cases consideredin Subsec. V C in which the needle is subject to the orderparameter profile induced by the boundaries ( i, j ) thereappear terms ∝ ( D/W ) / inside the logarithm deter-mining ∆ F which cannot be made sufficiently small insimulations with mesoscopic D (in order not to exceed atractable lattice size W × L ) and prevent a useful expan-sion (see the remarks above Eq. (2.16)). Actually, for thecase ( − [+] − ) the naive application of (ii) would predictthe wrong orientation of the needle near the strip cen-ter (compare the discussion in the paragraph containingEq. (5.4)).[62] For the case (+[+] − ) the simulation data and the an-alytic prediction indicate that ∆ F becomes positive , inagreement with the observation (i) in the Summary, forvalues v N /W which are closer to the + boundary (i.e.,which are more negative) than the ones shown in Fig. 15.[63] See, e.g., V. I. Smirnov, A Course of Higher Mathematics (Pergamon, Oxford 1964), Vol. 3.[64] Unlike in d = 2, in three spatial dimensions ( d = 3) thescaling dimension 2+ x φ of ∂ r Nk ∂ r Nl φ is smaller than thescaling dimension d of the stress tensor. Thus for small“normal” particles with uniaxial shape-symmetry (suchas a disk of diameter D or a needle of length D ) and lo-cated between infinitely extended parallel plates - a dis-tance W apart and with broken symmetry -, the leadingorientation dependent contribution within the small par-ticle expansion is proportional to D x φ times the secondderivative of the order parameter profile in the systemwithout the particle. Moreover with x φ ( d = 3) ≈ / d = 2) it might be possi-ble within a simulation to keep ( D/W ) x φ and ( D/a N ) x φ significantly smaller than 1 and to single out the leadingquasi-torque ∆ F l proportional to D x φ .[65] Needles in d = 2 and disks in d = 3 are “impenetrable”boundaries of finite extent D in the sense that spins closebut on different sides, separated by the width W of theboundary with W ≪ D , are coupled only via correlationsaround the ends of the boundary. (This also applies to theneedles in the present lattice model with nearest neigh-bor couplings and where W is of the order of the latticeconstant.) Thus the density profiles they induce at meso-scopic distances much larger than W are independent of W and depend only on D . For needles in d = 3, however,spins can communicate round about the needle axis. Forthe “normal” needle the issue arises to which extent itsproperties, which are independent of their width for small mesoscopic widths (see Ref. [66]), can be transferred tothe case of a microscopic width of a needle representedby spins fixed in a single line of the lattice only.[66] In d = 3 the insertion free energy of a thin “normal”needle is independent of its width W . This independencefollows from the corresponding property of a thin cylinderor needle of infinite length discussed in A. Hanke and S.Dietrich, Phys. Rev. E , 5081 (1999) and in A. Hanke,Phys. Rev. Lett. , 2180 (2000).[67] E. Eisenriegler, J. Chem. Phys. , 204903 (2006). [68] Moving away from the critical point leads to an evenricher diversity in the behavior of the orientation-dependent interactions of non-spherical particles, be-cause the finite correlation length ξ enters as anothermesoscopic length besides the lengths L , W , v N , and D (see the mean-field analysis in Ref. [14]). In the“small particle expansion” additional terms associatedwith isotropic and anisotropic operators appear which areaccompanied by powers of the particle size higher thanthe leading ones. In the context of the Gaussian model anexample is provided by the operator O III in Eq. (2.12)in E. Eisenriegler, A. Bringer, and R. Maassen, J. Chem.Phys. , 8093 (2003).[69] Equations (B1) and (B8) are unpublished expressions de-rived by T. W. Burkhardt using the approach presentedin T. W. Burkhardt and J.-Choi, Nucl. Phys. B , 447(1992).[70] The vanishing of f ( P ) ǫ in the infinite strip with periodicboundary condition follows from its vanishing in the un-bounded plane, the conformal mapping between thesetwo geometries, and from ǫ transforming as a so-called“primary” operator.[71] The relations given below Eq. (B1) between h T kl ( x, y ) i and h T ( z ) i are based on the normalization of T kl asspecified by Eq. (2.8) and consistent with Eq. (B4).Here T ( z ) is the usual [37] complex stress tensor whichwithin thermal averages is an analytic function of z and its normalization is specified by its transformation law.For two geometries G and ˜ G related by the conformalmapping z = z (˜ z ), the corresponding analytic func-tions h T ( z ) i G and h T (˜ z ) i ˜ G are related by h T (˜ z ) i ˜ G =( dz/d ˜ z ) h T ( z ) i G + S/
24 with the Schwartz derivative S = ( z ′′′ /z ′ ) − (3 / z ′′ /z ′ ) where the primes denotederivatives with respect to ˜ z . For example, for the map-ping z = z ( c ) = c /l considered above Eq. (B2) this re-lation allows one to obtain h T ( c ) i ˜ G ≡ h T ( c ) i in Eq. (B2)from h T ( z ) i G ≡ h T ( z ) i ([ O ][ O ]) according to Eq. (B1).[72] For the simple needle configuration x < = − /k, x > = − , x < = 1 , x > = 1 /k with0 ≤ k ≤ u + iv = i ( W/ R z dt/ p (1 − t )(1 − k t )] /K ( k ).This maps the upper and lower halves of the z planewith y > y < u < u >
0, respectively.Crossing the x axis outside of the interval where theneedles are, i.e., at | x | > /k corresponds to jumpingwithin the rectangle from ( u = − L/ , v ) to ( u = L/ , v )which amounts to the periodic boundary conditionin uu