arXiv subjects

Hendrik Hobrecht

Publications and source records attributed to Hendrik Hobrecht.

5 recordsLinked to original sources

arXiv2019

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

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.

cond-mat.stat-mech↗

arXiv2019

Anisotropic scaling of the two-dimensional Ising model I: the torus

We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and - if present - the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.

cond-mat.stat-mech↗

arXiv2017

Many-body critical Casimir interactions in colloidal suspensions

We study the fluctuation-induced Casimir interactions in colloidal suspensions, especially between colloids immersed in a binary liquid close to its critical demixing point. To simulate these systems, we present a highly efficient cluster Monte Carlo algorithm based on geometric symmetries of the Hamiltonian. Utilizing the principle of universality, the medium is represented by an Ising system while the colloids are areas of spins with fixed orientation. Our results for the Casimir interaction potential between two particles at the critical point in two dimensions perfectly agree with the exact predictions. However, we find that in finite systems the behavior strongly depends on whether the $Z_{2}$ symmetry of the system is broken by the particles. Eventually we present Monte Carlo results for the three-body Casimir interaction potential and take a close look onto the case of one particle in the vicinity of two adjacent particles, which can be calculated from the two-particle interaction by a conformal mapping. These results emphasize the failure of the common decomposition approach for many-particle critical Casimir interactions.

cond-mat.soft↗

arXiv2016

Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios

We present a systematic method to calculate the universal scaling functions for the critical Casimir force and the according potential of the two-dimensional Ising model with various boundary conditions. Therefore we start with the dimer representation of the corresponding partition function $Z$ on an $L\times M$ square lattice, wrapped around a torus with aspect ratio $ρ=L/M$. By assuming periodic boundary conditions and translational invariance in at least one direction, we systematically reduce the problem to a $2\times2$ transfer matrix representation. For the torus we first reproduce the results by Kaufman and then give a detailed calculation of the scaling functions. Afterwards we present the calculation for the cylinder with open boundary conditions. All scaling functions are given in form of combinations of infinite products and integrals. Our results reproduce the known scaling functions in the limit of thin films $ρ\to 0$. Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.

cond-mat.stat-mech↗

arXiv2014

Direct simulation of critical Casimir forces

We present a new Monte Carlo method to calculate Casimir forces acting on objects in a near-critical fluid, considering the two basic cases of a wall and a sphere embedded in a two-dimensional Ising medium. During the simulation, the objects are moved through the system with appropriate statistical weights, and consequently are attracted or repelled from the system boundaries depending on the boundary conditions. The distribution function of the object position is utilized to obtain the residual free energy, or Casimir potential, of the configuration as well as the corresponding Casimir force. The results are in perfect agreement with known exact results. The method can easily be generalized to more complicated geometries, to higher dimensions, and also to colloidal suspensions with many particles.

cond-mat.stat-mech↗