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:cond-mat/0106629 · Physical review. B, Condensed matter

Radial marginal perturbation of two-dimensional systems and conformal invariance

Loïc Turban

Published Oct 1, 1991• 13 citations
arXivHTMLPDF

Abstract

The conformal mapping w=(L/2\ensuremath{\pi})lnz transforms the critical plane with a radial perturbation \ensuremath{\alpha}${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}\mathit{y}}$ into a cylinder with width L and a constant deviation \ensuremath{\alpha}(2\ensuremath{\pi}/L${)}^{\mathit{y}}$ from the bulk critical point when the decay exponent y is such that the perturbation is marginal. From the known behavior of the homogeneous off-critical system on the cylinder, one may deduce the correlation functions and defect exponents on the perturbed plane. The results are supported by an exact solution for the Gaussian model.

DOI
10.1103/physrevb.44.7051

Related papers

Marginal extended perturbations in two dimensions and gap-exponent relations

Loïc Turban, Bertrand Berche

The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.

arXiv:hep-th/9305119•Journal of Physics A Mathematical and General•Jul 7, 1993•9 citations

Radial Fredholm perturbation in the two-dimensional Ising model and gap-exponent relation

Dragi Karevski, Loïc Turban, Ferenc Iglói

We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i.e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.

arXiv:cond-mat/9505111•Journal of Physics A Mathematical and General•Jul 21, 1995•2 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

Conformal invariance and linear defects in the two-dimensional Ising model

Loïc Turban

Using conformal invariance, the author shows that the nonuniversal exponent eta 0 associated with the decay of correlations along a defect line of modified bonds in the square-lattice Ising model is related to the amplitude A 0 = xi n /n of the correlation length/ at the bulk critical coupling K c , on a strip with width n, periodic boundary conditions and two equidistant defect lines along the strip through A 0 =( pi eta 0 ) -1 .

arXiv:cond-mat/0106596•Journal of Physics A Mathematical and General•Apr 21, 1985•43 citations

Conformal invariance and the Ising model on a spheroid

Youjin Deng, Henk W. J. Blöte

We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 14, 2003•14 citations

Off-diagonal density profiles and conformal invariance

Loïc Turban, Ferenc Iglói

Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.

arXiv:cond-mat/9612128•Journal of Physics A Mathematical and General•Mar 7, 1997•13 citations
The conformal mapping w=(L/2\ensuremath{\pi})lnz transforms the critical plane with a radial perturbation \ensuremath{\alpha}\ensuremathρ\ensuremath−y{\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}\mathit{y}}\ensuremathρ\ensuremath−y into a cylinder with width L and a constant deviation \ensuremath{\alpha}(2\ensuremath{\pi}/L)y{)}^{\mathit{y}})y from the bulk critical point when the decay exponent y is such that the perturbation is marginal. From the known behavior of the homogeneous off-critical system on the cylinder, one may deduce the correlation functions and defect exponents on the perturbed plane. The results are supported by an exact solution for the Gaussian model.
The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.
We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i.e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.
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.
We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.
Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.