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:0704.1949 · Journal of the Physical Society of Japan

Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

Kouji Ueda, Roman Krčmár, Andrej Gendiar, Tomotoshi Nishino

Published Aug 10, 2007• 42 citations
arXivHTMLPDF

Abstract

Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.

DOI
10.1143/jpsj.76.084004

Related papers

Ising model on a hyperbolic lattice studied by the corner transfer matrix renormalization group method

Roman Krčmár, Andrej Gendiar, K Ueda +1 more

We study two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as the tessellation of polygons with p>=5 sides, such as pentagons (p=5), hexagons (p=6), etc. Such lattices are on hyperbolic planes, which have constant negative scalar curvatures. We calculate critical temperatures and scaling exponents by use of the corner transfer matrix renormalization group method. As a result, the mean-field like phase transition is observed for all the cases p>=5. Convergence of the calculated transition temperatures with respect to p is investigated towards the limit p->infinity, where the system coincides with the Ising model on the Bethe lattice.

arXiv:0712.0461•Journal of Physics A Mathematical and Theoretical•Mar 10, 2008•43 citations

Phase Transition of the Ising Model on a Hyperbolic Lattice

Takatsugu Iharagi, Andrej Gendiar, Hiroshi Ueda +1 more

The matrix product structure is considered on a regular lattice in the hyperbolic plane. The phase transition of the Ising model is observed on the hyperbolic $(5, 4)$ lattice by means of the corner-transfer-matrix renormalization group (CTMRG) method. Calculated correlation length is always finite even at the transition temperature, where mean-field like behavior is observed. The entanglement entropy is also always finite.

arXiv:1005.3378•Journal of the Physical Society of Japan•Sep 27, 2010•22 citations

Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices

Andrej Gendiar, Roman Krčmár, Sabine Andergassen +2 more

The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.

arXiv:1205.3850•Physical Review E•Aug 6, 2012•23 citations

Phase transition of clock models on a hyperbolic lattice studied by corner transfer matrix renormalization group method

Andrej Gendiar, Roman Krčmár, Kouji Ueda +1 more

Two-dimensional ferromagnetic N -state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases N>or=3 up to N=30 . The model with N=3 , which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases N>or=4 . When N>or=5 we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as N(-2) . From these facts we conclude that the phase transition of the classical XY model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.

arXiv:0801.0836•Physical Review E•Apr 23, 2008•14 citations

Tricritical point of the J1−J2 Ising model on a hyperbolic lattice

Roman Krčmár, Takatsugu Iharagi, Andrej Gendiar +1 more

The ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains a ferromagnetic nearest-neighbor interaction J1 and a competing antiferromagnetic interaction J2 . A mean-field-like second-order phase transition is observed when the ratio kappa=J_{2}J_{1} is less than 0.203. In the region 0.203<kappa<14 , the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field-like.

arXiv:0807.0150•Physical Review E•Dec 17, 2008•19 citations

Tensor-network study of Ising model on infinite hyperbolic dodecahedral lattice

Matej Moško, Andrej Gendiar

We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous noncritical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents β=0.4999 and δ=3.007, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multistate spin models.

arXiv:2510.20939•Physical review. E•Apr 15, 2026•0 citations
Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.
We study two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as the tessellation of polygons with p>=5 sides, such as pentagons (p=5), hexagons (p=6), etc. Such lattices are on hyperbolic planes, which have constant negative scalar curvatures. We calculate critical temperatures and scaling exponents by use of the corner transfer matrix renormalization group method. As a result, the mean-field like phase transition is observed for all the cases p>=5. Convergence of the calculated transition temperatures with respect to p is investigated towards the limit p->infinity, where the system coincides with the Ising model on the Bethe lattice.
The matrix product structure is considered on a regular lattice in the hyperbolic plane. The phase transition of the Ising model is observed on the hyperbolic (5,4)(5, 4)(5,4) lattice by means of the corner-transfer-matrix renormalization group (CTMRG) method. Calculated correlation length is always finite even at the transition temperature, where mean-field like behavior is observed. The entanglement entropy is also always finite.
The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.
Two-dimensional ferromagnetic N -state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases N>or=3 up to N=30 . The model with N=3 , which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases N>or=4 . When N>or=5 we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as N(-2) . From these facts we conclude that the phase transition of the classical XY model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.
The ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains a ferromagnetic nearest-neighbor interaction J1 and a competing antiferromagnetic interaction J2 . A mean-field-like second-order phase transition is observed when the ratio kappa=J_{2}J_{1} is less than 0.203. In the region 0.203<kappa<14 , the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field-like.
We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous noncritical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents β=0.4999 and δ=3.007, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multistate spin models.