Multiparameter universality and conformal field theory for anisotropic confined systems: test by Monte Carlo simulations
MMultiparameter universality and conformal field theoryfor anisotropic confined systems: test by Monte Carlo simulations
Volker Dohm, Stefan Wessel, Benedikt Kalthoff, and Walter Selke
Institute for Theoretical Physics, RWTH Aachen University, 52056 Aachen, Germany (Dated: February 21, 2021)Analytic predictions have been derived recently by V. Dohm and S. Wessel, Phys. Rev. Lett. , 060601 (2021) from anisotropic ϕ theory and conformal field theory for the amplitude F c ofthe critical free energy of finite anisotropic systems in the two-dimensional Ising universality class.These predictions employ the hypothesis of multiparameter universality. We test these predictions bymeans of high-precision Monte Carlo (MC) simulations for F c of the Ising model on a square latticewith isotropic ferromagnetic couplings between nearest neighbors and with an anisotropic couplingbetween next-nearest neighbors along one diagonal. We find remarkable agreement between theMC data and the analytical prediction. This agreement supports the validity of multiparameteruniversality and invalidates two-scale-factor universality as F c is found to exhibit a nonuniversaldependence on the microscopic couplings of the scalar ϕ model and the Ising model. Our resultsare compared with the exact result for F c in the three-dimensional ϕ model with a planar anisotropyin the spherical limit. The critical Casimir amplitude is briefly discussed. I. INTRODUCTION
A widely-held belief in the traditional theory of criticalphenomena has been the general validity of two-scale-factor universality for the singular behavior of confinedthermodynamic systems with short-range interactions [1–8]. The most fundamental quantity is the free-energydensity f . For d -dimensional systems with a characteris-tic length L the asymptotic critical behavior of the singu-lar part f s of f (divided by k B T ) has been hypothesizedto be described by the scaling form for d < f s ( t, h, L ) = L − d F ( C tL /ν , C hL βδ/ν ) (1.1)for large L , small t = ( T − T c ) /T c and small orderingfield h , with universal critical exponents ν, β, δ . It wasasserted that, for given geometry and boundary condi-tions, the scaling function F ( x, y ) is universal, i.e., thatthe two metric factors C and C are the only nonuniver-sal parameters entering (1.1)[1–8]. The universal struc-ture of (1.1) is referred to as two-scale-factor universality[1, 5]. It was believed that (1.1) is valid for all systems ina universality class, i. e., for both isotropic and weaklyanisotropic systems, since it was argued that universalitycan be restored [9], reintroduced [10], or repaired [5] inthe latter systems by a suitable anisotropic scale trans-formation restoring asymptotic isotropy [2, 11, 12]. Cor-respondingly the same finite-size scaling function of thecritical Casmir force has been predicted for anisotropicsuperconductors and isotropic superfluids [13, 14]. How-ever, it has been shown [15–18] that restoring isotropydoes not eliminate nonuniversality since the transformedisotropic system still carries the nonuniversal anisotropyinformation of the original system both in its changedgeometry and in the nonuniversal orientation of its trans-formed boundary conditions.More recently a unified hypothesis of multiparameter uni-versality [19], originally introduced for critical bulk am-plitude relations [17], was formulated for bulk and con-fined anisotropic systems. Subsequently its exact validitywas proven for the critical bulk order-parameter correla-tion function and for a critical bulk amplitude relation in two dimensions [20]. For confined systems this hypothe-sis predicts that f s depends on d ( d +1) / L d at t = 0 , h = 0, it implies that thesingular part of the total free energy at criticality F c ( q, Ω) = L d f s (0 , , L ) = F (0 ,
