Fluctuations of correlation functions in disordered spin systems

Crisanti, A.; Nicolis, Stam; Paladin, Giovanni; Vulpiani, Angelo

doi:10.1088/0305-4470/23/13/042

Cited by 15 publications

(29 citation statements)

References 19 publications

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“…In previous work [6][7][8][9] this was done by generating strips of length N >> 1 and carrying out the actual transfer-matrix products, so the end result would presumably reflect the properties of a representative sample. While the average free energy per site is unambiguously related to the dominant Lyapunov exponent of the transfer-matrix product [16], the relationship of correlation functions to Lyapunov exponents is much more involved, because the probability distribution of the correlation functions themselves turns out to be rather complex [17,18]. It can be shown that the most probable correlation decay is given by the difference between the two largest Lyapunov exponents [17], which constitutes a straightforward generalisation of Eq.…”

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confidence: 88%

Transfer-matrix scaling from disorder-averaged correlation lengths for diluted Ising systems

Queiroz

Stinchcombe

1994

Phys. Rev. B

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A transfer matrix scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalisation procedures are then used to analyse aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration p close to p c , the extent of exponential behaviour of the T c × p curve is clearly seen for over two decades of variation of p − p c . Close to the pure-system limit, the exactly-known reduced slope is reproduced to a very good approximation, though with non-monotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent η remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.PACS numbers: 75. 10H, 75.40c, 05.50 Typeset using REVT E X 1 The transfer matrix is well known to provide exact solutions for a number of lowdimensional pure systems such as spin models etc [1]. Applied to finite-width strips and combined with finite size scaling [2], it has also proved to be extremely powerful and accurate for non-solvable pure cases, ranging from magnetic systems [3] to walks [4] and quantum Hamiltonians [5]. The application of such "strip-scaling" techniques to random systems has however been extremely limited, despite the enormous interest in such problems as the spin glass, random field and dilute magnets, ceramic superconductors etc. This is because the form so far utilized [6][7][8][9] applies the strip scaling to particular realisations of the random system which need not be representative unless appropriate averaging (or self-averaging) is employed, resulting in very large scale computing on extremely long strips and/or many realisations.Here, we provide a strip scaling approach for random systems, in which the disorder averaging is carried out as one proceeds along the strip, and which therefore does not have the deficiencies noted above. Results reported here extend, and give details of, those contained in a previous Rapid Communication [10]. Our scheme does rely on certain assumptions regarding the dominant contributions to the decay of correlations, which must be spelt out clearly and supported by numerical evidence whenever possible. This is one of our purposes in what follows.The main numerical application of the technique here is to the two-dimensional sitediluted Ising model [11]. The transfer-matrix descriptions of the special limiting cases of percolation [12] and pure Ising system [13] are well-understood, and are reproduced by t...

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confidence: 88%

Transfer-matrix scaling from disorder-averaged correlation lengths for diluted Ising systems

Queiroz

Stinchcombe

1994

Phys. Rev. B

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“…An analysis of the correlation function probability distribution is needed in order to ensure that self-averaging problems do not alter the mean values [44]. The methodology that we propose is to deduce the critical behavior from the decay of the correlation functions using conformal symmetry.…”

Section: B Probability Distribution Of the Correlation Functionmentioning

confidence: 99%

Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries

Chatelain¹,

Berche²

1999

Phys. Rev. E

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We present a numerical study of 2D random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc = 4 which discriminates between second and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques which were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q > 4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes which is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.

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“…= exp ln Q [9,10,11,12]. Depending on the underlying probability distribution, Q and Q typ may differ appreciably, as is the case when one considers correlation functions [9,11].…”

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confidence: 99%

“…13 we picked the averages ln G(R) ≡ ln ( σ 0 σ R ). The quantity exp ln G(R) is expected [11,12] to scale as σ 0 σ R typ . For each L the slope of a two-point semi-log plot of exp ln G(R) gave an approximate value for 1/ξ typ .…”

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confidence: 99%