Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2409.01434

The anisotropic square lattice Ising model with quenched surface disorder

Luca Cervellera, Oliver Oing, Jan Büddefeld, Alexander Hahn, Alfred Hucht

Published Sep 2, 2024• 1 citation
arXivHTMLPDF

Abstract

Using exact enumeration, the surface free energy as well as the critical Casimir amplitude and Casimir force are calculated for the square lattice Ising model with anisotropic couplings and quenched surface disorder on one surface in cylinder geometry at criticality. The system shape is characterized by the aspect ratio $L/M$, where the cylinder length $L$ can take arbitrary values, including $L\to\infty$, while the circumference $M$ is varied from $M=4$ to $M=54$, resulting in up to $2^{54}$ numerically exact free energy calculations. A careful $M\to\infty$ extrapolation shows that quenched surface disorder is irrelevant in two dimensions, but gives rise to logarithmic corrections.

DOI
10.48550/arxiv.2409.01434

Related papers

The square lattice Ising model on the rectangle II: finite-size scaling limit

Alfred Hucht

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.

arXiv:1701.08722•Journal of Physics A Mathematical and Theoretical•Apr 5, 2017•21 citations

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

Hendrik Hobrecht, Alfred Hucht

Abstract 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 × 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 × 2 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 → 0 . Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.

arXiv:1611.05622•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2017•15 citations

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

Hendrik Hobrecht, Alfred Hucht

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.

arXiv:1805.00369•SciPost Physics•Mar 2, 2020•8 citations

The boundary disorder correlation for the Ising model on a cylinder

Rafael L. Greenblatt

I give an expression for the correlation function of disorder insertions on the edges of the critical Ising model on a cylinder as a function of the aspect ratio (rescaled in the case of anisotropic couplings). This is obtained from an expression for the finite size scaling term in the free energy on a cylinder in periodic and antiperiodic boundary conditions in terms of Jacobi theta functions.

arXiv:2407.03100•SciPost Physics Core•Feb 10, 2025•1 citation

Extrapolated renormalization-group calculation of the surface tension in square-lattice Ising model

Evaldo M. F. Curado, Constantino Tsallis, Silvio V. F. Levy +1 more

By using self-dual clusters (whose sizes are characterized by the numbers $b=2,3,4,5$), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature ${T}_{c}$ is recovered for any value of $b$; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.

Physical review. B, Condensed matter•Feb 1, 1981•19 citations

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

Hendrik Hobrecht, Alfred Hucht

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.

arXiv:1803.10155•SciPost Physics•Sep 2, 2019•14 citations
Using exact enumeration, the surface free energy as well as the critical Casimir amplitude and Casimir force are calculated for the square lattice Ising model with anisotropic couplings and quenched surface disorder on one surface in cylinder geometry at criticality. The system shape is characterized by the aspect ratio L/ML/ML/M, where the cylinder length LLL can take arbitrary values, including L→∞L\to\inftyL→∞, while the circumference MMM is varied from M=4M=4M=4 to M=54M=54M=54, resulting in up to 2542^{54}254 numerically exact free energy calculations. A careful M→∞M\to\inftyM→∞ extrapolation shows that quenched surface disorder is irrelevant in two dimensions, but gives rise to logarithmic corrections.
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.
Abstract 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 × 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 × 2 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 → 0 . Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.
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.
By using self-dual clusters (whose sizes are characterized by the numbers b=2,3,4,5b=2,3,4,5b=2,3,4,5), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature Tc{T}_{c}Tc​ is recovered for any value of bbb; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.
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.