Correlation decay and conformal anomaly in the two-dimensional random-bond Ising ferromagnet

Abstract

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 strips. Estimates thus obtained are consistent with c = 1/2, the same as for the pure Ising model.

Critical behavior of the two-dimensional spin-diluted Ising model via the equilibrium ensemble approach

“…From this we can conclude that the exponent η does not depend on disorder, a result which has been known already 20,21,9,41,11 but without the precision shown above. Note that the correlation length ξ extracted from the ratio of TM eigenvalues (Eq.…”

Logarithmic corrections to gap scaling in random-bond Ising strips

“…Similar conclusions have been drawn for the corresponding quantities evaluated on strips [17,18], where the exact critical correlation functions for the pure Ising model are known from conformal invariance [19].…”

“…Details of the calculation are given in Ref. [17], where the data used here are displayed as well, in plots of (L/π)(…”

“…Since (i) disorder in the two-dimensional Ising model is believed to be a marginal operator, (ii) Lyapunov exponents of transfer matrices in random systems are the counterparts of eigenvalues in homogeneous ones, and (iii) numerical evidence shows that conformal-invariance results derived for uniform systems may be extended to random cases, provided that suitable averages are taken [2,17], we shall examine sequences of estimates…”

“…Table 1 shows results from least-squares fits of data in the range L = 6 − 13. This interval was chosen in order to minimise the accumulated standard deviation χ 2 per degree of freedom [17]. The extrapolated η is expected to be universal as long as r = 0; our error bar for r = 0.01 indeed includes η = 1/4, though admittedly at the edge.…”