Anisotropic scaling of the two-dimensional Ising model II: Surfaces and boundary fields

Based on the results published recently [arXiv:1803.10155], the influence of surfaces and boundary fields are calculated for the anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we apply first one and then two boundary fields along the perpendicular direction which can be homogeneous or staggered, representing open, symmetry-breaking, and the so called Brascamp-Kunz boundary conditions. Furthermore we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit.


Introduction
In the first part of this work [1], denoted as I in the following, we calculated the free energy of the two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions (BCs) in both directions for anisotropic couplings K ⊥ and K in perpendicular and parallel direction, respectively (both in units of k B T with Boltzmann constant k B ). We introduced an according scaling theory for the anisotropic case and calculated the corresponding finite-size scaling functions, as well as the surface tension induced by at least on antiperiodic BC. This second part is devoted to the influence of surfaces and boundary fields (BFs) and will pick up the thread of the surface tension again.
Twenty years after Onsager presented his famous solution of the infinitely large twodimensional Ising model [2], the theory of scaling laws got into the focus of research and with it the need to describe finite systems and surfaces as well as interfaces [3][4][5][6]. Such an interface arises in the Ising model when the order parameter is kept constant while the system transfers from the unordered to the ordered phase. Demixing transitions can be found in many systems, e. g., binary liquids like 2, 6.lutidin/water mixtures or ternary mixtures like 3-methyl-pyridine/water/heavy water [7][8][9][10], which both have a so called closed-loop phase diagram, i. e., they have an upper and a lower critical point. Other examples are colloidal suspensions immersed in such binary liquids and cell membranes [11,12]. Especially the binary and ternary systems are of special interest for the experimental measurement of the critical Casimir effect [13][14][15], especially for colloidal interactions, as the (lower) critical demixing point is near room temperature in contrast to liquid helium [16,17]. Additionally the surface preferences of the colloidal particles and the containers can be chemically tuned to be either hydrophilic or hydrophobic, which leads to a variety of experimental setups [18][19][20][21][22][23][24].
This rich behaviour can be explained theoretically as follows: Beneath its neat temperature sensitivity the Casimir force strongly depends on the BCs and the possible combinations of boundary fields (BFs) can be used to subclassify the universal behaviour further [25,26]. For the Ising universality class, Dirichlet BCs are the most simple kind of surface one can apply. On the finite lattice they can be implemented in two ways: Open boundaries, i. e., setting a line of couplings on the torus equal to zero, or applying a staggered BF to both ends of the cut, which leads to the Brascamp-Kunz BCs (BK BCs) [27]. The latter one is believed to be equivalent to the former one in the scaling limit and was subject to several studies [28,29].
The finite-size scaling functions of the two-dimensional Ising model with various BCs are usually investigated in the two contrary limits of either thin films [30,31], i. e., arbitrary temperature but restricted geometry, or due to conformal field theory (CFT) [32,33], i. e., at the critical temperature but arbitrary aspect ratio of the cylinder, see [34] for a more detailed calculation. In fact the latter one can be used to calculate the scaling functions for arbitrary geometries due to the conformal invariance [35]. Nevertheless, the connection between those two cases, i. e., for arbitrary temperature and arbitrary aspect ratio, is not so well investigated in the literature, and this work ought to fill this gap. The only exceptions are the results for the periodic torus [36] and the cylinder [37], while for open BCs in both directions this task was performed only recently [38,39].