0) (1.2)depends on d ( d + 1) / − q = ξ (1)0 ± /ξ (2)0 ± of the twoprincipal correlation lengths ξ ( β ) ± = ξ ( β )0 ± | t | − ν , β = 1 , T c (+) and below T c ( − ) with t = ( T − T c ) /T c where ν = 1 for Ising-like systems [20]. If correct, thishypothesis has serious consequences for the predictabilityof the critical amplitude F c ( q, Ω) and the ensuing criti-cal Casimir amplitude. As pointed out earlier [19, 21],contrary to the feature of two-scale-factor universality,the angle Ω generically depends in an unknown way onthe anisotropic interactions, thus F c ( q, Ω) remains to bean unpredictable amplitude for all anisotropic systemswhose orientation of the principal axes is unknown. Forthe same reason the critical Casimir amplitude remainsunpredictable for such anisotropic systems. This refutesthe claim [22] that the anisotropy encountered in weaklyanisotropic systems is of a fairly harmless kind. It alsorefutes common assertions (see, e.g., [23–27]) that thecritical Casimir amplitude is independent of microscopicdetails and thus depends only on a few global and gen-eral properties, such as the dimension d , the number ofcomponents of the order parameter, the shape of the con-finement, and the type of the boundary conditions (BC).Very recently it has indeed been proven [28] thattwo-scale-factor universality is violated for the weaklyanisotropic scalar ϕ model with periodic boundary con-ditions (PBC) in d = 2 dimensions. An exact result hasbeen derived for F c ( q, Ω) of the general ϕ model witharbitrary short-range pair interactions, i.e., for arbitraryorientations Ω of the principal axes, by means of confor- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b E e (2) E e (1) E x x FIG. 1: (Color online) Lattice points of the ”triangular-lattice” Ising model (1.3) on a square lattice with couplings E , E , E . Dotted arrows e (1) and e (2) denote the principaldirections for E > E > , E > < Ω Is < π/ mal field theory (CFT) where no assumption has beenmade other than the validity of two-scale-factor univer-sality for isotropic systems. Furthermore, on the basisof the hypothesis of multiparameter universality, quanti-tative predictions have been made for F Is c = F c ( q Is , Ω Is )of the fully anisotropic triangular Ising model with PBCfor the case of arbitrary orientations Ω Is of the principalaxes [28]. The Hamiltonian of this model reads [9, 20, 29] H Is = − (cid:88) 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) (1.3)where σ j,k = ± a = 1) with horizontal, vertical,and diagonal couplings E , E , E (see Fig. 1). Both theangle Ω Is ( E , E , E ) of the principal axes and the ra-tio of the principal correlation lengths q Is ( E , E , E ) = ξ (1)Is0 ± /ξ (2)Is0 ± are known functions of E , E , E [20]. Theangle is given by [20]tan 2Ω Is = 2 ˆ S ˆ S − ˆ S for E (cid:54) = E , (1.4)Ω Is = π/ E = E , (1.5)with ˆ S α = sinh 2 β Is c E α , α = 1 , , , β Is c = ( k B T Is c ) − .Exact agreement was reported between the predictionof [28] and the exact results [30, 31] for F Is c of theanisotropic Ising model with a ”rectangular anisotropy” E (cid:54) = E , E = 0 where the principal axes are paral-lel to the symmetry axes of the lattice, i.e., Ω Is = 0 [20],thus confirming multiparameter universality for this case.However, so far no proof exists for the validity of multi-parameter universality for finite anisotropic Ising modelswith a nonzero angle Ω Is of the principal axes.It is the purpose of this paper to test the predictions of[28] for F Is c for the case of a square geometry with a finiteangle Ω Is = π/
4, i.e where the principal axes are paral-lel to the diagonals of the square lattice (Fig. 1). Thiswill be achieved by means of high-precision Monte Carlo(MC) simulations for F Is c of the Ising model with theHamiltonian (1.3) on a square lattice with isotropic fer-romagnetic couplings E = E = E > E (cid:54) = 0between next-nearest neighbors (NNN) along one diago-nal in the ferromagnetic region E + E > ϕ model supports the validityof multiparameter universality but invalidates two-scale-factor universality as F Is c is found to exhibit a nonuni-versal dependence on the ratio E /E . That differs fromthe corresponding dependence of F c of the ϕ model al-though the latter model and the Ising model belong tothe same universality class. II. SHEAR TRANSFORMATION OF THEANISOTROPIC ϕ MODEL
