Abstract
Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. 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. Finally, we extend our results to the antiferromagnetic Ising model.
Highlights
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
We presented a systematic calculation of the universal free energy scaling functions for various anisotropic 2d Ising systems
Afterwards we showed how a boundary field can be introduced within the dimer approach and calculated the corresponding contributions in both the thermodynamic limit as well as the scaling limit for homogenous and staggered surface fields
Summary
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 kBT with Boltzmann constant kB). 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 reads ρ Θ(oo,p)(x , ρ) = ρ Θb(p)(x ) + Θs(oo,p)(x ) + Ψ(oo,p)(x , ρ). The terms are independent of whether we focus on the ordered or the unordered phase and 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 CKH around the logarithmic cut is relevant, see Fig. 11.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