We first recall the main results of part I: First we generalised the Kasteleyn-Fisher mapping between the two-dimensional Ising model and the problem of closest-packed dimers to the case of arbitrary couplings. Enforcing translationally invariant BCS in one direction the calculation of the partition function was reduced to a 2 × 2 transfer matrix method by two successive Schur reductions, making use of the dual couplings introduced by the self-duality of the twodimensional Ising model on the square lattice without any BF. To distinguish between periodic and antiperiodic BCs (denoted as (p) and (a), respectively) we introduced the parameters α and β for the perpendicular and parallel direction, respectively. In principle both parameters can take arbitrary values on the interval [−1, 1], where we focus on β = ±1 for (anti-)periodic BCs in parallel direction and α ∈ {−1, 0, 1} in perpendicular direction, where α = 0 accounts for open boundaries. The general scaling theory for two-dimensional systems with at least on translationally invariant direction of section I.3 will be used exhaustingly during this paper, as well as the anisotropic scaling theory. To calculate the scaling limit of the according free energy, i. e., the limit L → ∞, M → ∞ with ρ ≡ L/M = const., the hyperbolic structure of the scaling form (I.87) of the Onsager dispersion (I.68) was used together with suitable counting polynomials (I.89) to calculate the sums in terms of complex contour integrals. This paper will be structured as follows: First we introduce open BCs in perpendicular direction of the system, breaking its translational invariance. We will use this system to analyse the arise of (a) surface contributions to the scaling functions, and (b) a surface tension due to antiperiodic boundaries in parallel direction. Afterwards we will introduce a BF at one of the cylinders surfaces, following the procedure of McCoy and Wu [40] for a homogeneous field. Afterwards we will use the same procedure to emulate a staggered BF and show that it has no contribution to the scaling limit, which is a first hint towards the equivalence of open and the BK BCs. Subsequently we will return to the torus by coupling the BF to both surfaces of the cylinder to implement periodic (++) and antiperiodic (+−) symmetry-breaking BCs in perpendicular direction. Finally we will show the aforementioned equivalence between open and BK BCs. All our results are in perfect agreement with the conformal field theory (CFT) results [32,35].

The cylinder
For the cylindrical geometry with open BCs, denoted as (oo), we slightly modify the Hamiltonian of (I.1) such that there is no coupling between the rows m = 1 and m = M , with reduced couplings K ⊥ ,m and K ,m between nearest-neighbours in perpendicular and parallel direction, respectively, and Ising spin variables σ ,m ∈ {−1, +1} with periodic index in parallel direction σ ,m+M ≡ σ ,m . Again we can rewrite the partition function using the high-temperature formulation to obtain a form suitable for the Pfaffian method, where the non-singular part now reads For the singular part of the partition function we implement a procedure for our further calculations: We start with the matrix (I. 33), for which it is easy to see that, assuming β ∈ {+1, −1}, the open boundaries lead to a tridiagonal matrix as b L ≡ 0 due to the choice of α ≡ 0. On the other hand, if we want to apply a BF, we need an additional line of spins either left at = 0 or right at = L + 1, which is infinitely strong coupled. The according entry, e. g., at the right side of the system, b L+1 = 0, as we set t L+1 ≡ 0 in (I. 35). Thus all following matrices will be tridiagonal.
For a given tridiagonal matrix C m ) the determinant can be computed with a recursion formula based on the Laplace expansion, so the determinant of a matrix of the form may be calculated either by recursively following the diagonal upwards or downwards, where the latter one is more common. If we know the determinant of the according submatrices, the formulas read for the downward and the upward calculation, respectively, i. e., for C (α) m ) we delete the last row and column, while forC m ) we delete the first row and column instead. As we will assume the BC in parallel direction to be translationally invariant, i. e., β ∈ {p, a} the dependency on β is fully covered by the eigenvalues ϕ  m . As all boundary terms we will handle only appear in the entries a 1 , a L , and b L−1 , we may as well calculate the determinant of the submatrix representing the bulk without any surfaces (not even open ones) and then use the Laplace expansion to include the boundaries. The according matrix reads and its determinant may be simply calculated with the transfer matrix approach of section I.2.2, Eq. (I.39). Therefore we diagonalise the transfer matrix again with the unitary matrix and calculate the according power (cf. the dual expression (44) in [38]) to find Now we can use Eq. (4) to separate the determinant into the bulk, the surface and the finitesize contributions without any knowledge of the concrete boundaries as with p (α) (ϕ m ) and q (α) (ϕ m ) being variables representing the boundary terms, and An extended calculation, which combines both (4a) and (4b), as necessary for the (++) and (+−) BCs, is shown in detail in Appendix B. It is easy to see that the bulk contribution stems solely from the term e Lγm , as it is the only term which logarithm is explicitly linear in the length scale L (plus -of course -possible prefactors of the matrix). Additionally we can factorise anything that is not exponentially decaying with L to identify the surface contributions, namely leaving the remaining as residual finite-size contribution. Fortunately this last contribution then has the form which is remindful of the form of the scaling functions of thin films. Thus we only have to identify the boundary terms and insert them into Eq. (10) to obtain our desired results. For open boundaries the Schur reduction in Sec. I.2 follows the same steps as for the torus, but we choose α = 0 or analogously z L = 0, where both only appear as pair. As the cylindrical BCs forbid most kinds of transition circles on the oriented lattice, only one Pfaffian is needed to yield the correct partition function. This also follows naturally from the toroidal geometry; as we choose α = 0, two of the Pfaffians are equal as their only difference thus vanishes, additionally, the sign marked in Tab. I.1 between either the even or the odd part thus cancels the according Pfaffians out, leading to only the odd case for periodic and the even case for antiperiodic BCs in parallel direction. From Eq. (I.33) we conclude that the according matrix reads with the boundary term where µ ± (ϕ) = t + /t − ± cos ϕ. We can use Eq. (4a) to calculate its determinant. The relevant terms then read or equivalently by using the Onsager dispersion Eq. (I.68) Finally the determinant reads We conclude with a formula for the singular part of the partition function for β ∈ {p, a} as where the odd product gives periodic and the even product antiperiodic BCs.

