Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in d\geq 2 Dimensions
EExact Critical Casimir Amplitude of Anisotropic Systems from Conformal FieldTheory and Self-Similarity of Finite-Size Scaling Functions in d ≥ Dimensions
Volker Dohm and Stefan Wessel
Institute for Theoretical Physics, RWTH Aachen University, 52056 Aachen, Germany (Dated: January 08, 2021)The exact critical Casimir amplitude is derived for anisotropic systems within the d = 2 Isinguniversality class by combining conformal field theory (CFT) with anisotropic ϕ theory. Ex-plicit results are presented for the general anisotropic scalar ϕ model and for the fully anisotropictriangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic bound-ary conditions (PBC). These results demonstrate the validity of multiparameter universality forconfined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find anunexpected complex form of self-similarity of the anisotropy effects near the instability where weakanisotropy breaks down. This can be traced back to the property of modular invariance of isotropicCFT for d = 2. More generally, for d > O ( n )-symmetric systems with planar anisotropies and PBC both inthe critical region for n ≥ n ≥ Fluctuation-induced thermodynamic forces are ubiqui-tous in confined condensed matter systems [1]. They ex-ist in both isotropic systems such as fluids, superfluids,and binary liquid mixtures [2, 3] as well as in anisotropicsystems such as liquid crystals [1, 4], superconductors [5],and compressible solids [6]. Near a critical point, so-called critical Casimir forces [2, 3] arise from long-rangecritical fluctuations, which generate a universal finite-size critical behavior that can be classified in universal-ity classes with universal critical exponents [7]. Within auniversality class there exist subclasses [8, 9] of isotropicand weakly anisotropic d -dimensional systems – the lat-ter have d independent nonuniversal correlation-lengthamplitudes in d principal directions. While the Casimirforce amplitude at criticality is widely believed to be auniversal quantity [2, 3, 7, 10–14], this is not valid forweakly anisotropic O ( n )-symmetric systems with an n -component order parameter in 2 < d < d = 2 have remained unexplored in the literature.Two-dimensional systems are of fundamental theoreti-cal interest since conformal field theory (CFT) is capa-ble of deriving rigorous results for critical Casimir am-plitudes of isotropic systems on a strip [12–14, 19, 20]and for the partition function at the critical tempera-ture T c on a parallelogram [21–25]. In this Letter our fo-cus is on the critical Casimir force in weakly anisotropic( d = 2 , n = 1) Ising-like systems for which CFT has notmade any prediction so far. We show how to combinean exact result of CFT for the isotropic Ising model ona torus [22, 25] with an exact shear transformation ofanisotropic ϕ theory [16] which, on the basis of multi-parameter universality [8, 26], leads to exact predictionsfor all weakly anisotropic systems with periodic bound- ary conditions (PBC) in the ( d = 2 , n = 1) universalityclass. We discover unexpected self-similar structures inthe critical Casimir amplitude near the instability whereweak anisotropy breaks down. They can be traced backto the modular invariance of isotropic CFT. We alsodemonstrate the validity of multiparameter universalityfor confined systems. More generally, we find self-similarstructures in the O ( n )-symmetric ϕ theory with PBC for1 ≤ n ≤ ∞ in d > T c but also in the Goldstone-dominated