We first consider the anisotropic scalar ϕ lattice modelwith dimensionless continuous variables −∞ ≤ ϕ i ≤ ∞ on N = ( L/ ˜ a ) lattice points x i ≡ ( x i , x i ) on a squarelattice with lattice spacing ˜ a and a shape of an L × L square with PBC. We assume short-range pair interac-tions K i,j . The Hamiltonian and the free-energy densitydivided by k B T are defined by [16, 17] H = ˜ a (cid:34) N (cid:88) i =1 (cid:16) r ϕ i + u ϕ i (cid:17) + N (cid:88) i,j =1 K i,j ϕ i − ϕ j ) (cid:35) , (2.1) f ( t, L ) = − L − ln (cid:40)(cid:104) N (cid:89) i =1 (cid:90) ∞−∞ dϕ i (cid:105) exp ( − H ) (cid:41) . (2.2)The large-distance anisotropy is described by the 2 × A with matrix elementsdetermined by the second moments A αβ = lim N →∞ N − (cid:88) i,j ( x iα − x jα )( x iβ − x jβ ) K i,j . (2.3)We assume the following anisotropy matrix A = ( A αβ ) = (cid:18) a cc a (cid:19) (2.4)with c (cid:54) = 0 and positive diagonal elements A = A = a >
0. Weak anisotropy requires positive eigenvalues λ = a + c > , λ = a − c >
0, i.e. − a < c < a ,which ensures unchanged critical exponents of the two-dimensional Ising universality class [15]. The eigenvec-tors e (1) = 1 √ (cid:18) (cid:19) , e (2) = 1 √ (cid:18) − (cid:19) (2.5)are valid for both c > c <
0. The principal di-rections 1 and 2 are parallel and perpendicular to the(1 ,
1) direction corresponding to the angle Ω = π/
4. Near T c the asymptotic principal correlation lengths ξ ( β ) ± ( t ) = ξ ( β )0 ± | t | − have the ratio [17, 19] ξ (1)0 ± ξ (2)0 ± = (cid:16) λ λ (cid:17) / = (cid:16) a + ca − c (cid:17) / = q = (cid:26) > , c > ,< , c < . (2.6)The simplest version of this ϕ model is realized [16] byisotropic ferromagnetic NN couplings K = J/ ˜ a > K d = J d / ˜ a in the diagonal(1 ,
1) direction [Fig. 2 (a)]. This yields the dimension-less nondiagonal anisotropy matrix [16, 19] A = 2 (cid:18) J + J d J d J d J + J d (cid:19) . (2.7)with 0 < det A < ∞ and q = (1 + 2 J d /J ) / = (cid:26) > , J d > ,< , J d < , (2.8)in the range − J/ < J d < ∞ . We are interested inthe asymptotic (large L ) amplitude F c = L f sc where f sc ≡ f s (0 , L ) is the singular part of f ( t, L ) at T c . In thefollowing we specialize the general derivation of [28] tothe present case.From the structure of previous results for the anisotropic ϕ theory in 2 ≤ d < ¯A = A / (det A ) /d which inthe present two-dimensional case has the form ¯A ( q ) = 12 (cid:18) q + q − q − q − q − q − q + q − (cid:19) . (2.9)This matrix has the same eigenvectors (2.5) as A and thereduced eigenvalues ¯ λ β = λ β / (det A ) / , β = 1 , , withdet ¯A = ¯ λ ¯ λ = 1. It is expected that F c is a function of ¯A ( q ), F c ( q ) = F c [ ¯A ( q )] . (2.10)It has been shown [15–17] that a shear transformationcan be performed such that the anisotropic ϕ modelon a square is transformed to a ϕ model on a rhombus(Fig. 2) with changed second moments A (cid:48) αβ = δ αβ , A (cid:48) = , representing a system with isotropic critical correla-tions at large distances. The couplings K i,j , K , and K d and the temperature variable r are kept fixed inthis transformation which is constructed such that theHamiltonian is invariant. We describe the vertical sidesof the square and the corresponding transformed sides ofthe rhombus by the vectors L and L rh , respectively (Fig.2). They are related by the transformation L rh = λ − / U L = L / (cid:32) λ − / λ − / (cid:33) , L = L (cid:18) (cid:19) , (2.11) U = 12 / (cid:18) − (cid:19) , λ = UAU − = (cid:18) λ λ (cid:19) , (2.12)with the rotation matrix U and the diagonal rescalingmatrix λ . Here we have considered a clockwise rotation;equivalent results are obtained by a counterclockwise ro-tation. The ratio of the components of L rh determinesthe angle 0 < ω < π (Fig. 2 b) as ( λ /λ ) / = tan( ω/ ω ( q ) = 2 arctan (cid:104)(cid:16) λ /λ (cid:17) / (cid:105) (2.13)= 2 arctan ξ (2)0 ± /ξ (1)0 ± = 2 arctan q − . (2.14) (a) (b) K KK d KK d K e (1) e (2) α ω K d = − K/ L rh L FIG. 2: (a) Square lattice with isotropic NN couplings K and anisotropic NNN coupling K d . (b) For the case K d = − K/ λ /λ = 1 /
2, the trans-formed lattice with isotropic second moments A (cid:48) αβ = δ αβ hasthe angle ω = 2 arctan( √
2) and the complementary angle α = 2 arctan(1 / √
2) (compare Fig. 2 of [16]).