Surface contribution and scaling functions
First we will focus on the periodic case and, as done before for the torus, we may split up the (singular part of the) free energy according to its geometric contributions, i. e., into the bulk contribution, which is exactly the same as in Sec. I.4, the surface contribution for open boundaries, which we will focus on now, and the residual part, The separation of the last section gives us immediately the surface free energy of the finite system and to calculate its thermodynamic limit (z, t), see (23). The dashed line marks criticality for arbitrary anisotropy κ = K ⊥ /K . For fixed anisotropy the two couplings are connected by t = (z * ) κ and the black lines mark the run of the according curve, where the isotropic case is shown in red.
we apply the Euler-Maclaurin sum formula again, which yields Note that we were not able to solve this integral at the isotropic critical point exactly, nevertheless its value is known exactly in terms of Hurwitz zeta functions [39] and this form is numerical identical to both the exact value and the original expression by McCoy and Wu [40] with arbitrary precision. Fig. 1 shows the open surface free energy density for both open boundaries in contrast to the splitting in [40] with respect to each individual boundary. The scaling function of the residual surface contribution from open boundaries can be calculated in the same manner we already used to calculate the scaling function Θ b (x ). Therefore we need to make a series expansion of the according term (see Appendix A for details) and find ln η Figure 2: Complex structure for the odd sum over ln Γ m + x / (2Γ m ) in the hyperbolic ω-plane in the unordered phase (left, x µ > 0) and the ordered phase (right, x µ < 0) together with the according contours C ≷ . Going from one to the other phase leads to a shift of one half-period of the functions along the imaginary axis, thus the contour C < = C < + + C < − + C KH of the ordered phase is basically the reverse of the unordered phase C > = C > + + C > − plus the additional keyhole contour C KH around the logarithmic branch cut at [iπ/2, 3iπ/2] (left) or at [−iπ/2, +iπ/2] (right). Additionally the contours need to evade the branch points at ω = inπ/2 with n ∈ Z. Note that there is a phase jump along the real axis and between every half-period due to the switching between the two counting kernels K ± o .
To calculate the summation of (25) we will use the hyperbolic parametrisation (I.90) again, but as there is an explicit dependency on x now in the formula we need to make a distinction between the ordered (x < 0) and the unordered (x > 0) phase.
In the unordered phase we use x = |x | and find with the integration kernels (I.89); the integrand is shown on the left in Fig. 2. On the imaginary axis there is a π-periodic, logarithmic branch cut which will be crucial to the ordered phase for both the periodic and especially the antiperiodic case. For now we deform the contour with a semicircle around the two branch points and let their radius go to zero, which gives no contribution to the free energy. Thus we can calculate the contour integral again (see I.4.2) by a simple shift in the integration variable by ±iπ/2, and find again that the upper and lower contour are identical with its real part being an odd function and its imaginary part being an even function, leaving the integral real. Again we can resubstitute b (x ). For ρ = 1 the scaling function converges to ln(2) for x → −∞, which describes the difference between the two limiting procedures of the scaling and the thermodynamic limit.
