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:2104.00923 · Journal of Statistical Mechanics Theory and Experiment

Iterated star–triangle transformation on inhomogeneous 2D Ising lattices

H. J. Hilhorst

Published Jun 1, 2021• 1 citation
arXivHTMLPDF

Abstract

Abstract We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings K i ( r ⃗ ) , where r ⃗ and i indicate the bond position and orientation, respectively. Iterative application of the star–triangle transformation to an initial lattice T ( 0 ) with a set of couplings { K i ( 0 ) ( r ⃗ ) } generates a sequence of lattices T ( n ) , for n = 1, 2, …, with couplings { K i ( n ) ( r ⃗ ) } . When T ( 0 ) includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables 1 / sinh 2 K i ( n

DOI
10.1088/1742-5468/ac06bf

Related papers

Boundary magnetization and spin correlations in inhomogeneous two-dimensional Ising systems

Theodore W. Burkhardt, Ihnsouk Guim, H. J. Hilhorst +1 more

We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance $m$ from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large $m$ is examined in detail. For $y<1$ the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For $y=1$, $A>{A}_{c}$ (${A}_{c}$ being a positive critical value), and for $y<1$, $A>0$, there is a spontaneous surface magnetization at the bulk critical temperature.

Physical review. B, Condensed matter•Aug 1, 1984•38 citations

Transformations of Ising Models

Michael Ellis Fisher

The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than $S=\frac{1}{2}$. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.

Physical Review•Feb 15, 1959•401 citations

Nonuniversal and Anomalous Decay of Boundary Spin Correlations in Inhomogeneous Ising Systems

H. J. Hilhorst, J. M. J. van Leeuwen

A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $\ensuremath{-}\frac{A}{{m}^{y}}$ with the distance $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g(\mathcal{r})$ is studied for critical bulk coupling. For $y=1$ the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on $A$. For $0<y<1$ anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of $A$ and $y$ are given.

Physical Review Letters•Oct 26, 1981•80 citations

Phase diagrams of two inhomogeneous Ising models on the triangular lattice

Andrzej Pękalski

Phase diagrams of two inhomogeneous annealed Ising models on the triangular lattice are constructed in an exact way. The interimpurity exchange constants (bonds), of the same but arbitrary value and sign, can be located along the horizontal edges of the triangles. In the first model the remaining non-random couplings are ferromagnetic and in the second they are antiferromagnetic. The dependence of the correlation functions and magnetisation on the temperature is calculated and discussed. It is shown that, for certain values of the impurity bond couplings and a range of their concentrations, there is no net magnetic moment at T=0 and low temperatures. The magnetisation appears at some finite temperature and then vanishes at a higher one. This increase of the magnetisation with the temperature is connected with the role of entropy in determining the minimum of the free energy.

Journal of Physics A Mathematical and General•Jan 7, 1989•2 citations

Boundary critical behaviour of two-dimensional random Ising models

Ferenc Iglói, Péter Lajkó, Walter Selke +1 more

Using Monte Carlo techniques and a star-triangle transformation, Ising models with random, `strong' and `weak', nearest-neighbour ferromagnetic couplings on a square lattice with a (1,1) surface are studied near the phase transition. Both surface and bulk critical properties are investigated. In particular, the critical exponents of the surface magnetization, , of the correlation length, , and of the critical surface correlations, , are analysed.

arXiv:cond-mat/9711182•Journal of Physics A Mathematical and General•Mar 27, 1998•8 citations

Semi-infinite two-dimensional Ising model with marginally inhomogeneous couplings and conformal invariance

T W Burhardt, Ferenc Iglói

The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al -2 , where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al -1 inhomogeneity transforms into A((L/ pi )sin( pi L/L)) -1 . For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.

Journal of Physics A Mathematical and General•Jun 21, 1990•24 citations
Abstract We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings K i ( r ⃗ ) , where r ⃗ and i indicate the bond position and orientation, respectively. Iterative application of the star–triangle transformation to an initial lattice T ( 0 ) with a set of couplings { K i ( 0 ) ( r ⃗ ) } generates a sequence of lattices T ( n ) , for n = 1, 2, …, with couplings { K i ( n ) ( r ⃗ ) } . When T ( 0 ) includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables 1 / sinh 2 K i ( n
We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance mmm from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large mmm is examined in detail. For y<1y<1y<1 the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For y=1y=1y=1, A>AcA>{A}_{c}A>Ac​ (Ac{A}_{c}Ac​ being a positive critical value), and for y<1y<1y<1, A>0A>0A>0, there is a spontaneous surface magnetization at the bulk critical temperature.
The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than S=12S=\frac{1}{2}S=21​. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.
A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $\ensuremath{-}\frac{A}{{m}^{y}}$ with the distance mmm from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation g(r)g(\mathcal{r})g(r) is studied for critical bulk coupling. For y=1y=1y=1 the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on AAA. For 0<y<10<y<10<y<1 anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of AAA and yyy are given.
Phase diagrams of two inhomogeneous annealed Ising models on the triangular lattice are constructed in an exact way. The interimpurity exchange constants (bonds), of the same but arbitrary value and sign, can be located along the horizontal edges of the triangles. In the first model the remaining non-random couplings are ferromagnetic and in the second they are antiferromagnetic. The dependence of the correlation functions and magnetisation on the temperature is calculated and discussed. It is shown that, for certain values of the impurity bond couplings and a range of their concentrations, there is no net magnetic moment at T=0 and low temperatures. The magnetisation appears at some finite temperature and then vanishes at a higher one. This increase of the magnetisation with the temperature is connected with the role of entropy in determining the minimum of the free energy.
The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al -2 , where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al -1 inhomogeneity transforms into A((L/ pi )sin( pi L/L)) -1 . For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.