Instead of ω we may employ the complementary angle0 < α = π − ω < π which yields α ( q ) = 2 arctan q. (2.15)The singular part F iso c ( α ) of the free energy at T c of therhombus is a function of α . The total free energies of thesystems on the square and on the rhombus differ onlyby a nonsingular additive term [16]. Thus the singularpart F c of the anisotropic system is left invariant and isdetermined by F c ( q ) = F iso c (cid:0) α ( q ) (cid:1) . (2.16)However, it is unknown how to perform an exact cal-culation of F iso c ( α ) within the isotropic ϕ model on arhombus. III. EXACT RESULT FOR THE ANISOTROPIC ϕ MODEL
At this point we follow the reasoning of [28] where two-scale-factor universality for isotropic systems [1] is in-voked. This means that isotropic scalar ϕ models andisotropic Ising models have the same universal amplitude F iso c = F Is , iso c if they are defined on a rhombus with thesame angle α and with the same BC. From conformal fieldtheory (CFT) [34, 35] an exact asymptotic contribution Z CFT ( τ ) to the partition function of the isotropic Isingmodel at T c on a rhombus with PBC, i.e., on a torus, hasbeen derived where the rhombus is parameterized by acomplex torus modular parameter τ ( α ) = Re τ + i Im τ = exp( i α ) (3.1)with | τ | = 1 , Im τ >
0, and with the angle 0 < α < π (see Fig. 3), α = arctan(Im τ / Re τ ) . (3.2)The partition function is expressed in terms of Jacobi FIG. 3: Illustration of the two choices for mapping theanisotropic square onto an isotropic rhombus with a clock-wise rotation (compare Fig. 2) or a counterclockwise rotation.Indicated in panel (a) are the angle α = 2 arctan(1 / √
2) andthe modular parameter τ , and in panel (b) the complementaryangle π − α = 2 arctan( √
2) and ˜ τ = − /τ . theta functions θ i (0 | τ ) ≡ θ i ( τ ) (in the notation of [35],see App. A) as [34] Z CFT ( τ ) = (cid:0) | θ ( τ ) | + | θ ( τ ) | + | θ ( τ ) | (cid:1) / (cid:0) | η ( τ ) | (cid:1) , (3.3)with η ( τ ) = ( θ ( τ ) θ ( τ ) θ ( τ )) / , from which we obtain F CFT c ( τ ) = − ln Z CFT ( τ ) . (3.4)The singular part of the free energy of the isotropic Isingmodel at T c is F Is , iso c (cid:0) τ ( α ) (cid:1) = F CFT c (cid:0) τ ( α ) (cid:1) = F iso c (cid:0) α (cid:1) , (3.5)where, owing to two-scale-factor universality, the lastequation applies to the transformed ϕ model on therhombus. We define τ [ q ] = τ (cid:0) α ( q ) (cid:1) = exp (cid:0) i α ( q ) (cid:1) = q − q + 1 + i q q . (3.6)Then we obtain from (3.5) and (2.16) our exact result forthe critical amplitude F c of the anisotropic ϕ model as F c ( q ) = F CFT c (cid:0) τ [ q ] (cid:1) (3.7)where the nonuniversal expressions (2.6) or (2.8) for q ( c/a ) or q ( J d /J ) have to be inserted. So far our onlyassumption is the validity of two-scale-factor universalityfor isotropic systems.We note that a counterclockwise rotation in the sheartransformation replaces τ by (cid:101) τ = − /τ = − τ ∗ = − Re τ + i Im τ (Fig. 3), using that | τ | = 1. The modularinvariance Z CFT ( τ ) = Z CFT ( − /τ ) [34, 35] guaranteesthe independence of F CFT c (cid:0) τ [ q ] (cid:1) on this choice. This in-variance also implies that Z CFT only depends on | Re τ | under the condition | τ | = 1, 1 /τ = τ ∗ . Thus, the crit-ical amplitude F CFT c (cid:0) τ [ q ] (cid:1) is a function of s ( q ) , i.e., asymmetric function of s ( q ) ≡ Re τ [ q ] = ( q − / ( q + 1) (3.8)= (1 + J/J d ) − = (cid:26) > , J d > ,< , J d < , (3.9) FIG. 4: Exact analytic result F c ( q ), (3.7) for the anisotropic ϕ theory at T = T c as a symmetric function of s ( q ), (3.8)with maxima at s ( q ) = ± /
2, indicated by the verticallines. The horizontal line represents the isotropic value of F c (1) = − . . for q = 1 ( s ( q ) = 0 , J d = 0). Posi-tive (negative) values s ( q ) > s ( q ) <
0) correspond to therange of ferromagnetic (antiferromagnetic) couplings J d . Asymmetric representation is also possible as a function of ln q . which is shown in Fig. 4. Since q depends on the mi-croscopic couplings the amplitude F c ( q ) is a nonuniver-sal quantity violating two-scale-factor universality. Sincepositive (negative) values of s ( q ) correspond to J d > J d <