to Φ and find For the ordered Phase we use x = −|x |, which corresponds to shift of iπ in the ω-plane, thus moving the logarithmic branch cut within our integration contour, see Fig. 2. Nevertheless, because of the π-periodicity of the hyperbolic parametrization along the imaginary axes, we can reuse our result for the unordered phase, as we just change the direction of the according parts of the contour, which together with the change to negative x leaves Eq. (27) unchanged. Thus we get an additional contribution from the keyhole integral around the branch cut, see Fig. 2, Note that there is a jump in the phase along the real axis due to the switch between K + o (ω) and K − o (ω), which is beneficial for our calculation as the real part of the integrand is an odd function and its principal value with the logarithmic divergence at ω = 0 does not contribute at all. The imaginary part of the integrand can be simplified by the principal value of the complex logarithm to where we first used that the the contributions from the four quadrants are all the same and then that the counting polynomials K ± e/o (Φ) are defined as logarithmic derivative of the according characteristic polynomial P ± e/o (Φ), see (I.88) and (I.89). Again we see a contribution responsible for the effect due to the difference of the thermodynamic and the scaling limit caused by the broken symmetry of the system in the ordered phase. it gives a limiting value of ln 2 for x → −∞, which here solely stems from the keyhole integral around the logarithmic branch cut. Thus we find the scaling function of the residual contribution to the open BC free energy to be For the finite-size contribution, we proceed likewise and find for the series expansion see again Appendix A, with which we conclude with a product analogous to (I.98) and a residual strip free energy scaling function where we have already taken the square root by using only half of the eigenvalue spectrum. Combining all these three contributions, the scaling function for the open cylinder with periodic BCs thus reads and is depicted in Fig. 3 for different values of ρ as well as its counterpart for the perpendicular direction in Fig. 4.

Domain wall
The partition function of the cylindrical system with open boundaries in perpendicular direction and antiperiodic BCs in parallel direction differs from the one with periodic BCs only in the set over which the product is performed, namely the odd numbers for the latter and the even numbers for the former case. Thus we find where the additional factor z −L stems from the elimination of the square root; due to the symmetries of the product around ϕ = 0 and ϕ = π every term but the ones for ϕ = 0 and ϕ = π appears twice. Thus we have to calculate them separately and find det C (oo) In general the thermodynamic limits of the bulk and the surface contribution do not change despite the shifted summations. Thus the difference to the case of periodic BCs gives the additional energy due to the formation of a domain wall, and we can easily calculate the surface tension σ (oo,a) (L, M ) as and -just as usual -we can decompose it into the three parts according to bulk, surface, and finite-size contribution, which each fulfills a scaling relation according to with the total surface tension scaling function With the surface tension being the difference between the antiperiodic and the periodic case, the scaling functions of the further one are easily obtained as and since the surface tension decomposes into its three parts as well as the open cylinder the scheme applies here, too.
To calculate the three contributions, we use the scaling limits in combination with the hyperbolic parametrisation. The counting polynomial is simply the difference of the even and the odd one δK of (I.89c). We start with the bulk contribution, as it contains the additional boundary terms from ϕ = 0 and ϕ = π, which contains the characteristic linear divergence of the scaling function in the ordered phase. Afterwards we will take a close look to the surface contribution, as the calculation in the hyperbolic parametrisation involves another keyhole integral corresponding to a logarithmic correction to the linear divergence. The strip free energy scaling function then is easy again, as its best converging form is again an infinite product.
For the bulk contribution to the surface tension scaling function, we first notice the great similarity to the finite form of δΘ b (x ) from the calculation for the torus in section I. 4. Thus we rewrite the alternating sum as where we used the symmetry of the γ m . With the boundary values of γ, we find the correction to be Using (I.83) we can go to the scaling limit by a series expansion around M → ∞ to find with the Heaviside step function H(x) ≡ ∂ x max{0, x}, which is simply the linear diverging term we expected. Thus we can conclude for the bulk contribution with and the bulk contribution for the antiperiodic case reads b (x ) for ρ → ∞ with the linear divergence in the ordered phase for x < 0. Now we turn to the surface contribution and use (25) together with the alternating counting polynomial δK(Φ) in the hyperbolic parametrisation, but here we need to exclude any pole at ω = 0 as the sum only runs over the the positive numbers. Nevertheless, we can frankly expand the contour to the negative half-plane, which simply counts every pole twice. Because of the alternating character of the integration kernel δK the integrals are convergent, see Fig. 2, but, as for the open cylinder with periodic boundaries, we need to distinguish between the ordered and the unordered phase. For the unordered phase, there is no pole at ω = 0, as the integrand has an according zero, and thus, by the usual shift of the contour, we get where we substituted back to the Φ-plane like done before.