low-temperature region of anisotropic systemswith 2 ≤ n ≤ ∞ .We consider systems with short-range interactions in arectangular L d − (cid:107) × L geometry with PBC near an ordi-nary critical point. The total free energy F tot (dividedby k B T ) can be decomposed into singular and nonsin-gular parts. We are interested in the singular part F c of F tot at T c . It is well known that the critical free-energy density f c = F c / ( L d − (cid:107) L ) has the large- L behav-ior f c ( L (cid:107) , L ) = L − d F c ( ρ ) at fixed aspect ratio ρ = L/L (cid:107) [7, 10] with a finite amplitude F c ( ρ ), which implies that F c = ρ − d F c ( ρ ) (1)is a finite quantity in the large- L limit. The criticalCasimir force in the vertical direction is obtained as F Cas ,c = − ∂ ( Lf c ) /∂L = L − d X c ( ρ ) , (2)where the derivative is taken at fixed L (cid:107) . This yields thecritical Casimir amplitude [8, 27] X c ( ρ ) = ( d − F c ( ρ ) − ρ ∂F c ( ρ ) /∂ρ = − ρ d ∂ F c /∂ρ. (3)If two-scale-factor universality [7, 10, 11] is valid the am-plitudes F c , F c , and X c , for given geometry and BC,are universal. In this Letter we show that these ampli-tudes exhibit a nonuniversal dependence on microscopiccouplings with a complex self-similar structure if the sys-tems are anisotropic. From CFT we derive exact resultsfor d = 2 for both the scalar ϕ model and the Isingmodel which belong to the same universality class. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b FIG. 1: Steps of argumentation for the exact relationshipsbetween the critical free energies F c of finite anisotropic andisotropic ϕ and Ising models in d = 2 (see also Fig. 2). We outline our strategy in the schematic Fig. 1 for thecase ρ = 1. The anisotropic ϕ model is characterized bytwo important nonuniversal parameters (see also Fig. 2):the angle Ω describing the orientation of the two prin-cipal axes and the ratio q = ξ (1)0 ± /ξ (2)0 ± of the two prin-cipal correlation lengths [26] ξ ( β ) ± = ξ ( β )0 ± | t | − , β = 1 , t = ( T − T c ) /T c . For the anisotropic Ising model thecorresponding parameters are denoted by Ω Is and q Is .Step 1 uses a shear transformation of the anisotropic ϕ model on a square to an isotropic ϕ model on a paral-lelogram that leaves the critical free energy F c invariant[16]. Step 2 is based on two-scale-factor universality [7]implying that the critical free energy F iso c of the isotropic ϕ model is the same as F CFT c of the isotropic Ising modelon the same parallelogram described by CFT. Step 3 em-ploys the hypothesis of multipararmeter universality [8]predicting that F Is c of the anisotropic Ising model with ρ = 1 is obtained from F c ( q, Ω) of the anisotropic ϕ model by the substitution q → q Is , Ω → Ω Is . Overall,these steps are equivalent to an effective shear transfor-mation (dashed arrow in Fig. 1) between the isotropicIsing model on a parallelogram and the anisotropic Isingmodel on a square. Step
1: We first consider the anisotropic scalar ϕ latticemodel in a L (cid:107) × L rectangle on a square lattice with latticespacing ˜ a and PBC. The Hamiltonian and the total freeenergy divided by k B T on N = L (cid:107) L/ ˜ a lattice points x i ≡ ( x i , x i ) are defined by [9, 16] H = ˜ a (cid:104) (cid:88) i (cid:16) r ϕ i + u ϕ i (cid:17) + (cid:88) i,j K i,j ϕ i − ϕ j ) (cid:105) and by F tot = − ln { (cid:81) Ni =1 (cid:82) ∞−∞ dϕ i exp ( − H ) } . Thelarge-distance anisotropy is described by the symmetricanisotropy matrix A = ( A αβ ) = (cid:18) a cc b (cid:19) , A αβ = lim N →∞ N − (cid:88) i,j ( x iα − x jα )( x iβ − x jβ ) K i,j . (4) FIG. 2: (a) Square lattice with vertical (horizontal) side L ( L (cid:107) ) and aspect ratio ρ = | L | / | L (cid:107) | . Arrows represent theunit vectors e (1) , e (2) of the principal axes for Ω = π/