0) according to (3.9) we see from Fig. 4 that F c yields the same values in the range of ferromagnetic andantiferromagnetic couplings J d . This symmetry of F c only reflects the modular invariance in the presence ofPBC but does not yet imply the universality of the func-tional form of F c ( q ). In our present case, the function F c exhibits a two-peak structure where the two equal-height maxima are located at s ( q max ) = ± correspond-ing to q max = 3 ± / and J d = J, J d = − J/
3, respec-tively. Furthermore, we note that F c → −∞ for q → s ( q ) → −
1) and q → ∞ ( s ( q ) → τ → − τ → +1 respectively,in these limits. In terms of coupling parameters, theselimits correspond to J d → − J/ J d → ∞ , respec-tively. Hence, the CFT rhombus degrades towards bothends of the regime − / < J d /J < ∞ of weak anisotropy.By rewriting q = exp(ln q ) we note that one finds that F c ( q ) = (cid:101) F c (ln q ) is a symmetric function also of ln q , (cid:101) F c (ln q ) = (cid:101) F c ( − ln q ), as a consequence of modular in-variance. IV. ANALYTIC PREDICTION AND MONTECARLO SIMULATION FOR THE ANISOTROPICISING MODEL
We return to the Ising model defined by (1.3) in a squaregeometry. The free energy per site (divided by k B T ) fora square lattice with L lattice sites at β = 1 / ( k B T ) is f Is ( β, L ) = − L ln (cid:88) { σ } exp( − βH Is ) . (4.1)The hypothesis of multiparameter universality [19] forthe confined system predicts that, parallel to the case forthe bulk correlation function [20], the singular part F Is c ofthe total free energy L f Is ( β Is c , L ) at T Is c is obtained from F c ( q, Ω) of the scalar ϕ model at T c by the substitution q → q Is , Ω → Ω Is provided that both models have thesame geometry and the same BC. Thus the prediction is[28] F Is c = F c ( q Is , Ω Is ). Here we consider this Ising modelfor E = E = E > , E (cid:54) = 0 in the ferromagnetic range E + E > Is = π/ q Is = ξ (1)Is0 ± ξ (2)Is0 ± = 1sinh 2 β Is c E = (cid:26) > , E > ,< , E < . (4.2)Due to the condition of criticality [32]sinh (2 β Is c E ) + 2 sinh(2 β Is c E ) sinh(2 β Is c E ) = 1 , (4.3)2 β Is c E ≡ y is a function of E /E determined implic-itly by sinh ( y ) + 2 sinh( y ) sinh( yE /E ) = 1, thus q Is =1 / sinh( y ) is a function of E /E in the range −
3) exactly reduces tothe analytic results stated there, given that these latticescorrespond to τ = i and τ = e iπ/ , respectively, as notedin [28]. We also note that F Is c → −∞ for E → − E and E → ∞ (corresponding to q Is → s Is → − q Is → ∞ ( s Is → +1)) , as the modular parameterapproaches τ → − τ → +1. Thus, for the Isingmodel, the CFT rhombus degrades towards both endsof the regime − < E /E < ∞ of finite T c . While inthe upper limit E /E → ∞ , the system decouples intoone-dimensional chains, it does not exhibit order even at T = 0, due to frustration for E /E < − FIG. 5: Comparison of the exact analytic prediction F Is c ,(4.5) for the anisotropic Ising model,(1.3), to finite-size es-timates F Is c,L from MC simulations for L = 16 (circles) assymmetric functions of s ( q Is ), with maxima at s ( q Is ) = ± / s ( q Is ) = 0 and s ( q Is ) = 1 / F Is c (1) = − . q Is = 1( s ( q Is ) = 0 , E = 0). The positive (negative) values s ( q Is ) > s ( q Is ) <
0) correspond to the range of ferromagnetic (anti-ferromagnetic) couplings E . A symmetric representation isalso possible as a function of ln q Is . In order to assess the validity of our analytical predic-tion for general values of E /E , we performed MC sim-ulations of the Hamiltonian H Is on finite L × L latticeswith periodic BC, using local spin flips with the standardMetropolis update, combined (for E ≥
