Randomly branched polymers and conformal invariance

Miller, Jeffrey D.; De’Bell, K

doi:10.1051/jp1:1993211

Cited by 13 publications

(17 citation statements)

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“…A similar analysis of lattice trees con6ned to the surface of a cone has recently been performed by Miller and De'Bell (1991), who have also discussed the constraints on a conformal theory for lattice trees and concluded that a linear relation in 1/n is a necessary feature of such a theory. Unfortunately, the analysis of the data for small values of a is difficult, as the numerical estimates are in general less well behaved than for larger angles.…”

Section: Critical Exponentsmentioning

confidence: 75%

Surface phase transitions in polymer systems

1993

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Self-avoiding walks, lattice trees, and related geometrical models provide a link between the physics of polymers and the study of critical phenomena. In particular, these models in the presence of a surface provide insight into surface adsorption in dilute polymer systems in a good solvent. The theme of this review is the influence of polymer structure (topology) on the critical properties of these models. Emphasis is placed on recent results by rigorous methods, scaling theory, and conformal covariance theory. Numerical results that may be used to test the predictions of scaling and conformal covariance theories are also summarized. Related topics such as the adsorption of directed polymers, the semidilute regime, the theta point and theta solvents, and percolation (polymer gels) are briefly discussed in the final section.

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Section: Critical Exponentsmentioning

confidence: 75%

Surface phase transitions in polymer systems

1993

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“…0.155 (7) 0.153 (8) 0.152 (9) 0.151 (8) 0.150 (10) 0.150 (8) Owing to the unknown scaling function ψ(ω), the determination is not extremely accurate, since a few points are used for the fits. It can nevertheless be improved if one considers the magnetization profile inside a square with fixed boundary conditions.…”

Section: []mentioning

confidence: 99%

“…From the fundamental point of view, situations are known where a diverging correlation length does not guarantee the validity of CI. A few years ago, lattice animals were indeed found to be not conformally invariant although they display isotropic critical behavior with correlation lengths satisfying the usual scaling ξ ∼ L with the system size [7]. If conformal invariance works, on the other hand, its powerful techniques might be applied with no restriction to investigate the critical behavior of 2D random systems [8].In the 2D random-bond Ising Model, both analytic [9] and numerical results [10] are available.…”

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confidence: 99%

Tests of conformal invariance in randomness-induced second-order phase transitions

Chatelain¹,

Berche²

1998

Phys. Rev. E

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The conformal covariance of correlation functions is checked in the second-order transition induced by random bonds in the two-dimensional eight-state Potts model. The decay of correlations is obtained via transfer matrix calculations in a cylinder geometry, and large-scale Monte Carlo simulations provide access to the correlations and the profiles inside a square with free or fixed boundary conditions. In both geometries, conformal transformations constrain the form of the spatial dependence, leading to accurate determinations of the order parameter scaling index, in good agreement with previous independent determinations obtained through standard techniques. The energy density exponent is also computed.PACS numbers: 64.60. Cn, 05.50.+q,05.70.Jk, 64.60.Fr It is well known that quenched randomness can deeply affect the critical properties at second-order phase transitions [1], and is liable to smooth first-order transitions, eventually leading to continuous transitions [2,3].In random systems, owing to strong inhomogeneities inherent in disorder, the usual symmetry properties required by conformal invariance (CI) [4] do not hold. However, by averaging over disorder realizations (denoted by [. . .] av ), translation and rotation invariance are restored, and it is generally believed that conformal invariance techniques should apply in principle. Recent results based on this assumption have recently been obtained at randomness-induced second-order transitions [5,6], but clear evidence of the validity of conformal invariance is still missing. The question is of both fundamental and practical interest. From the fundamental point of view, situations are known where a diverging correlation length does not guarantee the validity of CI. A few years ago, lattice animals were indeed found to be not conformally invariant although they display isotropic critical behavior with correlation lengths satisfying the usual scaling ξ ∼ L with the system size [7]. If conformal invariance works, on the other hand, its powerful techniques might be applied with no restriction to investigate the critical behavior of 2D random systems [8].In the 2D random-bond Ising Model, both analytic [9] and numerical results [10] are available. Transfer matrix (TM) calculations were also used to study the correlation function decay along strips and to compute the conformal anomaly (defined as a universal amplitude in finite-size corrections to the free energy) [11] and, since disorder is marginally irrelevant in the 2D Ising model, conformal invariance techniques were indeed efficient. At randomness-induced second-order phase transitions, a direct comparison between the results deduced from conformal invariance, and standard techniques, such as finitesize scaling (FSS) Monte Carlo (MC) simulations, have nevertheless not yet been made. After the pioneering work of Imry and Wortis, the first large-scale MC simulations devoted to the influence of quenched randomness, in a system whose pure version undergoes a strong firstorder phase transition, were a...

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“…Extensions of the present work to branched polymers (lattice animals) [7] are currently being pursued. Though conformal invariance concepts are not applicable in the case [22], surface critical indices such as the crossover exponent φ = y s /y can be calculated and compared e.g. to series results [23], for which error bars are rather large at present.…”

Section: Square Triangularmentioning

confidence: 99%

On the surface properties of two-dimensional percolation clusters

Queiroz¹

1995

J. Phys. A: Math. Gen.

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The two-dimensional site percolation problem is studied by transfermatrix methods on finite-width strips with free boundary conditions. The relationship between correlation-length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, η s = 0.6664 ± 0.0008, consistent with the analytical value η s = 2/3. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value y s = −1 for the irrelevant exponent.

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Randomly branched polymers and conformal invariance

Cited by 13 publications

References 1 publication

Surface phase transitions in polymer systems

Surface phase transitions in polymer systems

Tests of conformal invariance in randomness-induced second-order phase transitions

On the surface properties of two-dimensional percolation clusters

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