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

Annalen der Physik

Effects of geometry on transfer matrices, spin chains and critical behaviour

Brian Davies, Ingo Peschel

Published Jan 1, 1993• 3 citations
Publisher

Abstract

Abstract We consider two‐dimensional Ising models bounded by general parabolic curves and study their transfer matrices and associated quantum spin chains. We derive their eigenvalue spectra numerically and analytically, both at the critical point and in its vicinity. From this we find how the geometrical form of the system is reflected in the spectrum and how it influences the critical behaviour near the tip.

DOI
10.1002/andp.19935050107

Related papers

Critical behaviour in parabolic geometries

I Peschel, Loïc Turban, Ferenc Iglói

The authors study two-dimensional systems with boundary curves described by power laws. Using conformal mappings they obtain the correlations at the bulk critical point. Three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions. For an Ising system of parabolic shape the behaviour of the order at the tip is also found.

arXiv:cond-mat/0106602•Journal of Physics A Mathematical and General•Oct 21, 1991•24 citations

Finite-size scaling of corner transfer matrices for the two-dimensional Ising model

Ingo Peschel

The eigenvalue spectra of generalised corner transfer matrices are calculated for an Ising model on a square lattice. The predictions of conformal invariance at the critical point are thereby verified.

Journal of Physics A Mathematical and General•Feb 7, 1988•9 citations

Geometric effects on critical behaviours of the Ising model

Hiroyuki Shima, Yasunori Sakaniwa

We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.

•May 5, 2006•29 citations

Random matrix theory and classical statistical mechanics: Spin models

H. Meyer, J C Anglès d'Auriac

We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper [H. Meyer, J.-C. Angl`es d'Auriac, and J.-M. Maillard, Phys. Rev. E 55, 5261 (1997)]. We show that the statistical properties of these spectra can serve as a criterion of integrability. It also provides an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality, considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra, which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.

arXiv:cond-mat/9612178•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1997•13 citations

Gradient critical phenomena in the Ising quantum chain

Thierry Platini, Dragi Karevski, Loïc Turban

We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg–Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour is in agreement with the predictions of the scaling theory.

arXiv:cond-mat/0611213•Journal of Physics A Mathematical and Theoretical•Jan 30, 2007•28 citations

Critical curve of the highly anisotropic two-dimensional spin-s Ising model

Leigh Sneddon, Robin B. Stinchcombe

The rectangular spin-s Ising ferromagnet with nearest-neighbour interactions is considered. The interaction energies are of the form s -2 J sigma i sigma j and s -2 J' sigma k sigma l , respectively for bonds in the two lattice directions, where the spin variables sigma i can take the values -s, -s+1,...,s-1,s. A transfer matrix formalism is used. By relating the largest eigenvalues of the transfer matrix for spin-s to those for spin- 1 / 2 , the leading order form for J'<<J of the spin-s critical curve is investigated. Subject to plausible assumption concerning the thermodynamic limit, the critical curve is shown to be given by K'=2s exp(-2K)(1+ epsilon ) where K=J/kT, K'=J'/kT and epsilon to 0 as K to infinity .

Journal of Physics C Solid State Physics•Oct 14, 1978•4 citations
Abstract We consider two‐dimensional Ising models bounded by general parabolic curves and study their transfer matrices and associated quantum spin chains. We derive their eigenvalue spectra numerically and analytically, both at the critical point and in its vicinity. From this we find how the geometrical form of the system is reflected in the spectrum and how it influences the critical behaviour near the tip.
We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.
We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper [H. Meyer, J.-C. Angl`es d'Auriac, and J.-M. Maillard, Phys. Rev. E 55, 5261 (1997)]. We show that the statistical properties of these spectra can serve as a criterion of integrability. It also provides an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality, considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra, which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.
We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg–Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour is in agreement with the predictions of the scaling theory.
The rectangular spin-s Ising ferromagnet with nearest-neighbour interactions is considered. The interaction energies are of the form s -2 J sigma i sigma j and s -2 J' sigma k sigma l , respectively for bonds in the two lattice directions, where the spin variables sigma i can take the values -s, -s+1,...,s-1,s. A transfer matrix formalism is used. By relating the largest eigenvalues of the transfer matrix for spin-s to those for spin- 1 / 2 , the leading order form for J'<<J of the spin-s critical curve is investigated. Subject to plausible assumption concerning the thermodynamic limit, the critical curve is shown to be given by K'=2s exp(-2K)(1+ epsilon ) where K=J/kT, K'=J'/kT and epsilon to 0 as K to infinity .