0) with Wolff-cluster updates [38]. We obtain the free energy per siteat T Is c , divided by k B T Is c , f Is ( β Is c , L ), from a thermody-namic integration f Is ( β Is c , L ) = (cid:90) β Is c u Is ( β, L ) dβ − ln 2 (4.8)of the inner energy per site, u Is ( β, L ), over a dense in-verse temperature grid, using the trapezoidal rule tonumerically perform the above integral. We used auniform β -grid with ∆ β = 0 . /E for all reporteddata. The above integration formula is obtained from u Is ( β, L ) = ∂f Is ( β, L ) /∂β and f Is (0 , L ) = − ln 2, as fol-lows from (4.1). As a finite-size estimate for F Is c we con-sider the quantity F Is c,L = 43 [ f Is ( β Is c , L ) − f Is ( β Is c , L )] L (4.9)which, by the asymptotic scaling (1.2), approaches F Is c =lim L →∞ F Is c,L in the thermodynamic limit. A comparisonof the MC data to our analytical prediction of F Is c isshown in Fig. 5 as a function of s ( q Is ). Here we showdata for L = 16 where we were able to obtain accuratevalues for F Is c,L . The residual finite-size effects are foundto be weak, as seen from comparing to the exact previousvalues at E = 0 and E = E [36, 37]. Aside from weakresidual finite-size effects, we obtain a remarkable agree-ment between the MC data and the analytical predictionover the full parameter range for which we could obtainhigh-precision MC data. The statistical uncertainty fromthe MC simulations significantly grows for E (cid:46) − . E ,corresponding to s ( q Is ) (cid:46) − . E < E = E = E > F Is c in the accessible range E (cid:38) − . E pro-vides substantial support for the validity of multiparame-ter universality for the case Ω Is = π/
4. The symmetry of F Is c ( q Is ) with respect to the ferromagnetic ( E >
0) andantiferromagnetic ( E <
0) range of the coupling E pre-dicted by the theory is well reproduced by the MC data.In particular, two equal-height maxima at s ( q Ismax ) = ± / E = E and E = − [arcsinh(1 / √ / arcsinh( √ E ≈ − . E .We note that this symmetric structure results from themodular invariance discussed in the context Fig. 4 andis not expected to be a generic feature of multiparameteruniversality for the general case of a rectangular geome-try and Ω Is (cid:54) = π/ V. VIOLATION OF TWO-SCALE-FACTORUNIVERSALITY
Due to multiparameter universality, the representationof F Is c as a function of the ratios q and q Is of principalcorrelation lengths is predicted [19, 28] to have a univer-sal character implying that F Is c ( q Is ) = F c ( q Is ), i.e., withthe same functional form of F c for the Ising model as forthe ϕ model. However, the functions F c and F Is c areexpected to differ if they are plotted as functions of thecoupling ratios J d /J and E /E , respectively. The phys-ical origin of this nonuniversal effect is a bulk propertyarising from the different dependence of the ratios of theprincipal correlation lengths q and q Is on the microscopiccoupling ratios. More specifically, Fig. 6 shows q and q Is as functions of J d /J and E /E , respectively. They differfor all values of the coupling ratios, except for two specialcases, namely for J d = E = 0, where q = q Is = 1, and for J d /J = E /E = 1, where q = q Is = √
3. While the for-mer case corresponds to asymptotically isotropic ϕ andIsing models with isotropic NN couplings on the samesquare lattice, the latter case corresponds to anisotropicsystems on a square lattice with equal NN and NNN cou-plings in which case both models can be transformed toasymptotically isotropic systems with isotropic NN cou-plings on the same triangular lattice (for the ϕ modelsee Fig. 2 (c) of [16], for the Ising model see [37]).The above expectation is confirmed by the nonuniver-sal plot of F c and F Is c as functions of the ratios J d /J and E /E , shown in Fig. 7. Thus F c and F Is c depend,contrary to the isotropic case, on microscopic details.This illustrates the severe breakdown of two-scale-factoruniversality for anisotropic systems, and the fact thatthe knowledge of F c or F Is c for only the isotropic casehas no predictive power for providing these quantities at FIG. 6: Nonuniversal dependence of the ratios q , (2.8),and q Is , (4.2), of the asymptotic principle correlation lengthsas functions of the coupling ratios J d /J and E /E for theanisotropic ϕ theory and Ising model, respectively. The in-set focuses on the lower limit of the regime of weak anisotropy.FIG. 7: Comparison of the exact analytic result F c , (3.7) forthe anisotropic ϕ model, (2.1),(2.7), and the prediction F Is c ,(4.5) for the anisotropic Ising model, (1.3), as functions of thecoupling ratios J d /J and E /E , respectively. Also includedare the finite-size estimates F Is c,L from MC simulations for theanisotropic Ising model with L = 16 (circles). The statisticaluncertainty of the MC data is smaller