5. (b)Transformed lattice for q = 1 / L p , L p (cid:107) , and theangle α and aspect ratio ρ p = | L p | / | L p (cid:107) | . (c) Parallelogramparametrized by the complex modular parameter τ , (11). Weak anisotropy requires positive eigenvalues λ > , λ > A , i.e., det A > T c are described bythe reduced anisotropy matrix ¯A = A / (det A ) /d whichfor d = 2 has the form [26] ¯A ( q, Ω) = (cid:18) q c + q − s ( q − q − ) c Ω s Ω ( q − q − ) c Ω s Ω q s + q − c (cid:19) (5)with q = ( λ /λ ) / = ξ (1)0 ± /ξ (2)0 ± and the abbreviations c Ω ≡ cos Ω , s Ω ≡ sin Ω where Ω determines the princi-pal axes described by the eigenvectors e (1) = ( c Ω , s Ω ) (cid:124) , e (2) = ( − s Ω , c Ω ) (cid:124) of A . The exact dependence of Ω and q on the couplings K i,j through a, b, c has been derived in[26]. A shear transformation can be performed such thatthe transformed ϕ model on a parallelogram (Fig. 2)has changed second moments A (cid:48) αβ = δ αβ representingan isotropic system [9, 15, 26, 30]. The transformations L p = λ − / U L and L p (cid:107) = λ − / U L (cid:107) of the verticaland horizontal sides L = L e v and L (cid:107) = L (cid:107) e h yield thecorresponding transformed sides L p and L p (cid:107) of the par-allelogram where the rotation and rescaling matrices U = (cid:18) cos Ω sin Ω − sin Ω cos Ω (cid:19) , λ = (cid:18) λ λ (cid:19) (6)are employed. The parallelogram is characterized by theangle 0 < α < π between L p and L p (cid:107) and the trans-formed aspect ratio ρ p = | L p | / | L p (cid:107) | . We findcot α ( q, Ω) = − ¯A ( q, Ω) = ( q − − q ) cos Ω sin Ω , (7)[ ρ p ( ρ, q, Ω)] = ρ ¯A ( q, Ω) ¯A ( q, Ω) = ρ tan Ω + q q tan Ω , (8)for arbitrary ρ, q, Ω which is valid for arbitrary BC. Thesingular part F iso c ( α, ρ p ) of the total free energy at T c of the isotropic parallelogram is a function of α and ρ p .The shear transformation leaves both the Hamiltonianand the singular part F c of the total free energy F tot at T c invariant [9, 16], thus F c is determined by F c ( ρ, q, Ω) = F iso c (cid:0) α ( q, Ω) , ρ p ( ρ, q, Ω) (cid:1) = ρ − F c ( ρ, q, Ω) . (9)In the strip limit the shear transformation yields [31]lim ρ → X c ( ρ, q, Ω) = ( q cos Ω + q − sin Ω) − X iso c, strip , (10)where X iso c, strip is the amplitude on an isotropic strip. Eqs.(9) and (10) demonstrate that F c , F c ,and X c depend onmicroscopic details via q ( a, b, c ) and Ω( a, b, c ), thus vio-lating two-scale-factor universality. So far it is unknownhow to calculate the dependence of F iso c on α and ρ p . Step
2: At this point we invoke two-scale-factor uni-versality for isotropic systems [8, 10] which means thatisotropic ϕ and Ising models have the same singularparts F iso c and F Is , iso c . For the Ising model exact infor-mation is available from CFT [22, 25]. Via an isotropiccontinuum description in terms of a free fermion field anexact contribution Z CFT ( τ ) to the partition function ofthe d = 2 isotropic Ising model on a torus at T c has beenderived. We choose the same parameters α and ρ p asfor the isotropic ϕ model. The Ising parallelogram isdescribed by a complex torus modular parameter [25] τ ( α, ρ p ) = Re τ + i Im τ = ρ p exp( i α ) (11)where α is the angle shown in Fig. 2(c) and ρ p = | τ | isthe aspect ratio of the Ising parallelogram. The partitionfunction