In the ordered phase we have to face not only the logarithmic cut with the logarithmic divergence, but due to the integration kernel δK(Φ) = csc(Φ) there is now an additional pole at the very same position. Additionally the other poles shift towards the origin for growing |x |, forming a branch cut for |x | → ∞, and in order to exclude this crude construct from the desired integral we need to make a keyhole integral again. Fortunately the integral over the rest of the contour C < is again the same as for the unordered phase because of the cancelation of the switching signs. To calculate the keyhole integral we use the series expansion of the and separate the integrations with δK(Φ) − Φ −1 and Φ −1 as integration kernels along the keyhole contour. Without the pole, the logarithmic divergence again cancels out and we find 1 4iπ where we again used the symmetry along the keyhole contour. A careful study of the remaining integral leaves us with 1 2iπ which we can combine to Hence we find for the surface tension simply without any additional divergence. Eventually we come to the strip contribution and, following our previous calculations, we use (32) to introduce Thus the scaling function for the surface tension reads For the antiperiodic strip contribution we thus find The total surface tension scaling function is shown in Figs. 6 where we allow the fields to vary depending on its position so we can dictate the BCs (α). This gives an additional term in both the singular and the non-singular part of the partition function, and following the same procedure as before, the latter one simply reads The singular part contributes as an additional factor within the sum over all spin configurations with and z (l/r) m = tanh h (l/r) m . Again we will assume homogeneous anisotropy, leaving us with the BF variables z (l/r) m ≡ z l/r ∀ m. We will start by applying only one boundary field to the cylinder and, following McCoy and Wu, we do so by adding a line of infinitely strong coupled spins either to the left or the right column. If we add a column to the left side of the system, i. e., at = 0, the according matrix reads with the boundary entries for a homogeneous boundary field z l = z h , where we have introduced the boundary terms as t = 0 corresponds to an infinitely strong coupling in parallel direction. Thus we have to apply (4a) twice and obtain η (+o) with the boundary field contribution Thus the final determinant using Eq. (10) reads det C (+o) If instead we add an additional column to the right side of the system, i. e., at = L + 1, we find the matrix with z r = z h and the according entries Here we have to apply both recursion formulas (4a) and (4b) and thus the calculation is a little bit nasty as it includes some additional symmetry considerations; it is shown in detail in Appendix B. The final result then reads, of course, because of the symmetry of the system with respect to reversing the perpendicular direction. Finally the partition function reads Note that we only assume periodic BCs in the parallel direction, since the BF and the antiperiodic BCs in combination need a more precise survey.

Surface field contribution and scaling limit
The previous calculation shows that the additional surface field can be easily separated as a correction to the open surfaces. Additionally, we obtain a factor µ − 0 (ϕ), which corresponds directly to the additional line of infinitely-strong coupled spins. We may calculate the according contribution to the partition function for the periodic case exactly as The field contribution to the free energy in the thermodynamic limit is again given by the integral over the corresponding contribution η and is depicted in Fig. 10. Its critical value for the isotropic case and infinitely strong field can be calculated exactly in terms of Catalan's constant G and the critical coupling z c , and reads For the scaling function it is necessary to consider the case of an infinitely strong surface field, i. e., z h = 1, otherwise its effect would vanish in the desired limit, and more important, we could not use our likewise simple methods. An according scaling theory for the strip geometry, i. e., M → ∞ for varying BFs was given in [31]. Such a field breaks the Z 2 -symmetry of the system and the scaling function goes to zero in both the unordered and the ordered phase in contrast to the scaling functions for the fully periodic torus or the periodic open cylinder. Indeed this reflects the duality of the two present BCs; under a duality transformation, an open boundary transforms into a symmetry-breaking one and vice versa, while additionally the high-and the low-temperature phase interchange. Nevertheless, this symmetry is only exact for at least in one direction infinitely large systems, i. e., the thin film limit, as otherwise the corrections do not vanish. The next two blocks we need to calculate in the scaling limit thus are the two η (h) ± (z h =1; ϕ). We start with a series expansion of η (h) − (z h =1; ϕ) at criticality and find not only a logarithmic divergence in M but also a constant term which still contains the anisotropy variable Figure 11: Complex structure for the odd sum over the ln (2Γ m )−ln Φ 2 m /2 in the hyperbolic ω-plane together with the contour C = C + + C − + C KH . Since this is the correction of the surface field to the open boundaries, the some kind of regular part from the contours C ± is zero and only the non-trivial keyhole contour C KH contributes. Note that there is no explicit dependency on x and thus we do not need to distinguish between the ordered an unordered phase.