than the symbol size.The crosses at J d = E = 0 and J d /J = E /E = 1 are theanalytical results from Refs. [36] and [37], respectively. Theinset focuses on the crossing points of both curves for thespecial coupling ratios 0 and 1. any generic value of the microscopic couplings. For thispurpose, further (nonuniversal) knowledge of the corre-sponding asymptotic principle correlation lengths ratio ismandatory (as well as of the orientation angle Ω of theprinciple axes in the general case [28]).Nevertheless the feature of multiparameter universalityensures that the exact result shown in Fig. 4, based onCFT and ϕ theory, should be valid, as a function ofthe appropriate ratio q of principal correlation lengths,for all weakly anisotropic systems of the ( d = 2 , n = 1)universality class with the same principal axes in a squaregeometry with periodic BC. VI. CRITICAL CASIMIR AMPLITUDE
We turn to the question as to the consequences of theseresults for the critical Casimir amplitude of the modelsanalyzed above. For this purpose we need to extend thesquare geometry to a rectangular L (cid:107) × L geometry withPBC where the singular part F c ( ρ, q, Ω) of the total freeenergy at T c now becomes a function of the aspect ratio ρ = L/L (cid:107) [28]. The explicit expression for F c at Ω = π/ ϕ model in terms of the modular parameter(3.6) is [28] F c ( ρ, q, π/
4) = F CFT c ( ρ τ [ q ]) = − ln Z CFT c ( ρ τ [ q ]) . (6.1)The amplitude X c ( ρ, q, π/
4) of the critical Casimir forcein the vertical direction is obtained as [28] X c ( ρ, q, π/
4) = − ρ ∂ F c ( ρ, q, π/ /∂ρ. (6.2)This yields X c ( ρ, q, π/
4) = ρ ∂Z CFT c ( ρ τ [ q ]) /∂ρZ CFT c ( ρ τ [ q ]) . (6.3)While the above results lead to a finite critical Casimiramplitude for any values of ρ [28] in general, we can provethat X c ( ρ, q, π/
4) vanishes for the special case of a squaregeometry, ρ = 1, irrespectively of the value of q . Namely,as shown in App. B, the derivate ∂Z CFT c ( ρ τ [ q ]) /∂ρ van-ishes exactly at ρ = 1, due to the properties of the Jacobitheta functions that enter Z CFT , such that X c (1 , q, π/
4) = 0 , (6.4)for all values of q (compare Fig. 3 (a) of [28] forΩ = π/
4, see also [25] for q = 1). Thus, for the ϕ theory on a square geometry, the critical Casimir ampli-tude vanishes for all values of J d /J within the regime ofweak anisotropy. The same result applies to the criticalCasimir amplitude of the anisotropic Ising model on thesquare geometry for all values of E /E within the regimeof weak anisotropy, provided that multiparameter uni-versality is valid also for ρ (cid:54) = 1 , Ω = π/
4. The latterassumption is needed since the definition of X c containsthe derivative with respect to ρ . More generally, multi-parameter universality predicts the critical Casimir am-plitude to vanish for all weakly anisotropic systems on asquare geometry in the ( d = 2, n = 1) universality class ifthe orientation angle Ω of the principle axes equals π/ X c for general q , Ω and ρ we refer to[28]. VII. PLANAR ANISOTROPY IN THREEDIMENSIONS
It has been pointed out [28] that, in the presence ofPBC, modular invariance plays an important role not
FIG. 8: Exact analytic result ∆ F c, ∞ , (7.4), for the d = 3 ϕ theory at T = T c for n = ∞ shown as symmetric function of s ( q ), (3.8), with maxima at s ( q ) = ± /
2. For comparison, ourexact CFT-based result ∆ F c , for the d = 2, n = 1 ϕ theoryin Fig. 4 is also shown. The inset illustrates the unit cell ofa simple-cubic lattice with a planar anisotropy as consideredhere. only in the finite-size effects of weakly anisotropic Ising-like systems in two dimensions but more generally in thethree-dimensional O( n )-symmetric ϕ theory with planaranisotropies. Here we incorporate the two-dimensionalanisotropy discussed above in the three-dimensional ϕ model with a L × L × L cubic geometry (see the insetin Fig. 8, see also Fig. 11 of [17]) in the large- n limitand compare the ensuing exact result for F c with thatpresented above in two dimensions.Three-dimensional anisotropy and rescaling matrices ofthis ϕ model can be chosen as (compare Eqs. (6.19)and (8.4) of [19]) A ( x,y )3 = (cid:32) A 00 T (det A ) / (cid:33) , λ = λ λ
