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

Phase Transitions

Renormalization on the triangular lattice with inhomogeneous interaction

A. Drzewiński, J. Szṅajd

Published Jun 1, 1996• 2 citations
Publisher

Abstract

Using a quantum version of the Niemeijer-van Leeuwen scheme we perform a real-space renormalization-group analysis for the Ising and XY models on a triangular lattice with inhomogeneous interactions. Due to the presence of different interactions we are able to consider as well the Kagomé lattice as a decorated triangular lattice in the limited cases. It is found that in this way we can investigate the flow diagrams and some thermodynamic properties of the models. For the Ising model the peaks in the specific heat are always connected with the presence of critical lines and are observed at the critical temperatures, while for the XY model their existence is not yet settled. For both models our results confirm universality with respect to inhomogeneous interactions.

DOI
10.1080/01411599608214638

Related papers

Critical Phenomena in the Two-Dimensional XY Model Using a Real-Space Renormalization Group

Douglas M. Lublin

The phase transition of the classical $\mathrm{XY}$ model on a triangular lattice is studied with a renormalization-group technique based on the cumulant-expansion method of Niemeijer and van Leeuwen. By utilization of some approximations, a nontrivial fixed point is located, and this yields the following critical values: $\frac{J}{k{T}_{c}}=0.25$, $\ensuremath{\nu}=1.35$, and $\ensuremath{\eta}=0.21$.

Physical Review Letters•Mar 10, 1975•31 citations

Renormalization-group study of the ferromagnetic Ising model on the triangular lattice

Chris Unger

The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature ${T}_{c}$ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for $T<{T}_{c}$. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.

Physical review. B, Condensed matter•Aug 1, 1984•0 citations

Application of renormalization group method to s=1/2xy model on triangular lattice

Jos Rogiers, Donald D. Betts

The two-dimensional spin one half XY model may undergo an unusual type of phase transition. A formulation of the renormalization group approach of Niemeijer and van Leeuwen suitable for quantum-mechanical systems is presented. The method is applied to the spin one half XY model on the triangular lattice using a second order cumulant expansion based on division of the triangular lattice into seven spin cells. The resulting free energy curve is in excellent agreement with that obtained from the high temperature series expansion. An unstable fixed point is found but the corresponding critical temperature is not in good agreement with the series expansion value. Pathological values are found for the critical exponents.

•Jan 1, 1976•0 citations

A Renormalization Group Study of the Ising Model on a Triangular Lattice with Long-Range Interactions

Sze-Shiang Feng, Qiu Xijun

The critical exponents of the triangular lattice Ising model with long-range interactions are calculated by the real space renormalization group. Using the simplest Kadanoff blocks and the lowest approximation of cumulant expansion, it is shown that there exists a finite critical temperature when .

Communications in Theoretical Physics•Jul 30, 1998•0 citations

Critical interactions for the triangular spin- s Ising model by a spin-restructuring transformation

A. Nihat Berker

A transformation is constructed which maps the spin-$s$ Ising model onto a spin-\textonehalf{} Ising model with furtherneighbor and many-site interactions. Renormalization-group methods may thus be applied directly to the spin-$s$ problem by prefacing the usual sequence of position-space rescaling transformations with this spin-restructuring transformation. We demonstrate the method by calculating the critical interactions (i.e., the inverse critical temperatures) of the spin-$s$ Ising model on the triangular lattice, using the recent results of Niemeijer and van Leeuwen for the critical subspace of the corresponding spin-\textonehalf{} Ising model. Our evaluations agree very well with recent series data.

Physical review. B, Solid state•Oct 1, 1975•29 citations

Spin correlations in the two-dimensional quantums=1/2XYmodel

J. Szṅajd

A quantum version of the Niemeijer--van Leeuwen real-space renormalization-group method is used to study the temperature dependence of the two- and four-spin correlations in the quantum XY model on the triangular lattice. The first-order cumulant expansion results suggest, similarly to other methods, a low-temperature phase of an essentially different kind from that predicted for the classical model. The possible explanation of the origin of the spurious 2D Heisenberg-like nontrivial fixed point in some renormalization-group calculations is also proposed.

Physical review. B, Condensed matter•Aug 1, 1995•3 citations
Using a quantum version of the Niemeijer-van Leeuwen scheme we perform a real-space renormalization-group analysis for the Ising and XY models on a triangular lattice with inhomogeneous interactions. Due to the presence of different interactions we are able to consider as well the Kagomé lattice as a decorated triangular lattice in the limited cases. It is found that in this way we can investigate the flow diagrams and some thermodynamic properties of the models. For the Ising model the peaks in the specific heat are always connected with the presence of critical lines and are observed at the critical temperatures, while for the XY model their existence is not yet settled. For both models our results confirm universality with respect to inhomogeneous interactions.
The phase transition of the classical XY\mathrm{XY}XY model on a triangular lattice is studied with a renormalization-group technique based on the cumulant-expansion method of Niemeijer and van Leeuwen. By utilization of some approximations, a nontrivial fixed point is located, and this yields the following critical values: JkTc=0.25\frac{J}{k{T}_{c}}=0.25kTc​J​=0.25, $\ensuremath{\nu}=1.35$, and $\ensuremath{\eta}=0.21$.
The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature Tc{T}_{c}Tc​ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for T<TcT<{T}_{c}T<Tc​. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.
The two-dimensional spin one half XY model may undergo an unusual type of phase transition. A formulation of the renormalization group approach of Niemeijer and van Leeuwen suitable for quantum-mechanical systems is presented. The method is applied to the spin one half XY model on the triangular lattice using a second order cumulant expansion based on division of the triangular lattice into seven spin cells. The resulting free energy curve is in excellent agreement with that obtained from the high temperature series expansion. An unstable fixed point is found but the corresponding critical temperature is not in good agreement with the series expansion value. Pathological values are found for the critical exponents.
A transformation is constructed which maps the spin-sss Ising model onto a spin-\textonehalf{} Ising model with furtherneighbor and many-site interactions. Renormalization-group methods may thus be applied directly to the spin-sss problem by prefacing the usual sequence of position-space rescaling transformations with this spin-restructuring transformation. We demonstrate the method by calculating the critical interactions (i.e., the inverse critical temperatures) of the spin-sss Ising model on the triangular lattice, using the recent results of Niemeijer and van Leeuwen for the critical subspace of the corresponding spin-\textonehalf{} Ising model. Our evaluations agree very well with recent series data.
A quantum version of the Niemeijer--van Leeuwen real-space renormalization-group method is used to study the temperature dependence of the two- and four-spin correlations in the quantum XY model on the triangular lattice. The first-order cumulant expansion results suggest, similarly to other methods, a low-temperature phase of an essentially different kind from that predicted for the classical model. The possible explanation of the origin of the spurious 2D Heisenberg-like nontrivial fixed point in some renormalization-group calculations is also proposed.