is expressed in terms of Jacobi theta functions θ i (0 | τ ) ≡ θ i ( τ ) (in the notation of [25], see [31]) as [22] Z CFT ( τ ) = (cid:0) | θ ( τ ) | + | θ ( τ ) | + | θ ( τ ) | (cid:1) / (cid:0) | η ( τ ) | (cid:1) , (12)with η ( τ ) = ( θ ( τ ) θ ( τ ) θ ( τ )) / , from which we obtain F CFT c ( τ ) = − ln Z CFT ( τ ) . The singular part of the totalfree energy of the isotropic Ising model at T c is F Is , iso c (cid:0) τ ( α, ρ p ) (cid:1) = F CFT c (cid:0) τ ( α, ρ p ) (cid:1) = F iso c (cid:0) α, ρ p (cid:1) , (13)where, owing to two-scale-factor universality, the lastequation applies to the transformed ϕ model on the par-allelogram. We define τ ( ρ, q, Ω) = τ (cid:0) α ( q, Ω) , ρ p ( ρ, q, Ω) (cid:1) with α ( q, Ω) and ρ p ( ρ, q, Ω) given by (7) and (8). Thenwe obtain from (13), (9), (1), and (3) our exact result forthe Casimir amplitude X c of the anisotropic ϕ model as X c ( ρ, q, Ω) = − ρ ∂ F c ( ρ, q, Ω) /∂ρ (14)with F c ( ρ, q, Ω) = F CFT c (cid:0) τ ( ρ, q, Ω) (cid:1) where the nonuni-versal expressions for q ( a, b, c ) and Ω( a, b, c ) [26] have tobe inserted. In the strip limit we obtain [31] X c (0 , q, Ω) = − π/ [12( q cos Ω + q − sin Ω)] , (15)in accord with the CFT result − π/
12 [12, 13] for q = 1.In Fig. 3 we present contour plots of X c in the com-plete Ω − q plane for ρ = 1 and ρ = 0 .
5. Contraryto the simple (Ω , q ) dependence (15) in strip geometryand to the claim that the effects of weak anisotropy arefairly harmless [32], we find unexpectedly complex struc-tures exhibiting the feature of self-similarity in the re-gions q (cid:28) q (cid:29) q = 0 and q = ∞ , where det A = 0 or λ α = 0, i.e., where weakanisotropy breaks down [28]. This self-similarity can betraced back to the property of modular invariance [25] Z CFT ( τ ) = Z CFT ( τ + 1), Z CFT ( τ ) = Z CFT ( − /τ ) forthe partition function of the isotropic Ising model at T c in a parallelogram geometry with PBC, i.e., on a torus,which implies F Is , iso c ( τ ) = F Is , iso c ( − /τ ) = F Is , iso c ( τ + 1) . FIG. 3: Universal contour plot of the critical Casimir ampli-tude X c of the anisotropic d = 2 ϕ model from (14) for ρ = 1in (a) and for ρ = 0 . d = 2 Isingmodel, (16) yields the same plots if q → q Is , Ω → Ω Is .FIG. 4: Critical amplitude F Is , iso c , (13), of the isotropic Isingmodel in parallelogram geometry (a) in the complex τ planeand (b) in the ρ p − α plane. The same plot (b) holds for F iso c ,(13), of the transformed isotropic ϕ model. This is illustrated by the periodic structure of F Is , iso c in the complex τ plane [Fig. 4(a)] which generates aself-similar structure in the ( ρ p , α ) plane [Fig. 4(b)].The modular transformation τ → − /τ corresponds to ρ p → /ρ p , α → π − α which yields equivalent paral-lelograms. The Dehn twist [25] τ → τ + 1 yields par-allelograms with different ρ p and α , but the invarianceof F Is , iso c can be understood geometrically since a giventorus can be cut in different ways such that different par-allelograms with PBC are generated which all have thesame critical free energy on the same torus. By two-scale-factor universality, the same result applies to F iso c of the isotropic ϕ model on the same torus. The depen-dence on ( ρ p , α ) for the isotropic system in Figs. 2(b) and4(b) is transferred by the shear transformation (7), (8)and by (14) to a corresponding dependence of X c ( ρ, q, Ω)on ( q, Ω) as is shown in Fig. 3. We note that so far noassumption has been made other than the validity of two-scale-factor universality for isotropic systems.