00 0 ( λ λ ) / (7.1)where = (cid:18) (cid:19) and T = (0 0) and where A is the 2 × x, y ) plane. The three eigenvalues are λ = 2( J + 2 J d ) , λ = 2 J, λ = ( λ λ ) / = (det A ) / .The reduced matrices ¯ A ( x,y )3 = A ( x,y )3 / (cid:0) det A ( x,y )3 (cid:1) / and ¯ λ = λ / (det λ ) / are ¯A ( x,y )3 ( q ) = (cid:32) ¯A ( q ) T (cid:33) , ¯ λ ( q ) = ¯ λ λ
00 0 1 (7.2)with ¯A ( q ) given by (2.9) and with ¯ λ = ( λ /λ ) / = q = ξ (1)0+ /ξ (2)0+ and ¯ λ = q − where ξ ( α )0+ are the planar princi-pal correlation-length amplitudes above T c . This repre-sents a three-dimensional ϕ model with the same planaranisotropy as for the d = 2 models discussed above, butwith no anisotropy of the bulk correlation function at T c G b,c ( x ) ∝ ( ... + x ) − (1+ η ) / [19] in the z -direction, owingto the particular choice of the principal correlation length ξ (3)0+ = (cid:2) ξ (1)0+ ξ (2)0+ (cid:3) / (7.3)determined by the mean correlation length in the plane.Substituting our matrix ¯A ( x,y )3 ( q ), (7.2), into Eq. (6.39)of [17] (see also the expressions in [39], Sec. IV for ρ = 1) yields the exact critical amplitude F c, ∞ ( q ) =lim n →∞ F c ( q ) /n in the large- n limit for PBC which isshown in Fig. 8 as∆ F c, ∞ ( q ) = F c, ∞ ( q ) − F c, ∞ (1) , (7.4)i.e., relative to the isotropic case q = 1, with F c, ∞ (1) = − . s ( q ), (3.8). The correspond-ing quantity ∆ F c ( q ) = F c ( q ) − F c (1) is also shown forour exact CFT-based result for the d = 2, n = 1 case ofFig. 4. In the large- n limit the anisotropy enters F c, ∞ ( q )through the function K defined in Eq. (17) of [28] whichdisplays the property of modular invariance parallel tothat of Z CFT . This explains the symmetry of the n = ∞ curve in Fig. 8. In particular we again find a two-peakstructure with equal-height maxima, with the same posi-tions q max = 3 ± / of the maxima as for the anisotropic d = 2 , n = 1 model discussed above. The symmetry withrespect to s was also found in [17] where, however, theorigin from modular invariance was not yet recognized[40].The same symmetry persists at finite n , as we have ver-ified on the basis of the approximate results of [19] forthe d = 3 O( n )-symmetric ϕ theory with the planaranisotropy defined above. Invoking multiparameter uni-versality for general 1 ≤ n ≤ ∞ we predict the same re-sults for three-dimensional O ( n )-symmetric fixed-lengthspin models, after substituting q → q spin , e. g., for XY models ( n = 2) or Heisenberg models ( n = 3). Appendix A: Jacobi theta functions from [35]
We follow [35] for the notation of the Jacobi theta func-tions θ i ( τ ) used in (3.3). For a complex number τ withIm( τ ) > θ ( τ ) = ∞ (cid:88) n = −∞ q ( n +1 / / , (A.1) θ ( τ ) = ∞ (cid:88) n = −∞ q n / , (A.2) θ ( τ ) = ∞ (cid:88) n = −∞ ( − n q n / , (A.3) with q = exp(2 πiτ ). Appendix B: Proof of ∂Z CFT c ( ρ τ [ q ]) /∂ρ | ρ =1 = 0 Besides the modular invariance of Z CFT c ( τ ), which gives Z CFT c ( τ ) = Z CFT c ( − /τ ) (B.1)we obtain the following property of Z CFT c ( τ ) with respectto complex conjugation of τ , Z CFT c ( τ ) = Z CFT c ( − τ ∗ ) (B.2)from the fact that | θ i ( τ ) | = | θ i ( − τ ∗ ) | , for i = 2 , , ϕ theory and Ising model the modular parameter hasunit length, | τ [ q ] | = 1, we get − /τ [ q ] = − τ [ q ] ∗ . (B.3)We next calculate ∂Z CFT c ( ρ τ [ q ]) /∂ρ | ρ =1 aslim (cid:15) → Z CFT c ((1 + (cid:15) ) τ [ q ]) − Z CFT c ((1 − (cid:15) ) τ [ q ])2 (cid:15) . (B.4)Due to modular invariance, Z CFT c ((1 + (cid:15) ) τ [ q ]) = Z CFT c ( − / ((1 + (cid:15) ) τ [ q ]))= Z CFT c ((1 − (cid:15) ) ( − /τ [ q ])) + O ( (cid:15) ) . Now using the identities (B.2) and (B.3), we obtain: Z CFT c ((1 − (cid:15) )( − /τ [ q ])) = Z CFT c ((1 − (cid:15) )( − τ [ q ] ∗ ))= Z CFT c ((1 − (cid:15) ) τ [ q ])) , such that we find: Z CFT c ((1 + (cid:15) ) τ [ q ])) = Z CFT c ((1 − (cid:15) ) τ [ q ])) + O ( (cid:15) ) . Using this in (B.4), we thus find that ∂Z CFT c ( ρ τ [ q ]) /∂ρ | ρ =1 = 0 . (B.5) [1] V. Privman and M.E. Fisher, Phys. Rev. B , 322(1984).[2] J. L. Cardy, in Phase Transitions and Critical Phenom- ena , edited by C. Domb and J. L. Lebowitz (Academic,New York, 1987), Vol. 11, p. 55.[3] J. L. Cardy,
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