Density profiles are investigated arising in a critical Ising model in two dimensions which is confined to a rectangular domain with uniform or mixed boundary conditions and arbitrary aspect ratio. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions - or with free-spin boundary conditions f, exact results are presented for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The new results follow by means of conformal transformations from known results in the half plane with +-+-+ and +f+f+ boundary conditions. The corners with mixed boundary conditions lead to interesting behavior, even in the limit of a half-infinite strip. The behavior near these corners can be described by a "corner-operator-expansion," which is discussed in the second part of the paper. The analytic predictions agree very well with simulations, with no adjustable parameters.
Density profiles are investigated arising in a critical Ising model in two dimensions which is confined to a rectangular domain with uniform or mixed boundary conditions and arbitrary aspect ratio. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions - or with free-spin boundary conditions f, exact results are presented for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The new results follow by means of conformal transformations from known results in the half plane with +-+-+ and +f+f+ boundary conditions. The corners with mixed boundary conditions lead to interesting behavior, even in the limit of a half-infinite strip. The behavior near these corners can be described by a "corner-operator-expansion," which is discussed in the second part of the paper. The analytic predictions agree very well with simulations, with no adjustable parameters.
Density profiles are investigated arising in a critical Ising model in two dimensions which is confined to a rectangular domain with uniform or mixed boundary conditions and arbitrary aspect ratio. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions − or with free-spin boundary conditions f, exact results are presented for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The new results follow by means of conformal transformations from known results in the half plane with +−+−+ and +f+f+ boundary conditions. The corners with mixed boundary conditions lead to interesting behavior, even in the limit of a half-infinite strip. The behavior near these corners can be described by a ``Corner-Operator-Expansion'', which is discussed in the second part of the paper. The analytic predictions agree very well with simulations, with no adjustable parameters.
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.
In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles 〈σ(r_{1})〉 and 〈ε(r_{2})〉, respectively. Does increasing the order at r_{1} always decrease the disorder at r_{2}? Does increasing the disorder at r_{2} always decrease the order at r_{1}? The answer to these questions is contained in the cumulant response function 〈σ(r_{1})ε(r_{2})〉^{(cum)}. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of σ(r_{1})ε(r_{2}) and exact results for the Ising model, we analyze 〈σ(r_{1})ε(r_{2})〉^{(cum)} in two-dimensional critical systems defined on the x-y plane with mixed + and - boundary conditions. Particularly interesting behavior is found when either of the operators σ or ε is located on a "zero line" in the x-y plane, along which 〈σ(r)〉 vanishes. Results for half-plane, triangular, and rectangular geometries are presented.
Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\ifmmode\times\else\texttimes\fi{}N$ with free boundaries. For five aspect ratios, $\ensuremath{\rho}=M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\ifmmode\times\else\texttimes\fi{}N=64\ifmmode\times\else\texttimes\fi{}{10}^{6}$. The accuracy of the free energy reaches ${10}^{\ensuremath{-}26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by ${f}_{\ensuremath{\infty}}=0.92969539834161021499(1)$, compared with Onsager's exact result ${f}_{\ensuremath{\infty}}=0.929695398341610214985\ensuremath{\cdots}.$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, compared with the CFT result c=0.5. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.
Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power $\frac{1}{2}\ensuremath{\eta}.$ The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of $\ensuremath{\eta}.$ We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.