with the inverse critical coupling We can use the critical value to regularise its non-critical counterpart, as all the diverging and non-universal contributions cancel each other out. Then we need to calculate the sum over (73) in a regularised form; therefor we can use the zeta-regularisation. First we notice that the summation over the odd numbers is symmetric and thus we can write by assuming Φ > 0. The zeta-regularised sum over the odd numbers to the power of (−s) Figure 12: Scaling functions Θ (+o,p) (x , ρ) of the free energy for different values for aspect ratios ρ ≥ 1. For ρ = ∞ there is a kink at x µ = 0 present and, in contrast to the periodic systems we discussed before, the scaling function converges to zero for all aspect ratios in both the ordered and the unordered phase due to the unambiguously broken Z 2 -symmetry.
with the Riemann zeta function ζ(s). For the constant part we take s → 0 and find that is, these terms do not contribute to the scaling limit, as they cancel out with the thermodynamic limit. The logarithmic term can be calculated by a derivative of (76) as Now we can regularise the temperature depending form of η Since we already calculated the sum over ln [(Γ m + x )/(2Γ m )] in section 2 for the open cylinder, we decompose (79) into Figure 13: Scaling functions Θ (+o,p) ⊥ (x ⊥ , ρ) for different values for the aspect ratio ρ ≤ 1. The scaling function converges rather fast agains its limiting case Eq. (88) and changes its sign for ρ ≈ 1 depending on the temperature. and thus we only have to calculate the sum over the first two summands. The terms are independent of whether we focus on the ordered or the unordered phase and thus we only have to do the calculation once, which is especially interesting as the integrals over the shifted contours C + and C − vanish and only the contribution of the keyhole contour C KH around the logarithmic cut is relevant, see Fig. 11. The logarithmic divergence cancels out again due to the symmetries around the origin and we find 1 4iπ which leaves us with for the surface field scaling function. For the strip residual free energy, we only need to calculate the fraction which regularises itself and thus we can write down the according product where the contributions from the open boundaries and the surface field contribute as the inverse of each other and thus cancel out. The strip scaling function then is, analogous to (33), simply and the complete scaling function reads and is depicted in Figs. 12 and 13. We can see that for systems with sufficiently small ρ there is a kink at x µ = 0 and, depending on the temperature and the aspect ratio, the scaling function changes it sign. Subsequently we can identify the two limiting cases for ρ = 0 and ρ = ∞; the latter one is again dominated by the periodicity and thus we find while the former one is approached rather fast, almost already for ρ = 1/2 and reads Θ (+o) For growing aspect ratio the scaling function changes its sign from an all negative to an all positive function with a temperature-dependent change for ρ ≈ 1.

Staggered fields
Instead of a homogeneous field, we may impose a staggered field, which is believed to be equivalent to Dirichlet boundaries in the thermodynamic limit. Therefore we couple the spins in the additional line again infinitely strong, but this time with the opposite sign, i. e., z L+1 = − 1. This changes every µ ± 0 (ϕ) → −µ ∓ 0 (ϕ) in (60), and thus we have to deal with the matrix with the boundary entries Thus we can calculated the contribution for the staggered surface field as . (92) for a staggered field. Additionally we find the contribution of the infinitely strong coupled line of spins to be which is equal to the one for a homogeneous field, as it gives the correction to the surface and a factor of two due to the Z 2 symmetry of this macro spin. The according free energy contribution of a staggered surface field is given by Figure 14: Surface free energy density of a staggered field f st (z st ; z, t), see (94). The dashed line marks criticality for arbitrary anisotropy κ = K ⊥ /K . For fixed anisotropy the two couplings are connected by t = (z * ) κ and the black lines mark the run of the according curve, where the isotropic case is shown in red. Note that we have chosen z st = √ 3/2 as it has a maximal value of ln (2). For z st =1 the density diverges logarithmically for t → 1.
For the scaling function we choose z st =1 and find the scaling form of η (st) Thus we see that both finite-size scaling function of the system with open BCs as well as the one with an infinitely-strong staggered field are the same, Numerically, i. e., for the finite systems, we see that they converge against the same scaling function with growing system size from opposite directions. We will later resume to this point, when we examine the BK BCs in section 5.

