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Physical Review Letters

Critical behavior near a smooth inhomogeneity in the planar Ising model and conformal invariance

Ferenc Iglói, Bertrand Berche, Loïc Turban

Published Oct 1, 1990• 29 citations
Publisher

Abstract

We consider a square-lattice Ising model with a smooth inhomogeneity in the bulk, where the couplings deviate from the bulk critical coupling by ${\mathit{Al}}^{\mathrm{\ensuremath{-}}1}$, l being the distance from the center of the inhomogeneity. Scaling exponents describing decay of correlations along the inhomogeneity have been determined exactly by conformal invariance and finite-size scaling. It is shown that the homogeneous model (A=0) with bulk Ising exponents separates the first-order transition regime (A>0) from the region of nonuniversal criticality (A0).

DOI
10.1103/physrevlett.65.1773

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We consider a square-lattice Ising model with a smooth inhomogeneity in the bulk, where the couplings deviate from the bulk critical coupling by Al\ensuremath−1{\mathit{Al}}^{\mathrm{\ensuremath{-}}1}Al\ensuremath−1, l being the distance from the center of the inhomogeneity. Scaling exponents describing decay of correlations along the inhomogeneity have been determined exactly by conformal invariance and finite-size scaling. It is shown that the homogeneous model (A=0) with bulk Ising exponents separates the first-order transition regime (A>0) from the region of nonuniversal criticality (A0).
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 consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large mmm, mmm being the distance from the edge. The case A<0A<0A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when A>0A>0A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
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.
A semi-infinite nearest-neighbor square Ising system whose couplings at a distance lll from the boundary differ from homogeneity by an amount $\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}$ is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$, ${\ensuremath{\beta}}_{1}$, ${\ensuremath{\gamma}}_{11}$, and ${\ensuremath{\delta}}_{11}$ display rich nonuniversal behavior as a function of the amplitude AAA. For AAA below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk ($l=\ensuremath{\infty}$) is critical.