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Journal of Physics A Mathematical and General

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

T W Burhardt, Ferenc Iglói

Published Jun 21, 1990• 24 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/23/12/009

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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.
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .
We consider Ising strips with width N and periodic boundary conditions and Ising squares with edge length N and special partially periodic boundaries. Assuming invariance of the spin correlations under conformal mappings of the infinite plane onto the strip and square, we determine the second and fourth moments of the total magnetization M from the known bulk two- and four-spin correlation functions at criticality. For both geometries the predictions of conformal invariance for universal asymptotic forms involving the ratio 〈M4{M}^{4}M4〉〈M2{M}^{2}M2${〉}^{\mathrm{\ensuremath{-}}2}$ as N\ensuremath{\rightarrow}\ensuremath{\infty} are in excellent agreement with transfer-matrix results.
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 spectrum of the transfer matrix of a semi-infinite two-dimensional Isin model with marginally inhomogeneous couplings has been determined exactly in the finite-size-scaling limit at the bulk critical point. Depending on the local temperature at the boundary the surface phase transition of this system is either of first order or of second order with nonuniversal critical exponents. For a second-order transition, the excitation energies of the transfer matrix are inversely proportional to the linear size (N) of the system, but the levels are not equidistantly spaced. On the other hand, for a first-order transition the lowest gap vanishes algegraically, but faster than 1/N.
A semi-infinite nearest-neighbour square Ising lattice is investigated whose couplings J(l) at a distance l from the boundary differ from homogeneity by an amount j(l)-J(infinity) approximately -A/l. On the basis of the Pfaffian method the critical behaviour at the surface of this system is investigated. The exponents eta 11 , nu 11 , beta 1 , gamma 11 and delta 11 all display rich non-universal behaviour as a function of the amplitude A. For A below a critical value there is a spontaneous surface magnetisation when the bulk (l = infinity) is critical and an asymmetry between the exponents on either side of the critical point.