Symmetry-breaking BCs
For symmetry-breaking boundary conditions we follow the same procedure as for one boundary field, but we couple the additional infinitely-strong coupled line to both the first and the last row, introducing the two boundary fields z l and z r and again forming a torus. It is easy to see that a b k on a boundary is either 0 if a perpendicular coupling is cut open, or ±1 if a row in parallel direction is coupled infinitely strong either ferro-or anti-ferromagnetic. The difference between the (++)-and the (+−) boundary conditions is solely yielded by the choice between periodic and antiperiodic continuity in the perpendicular direction, respectively. This makes it necessary to switch back to the formulation of the torus with the sum over four Pfaffians. The final matrix reads A detailed calculation of the determinant can be found in Appendix B, however, the final determinant reads det C (++) Following the procedure we used for the torus, we find the singular part of the partition function to be as due to the infinitely strong coupled line of spins each pair of Pfaffians is identical and we only have an even and an odd contribution left.

The alternating contribution
For the scaling function we first set z l = z r = z h and then factorise the odd contribution from the partition function like we did for the torus, see I.4, leaving us with the odd part, for which we already know all scaling functions but an alternating part from the BF contributions, which we will discuss now. Therefore we will proceed in the same manner as we did for the surface tension of the antiperiodic open cylinder; we take the square root by using the symmetry of the eigenvalues γ m as every factor but the two symmetry points at ϕ = 0 and ϕ = π appear exactly twice. Its values at those points are For convenience we change to the logarithm and find for the alternating sum so we just have to handle sums instead of products. We start with the bulk contribution, which we identify as all terms linear in L as where we reformulated the sum in such a way that we can identify the scaling limit we have already calculated for the torus. The additional term is slightly different to Eq. (42), nevertheless its scaling form is given by that is, the linear divergence is in contrast to Eq. (45) in the unordered phase.
For the surface contribution we find where we used To calculate the scaling limit we use the same fruitful approach as for one surface field, i. e., the zeta regularisation for the critical value, but this time we need to do it for an alternating sum. We use Eq. (73) and and write for the scaling form of the critical case The ln(M ) is just a constant as we handle it as independent term from some kind of cutoff, and additionally we have The zeta regularisation of the alternating sum reads and thus we find for the constant terms with s = 0 For the Φ-dependent term we need to calculate the derivative with respect to s again and afterwards let s → 0, thus the alternating sum reads in zeta regularised form Combining all these terms leads to as all other terms cancel out, i. e., the additional terms for ϕ = 0 and ϕ = π together correspond in the scaling limit of the critical value of the boundary contribution to this alternating sum. The scaling function converges rather fast agains its limiting case Eq. (118) and its minimum is minimal for ρ ≈ 2.
Fortunately we already calculated the contribution for the alternating sum over ln[(Γ m + x )/(2Γ m ) for the surface tension in section 2. Therefore we are left with the alternating sum over ln(2Γ m /|Φ m |) = ln(2Γ m ) − ln(Φ 2 m )/2 analogous to the calculation for one boundary field, which we will handle by the hyperbolic parametrisation and the corresponding contour integration. The complex structure is shown in Fig. 15. As before, the integration along C ± vanish, as their real part is an odd function and their imaginary part is zero and we are left with the keyhole contour integration around the logarithmic cut from −iπ/2 to iπ/2. We can proceed in the same manner as we did for the antiperiodic open cylinder in section 2 and split the integration kernel δK into its simple pole at ω = 0 and the rest. A careful study of the two integrals gives the final result 1 4iπ and in the style of the alternating contributions we saw before, we thus end up with as term from the surface field. For the strip contribution of symmetry-breaking BF, denoted as (sb), we simply find

Scaling functions
Combining the results of the last section, we find the final form of the scaling function for both the symmetric and the antisymmetric symmetry-breaking BCs to be which is remarkably remindful of the formulas obtain by CFT [35,41,42] and agrees with the results of Evans and Stecki for the strip geometry [30]. Let us first take a closer look at the symmetric symmetry-breaking BCs, i. e., (++) boundaries. As we have seen it before for the several other systems, the scaling function converges against Θ (p) b (x ) for ρ → ∞, but as it approaches ρ ≈ 2 the behaviour changes dramatically; the minimum shifts towards the unordered phase, and grows between ρ ≈ 2 and ρ ≈ 1 again to converge against the scaling function of the (++)-film, see Figs. 16 and 17. As this limit is already approached at ρ ≈ 1, consequently the systems behaviour does not change for 0 ≤ ρ ≤ 1, as shown in Fig. 18, where only Θ (++,p) ⊥ (x ⊥ ) for ρ = 1 and ρ = 1/2, and the thin-film-limit Θ (++) ⊥ (x ⊥ ) are shown. Because of the polarisation effect of the surfaces, the scaling functions go to zero for |x | → ∞, as the Z 2 -symmetry of the system is broken and no spontaneous flip of the magnetisation is possible.
