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arXiv:cond-mat/0106596 · Journal of Physics A Mathematical and General

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

Loïc Turban

Published Apr 21, 1985• 43 citations
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Abstract

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 .

DOI
10.1088/0305-4470/18/6/012

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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 .
We consider the two-dimensional (2D) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, J1>J2, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip xi(L) for different widths of the strip L</=21. The ratio of the two lengths xi(L)/L=A is found to approach the universal value A=2/pi for large L, independent of the dilution parameter J(1)/J(2). With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2D dilute Ising model is self-averaging, in the critical point conformally covariant and the corresponding decay exponent is eta( ||)=1.
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 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.
The two-dimensional random-bond Ising model is numerically studied on long strips by transfer-matrix methods. It is shown that the rate of decay of correlations at criticality, as derived from averages of the two largest Lyapunov exponents plus conformal invariance arguments, differs from that obtained through direct evaluation of correlation functions. The latter is found to be, within error bars, the same as in pure systems. Our results confirm field-theoretical predictions. The conformal anomaly c is calculated from the leading finite-width correction to the averaged free energy on the strips. Estimates thus obtained are consistent with c=1/2, the same as for the pure Ising model.