Step
3: We proceed to the anisotropic triangular Isingmodel on an L (cid:107) × L rectangle with the Hamiltonian[26, 33] H Is = − (cid:80) j,k (cid:2) E σ j,k σ j,k +1 + E σ j,k σ j +1 ,k + E σ j,k σ j +1 ,k +1 (cid:3) with spin variables σ j,k = ± Is ( E , E , E )of the principal axes and the ratio of the principal cor-relation lengths q Is ( E , E , E ) = ξ (1)Is0 ± /ξ (2)Is0 ± are knownfunctions of E , E , E [26]. Multiparameter universal-ity was proven for bulk systems in [26], thus the exactcritical bulk correlation function is governed by the Ising FIG. 5: Nonuniversal critical Casimir amplitudes X c and X Is c for ρ = 1 of the d = 2 ϕ and Ising models for (a) det A = ab − c >
0, (b) E + E > , E + E > , E + E > anisotropy matrix ¯A Is = ¯A ( q Is , Ω Is ) with the same ma-trix ¯A as in (5) for the ϕ model, but with q and Ωreplaced by q Is and Ω Is . Since bulk and finite-size prop-erties are governed by the same anisotropy matrix [8] wepredict that the exact critical Casimir amplitude of theanisotropic Ising model is given by X Is c ( ρ, q Is , Ω Is ) = − ρ ∂ F c ( ρ, q Is , Ω Is ) /∂ρ, (16)where F c ( ρ, q Is , Ω Is ) is the same function as in (14) butnow the results for q Is ( E , E , E ) and Ω Is ( E , E , E ) ofthe Ising model [26] have to be inserted. Our predictionsgo far beyond all previous special results [12, 13, 34–39]for confined isotropic and anisotropic Ising models. Herewe have succeeded in treating the general anisotropiccase of an arbitrary direction of the principal axes de-scribed by a nonzero angle Ω Is in a finite geometry withan arbitrary aspect ratio ρ . This is of physical relevancefor general anisotropic systems with more complicatedinteractions whose principal axes generically have skewdirections relative to the symmetry axes of the underly-ing lattice. This advance is made possible by our newapproach of combining exact relations of anisotropic ϕ lattice theory with exact results of CFT. Specifically ourpredictions agree with Ising-model results for isotropicstrips [12, 13, 34], rectangles [35], and parallelograms [36]as well as for anisotropic strips [37] and rectangles [38, 39]which constitutes a direct confirmation of multiparame-ter universality for confined systems. Thus we predictthat the results in Fig. 3 for the ϕ model are valid alsofor the Ising model after substituting q → q Is , Ω → Ω Is ,i.e., the ( q, Ω) representation has a universal characterthat is applicable to all weakly anisotropic systems inthe ( d = 2 , n = 1) universality class. It becomes nonuni-versal if the dependence of ( q, Ω) and ( q Is , Ω Is ) on a, b, c and E , E , E is inserted. We denote these Casimiramplitudes by X c [ ρ, a/c, b/c ] and X Is c [ ρ, E /E , E /E ].They are shown in Fig. 5 for ρ = 1. The nonuniversaldifferences between X c and X Is c confirm the prediction[8, 9, 15, 16] that the Casimir amplitude X c for weaklyanisotropic systems is not a universal quantity. Even ifit is known for one anisotropic system it cannot be pre-dicted for other anisotropic systems of the same univer-sality class since Ω generically depends in an unknownnonuniversal way on the anisotropic interactions [26].In the following we demonstrate that self-similar struc- FIG. 6: Universal contour plots of critical and low-temperature Casimir amplitudes of the d = 3 ϕ theory for n = ∞ [8, 31] with planar anisotropies (18): ˆ X c for ρ = 1with ¯A ( x , y ) in (a) and ˆ X for ρ = 0 . ¯A ( y , z ) in (b). tures exist more generally for O ( n )-symmetric systemswith PBC for d = 3 , n ≥ × A . Consider a L (cid:107) × L geometry with ρ = L/L (cid:107) .In [8] the scaling function X (ˆ x, ρ, ¯A ) of the Casimirforce of the ϕ model has been derived for n ≥
1, whereˆ x ∝ ( T − T c ) L /ν is the scaling variable. This includes thelow-temperature amplitude X ( ρ, ¯A ) ≡ X ( −∞ , ρ, ¯A )due to the Goldstone modes for n ≥
2. In particular,the exact result ˆ X = lim n →∞ X/n has been derivedin the large- n limit. At fixed ˆ x the anisotropy effect iscompletely contained in the function K ( y, (cid:98) C ) = (cid:88) m exp( − y m · (cid:98) Cm ) (17)where y is independent of ¯A . The sum (cid:80) m runs over m = ( m , m , m ) , m α = 0 , ± , ..., ±∞ and the 3 × (cid:98) C has the elements (cid:98) C αβ = ρ α ρ β ( ¯A ) αβ with ρ = ρ = ρ, ρ = 1. We consider two types of planaranisotropies as described by the anisotropy matrices ¯A ( x , y ) = (cid:18) ¯B h
00 1 (cid:19) , ¯A ( y , z ) = (cid:18) ¯B v (cid:19) . (18)In (18) the 2 × ¯B h and ¯B v describeanisotropies in the “horizontal” x − y and “vertical” y − z planes, respectively. The difference between these casesis that the Casimir force defined in (2) is perpendicular tothe x − y anisotropy in case of ¯A ( x , y ) whereas it is parallelto the y − z anisotropy in case of ¯A ( y , z ) . Both ¯B h and ¯B v have the same form as in (5), with ( q, Ω) replaced by( q ¯B , Ω ¯B ). This suggests that self-similar structures ex-ist for d = 3 like those found for d = 2. This is indeedverified by evaluating the exact results for ˆ X c and ˆ X of[8] for d = 3 , n = ∞ with the planar anisotropies (18) asshown in Fig. 6. We find similar structures from [8] forany finite n and ˆ x . The self-similar structures of Fig. 6disappear in the film limit ρ → L but aremaintained in the cylinder limit ρ → ∞ [6] at finite L (cid:107) .We find that, to some extent, this self-similarity can betraced back to the modular invariance of K ( y, (cid:98) C ), (17).Since a symmetric matrix ¯B with det ¯B = 1 contains onlytwo independent matrix elements, it can be expressed as ¯B = 1Im( τ ¯B ) (cid:18) | τ ¯B | − Re( τ ¯B ) − Re( τ ¯B ) 1 (cid:19) , (19)where τ ¯B = Re( τ ¯B ) + i Im( τ ¯B ) is a complex numberwith Im( τ ¯B ) >
0. Based on the one-to-one relation (19)between ¯B and τ ¯B , we can relate a modular transforma-tion τ ¯B → τ ¯B (cid:48) to a corresponding matrix ¯B (cid:48) , with, e.g., τ ¯B (cid:48) = τ ¯B +1 for the Dehn twist. The function K remainsinvariant under such transformations for ¯A = ¯A ( y , z ) , ρ = 1 and for ¯A = ¯A ( x , y ) and arbitrary ρ [31]. Thisis parallel to the modular invariance of Z CFT ( τ ). Moregenerally, we expect self-similar structures also for d = 3systems with non-planar anisotropies and PBC. Conclusion and outlook -
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