For the (+−) BCs the behaviour is very similar to any other system that forms a domain wall along one direction, the only difference here being that it is imposed by boundary fields and not an antiperiodic boundary. Nevertheless there is a linear divergence in the ordered phase due to the aforementioned domain wall, but additionally the surface field give a logarithmic correction. Just as for the (++) boundaries, the scaling function approaches its thin-film-limit Θ (+−) ⊥ almost already for ρ ≈ 1, see Fig. 20, while in the contrary limit the system is dominated by Θ (p) b (x ) again, that is the periodicity in the parallel direction as shown in Fig. 19.

Brascamp-Kunz BCs
Finally we impose the so called Brascamp-Kunz boundary conditions to our system by applying staggered boundary fields to the upper and lower boundary of the system. They are believed to be equivalent to Dirichlet boundary conditions in the scaling limit.
We implement them by an infinitely-strong negative coupling in the according additional line and a construction analogous to the (++) and (+−) BCs. The negative coupling within this additional line is equivalent to exchanging with the boundary entries The determinant can thus be calculated with the method of Appendix B and reads det C and as due to the infinitely-strong coupled line of spins the matrices for periodic and antiperiodic BCs in perpendicular direction do not differ, only two determinants are left for the non-regular part of the partition function. Thus the form is remindful of the (++) and (+−) BCs and reads where the ± denotes the case of an aligned (+) or an opposite (−) surface field in each row.
To show that the Brascamp-Kunz BCs are equivalent to open boundaries in the scaling limit, we proceed as we did for the symmetry-breaking BCs in the last section. First we factories the odd contribution, which is, analogously to the system with only one staggered surface field, equal to the scaling function of the open cylinder. Thus we have to show that the remaining terms vanish in the scaling limit, i. e. the alternating contribution is at least exponentially suppressed with growing system size. We do so by a closer look at the bulk contribution, which reads The terms that differ in the ordered and the unordered phase have the same scaling form L ln(zt) −2L ln a ξ , 2L ln t −2L ln a ξ (125b) due to the continuity of the free energy at this point. Nevertheless, the leading contribution diverges linear in L, which gives an exponential decay in the according contribution to the free energy scaling function, ln 1 ± e L ln t+ln(2t − z 2 st /z 2 )+ M −1 and thus the free energy of the a system with Brascamp-Kunz BCs converges either from above or below against the scaling function for the open cylinder, i. e., with δ ∈ {⊥, }, depending on whether the staggered fields are shifted against each other or not.

Conclusions
We presented a systematic calculation of the free energy scaling functions for various anisotropic 2d Ising systems. Therefor we started with the scaling functions for the periodic, open cylinder. Then we turned to the antiperiodic Cylinder and calculated the surface tension of the induced domain wall within our formalism. Afterwards we showed how a boundary field can be introduced within the dimer approach and calculated the according contributions in both the thermodynamic limit as well as the scaling limit for homogenous and staggered surface fields. Using the results from the previous calculations for the homogenous boundary field together with the schemes for periodic and antiperiodic boundary conditions on the torus, we introduced the aspect-ratio-depended form of the free energy scaling function for (++) and (+−) boundary conditions. Finally we translated those results to the case of staggered surface fields, the Brascamp-Kunz boundary conditions, and showed that their scaling limit is indeed equivalent to the one for open boundaries. During those calculations we saw how the two-dimensional Ising model decays into different blocks and thus that the scaling functions decay further as previously assumed. This behaviour should indeed translate to more complex systems, e. g., higher spacial dimensions, more complex geometric constrains, and other universality classes. Especially the question if the conformal invariance at criticality can be expanded towards or at least used as approximation of the scaling limit can now be tangled by a comparison with colloidal suspensions.

Homogeneous boundary fields
For the homogeneous BF the only new term is with which the according BF contribution can be expanded to with the inverse critical coupling a ξ ≡ r ξ 1 + r 2 We now face the problem, that the η with the series expansion ln η For the residual strip contribution we thus conclude with η (h)

B Recursive determinant simplifications
Some calculations necessary for the determinants are neither especially complicated nor particularly enlightening, but nevertheless confusing and deserve a detailed presentation. These calculations are shown here in the following appendices. Our starting point is the determinant det C