arXiv subjects

John Cardy

Publications and source records attributed to John Cardy.

At least 91 records · Page 5Linked to original sources

arXiv2001

The Stress Tensor in Quenched Random Systems

The talk describes recent progress in understanding the behaviour of the stress tensor and its correlation functions at a critical point of a generic quenched random system. The topics covered include:(i) the stress tensor in random systems considered as deformed pure systems; (ii) correlators of the stress tensor at a random fixed point: expectations from the replica approach and c-theorem sum rules; (iii) partition function on a torus; (iv) how the stress tensor enters into correlation functions: subtleties with Kac operators.

cond-mat.stat-mech↗

arXiv2001

Exact Scaling Functions for Self-Avoiding Loops and Branched Polymers

It is shown that a recently conjectured form for the critical scaling function for planar self-avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem, combined with the dimensional reduction arguments of Parisi and Sourlas. The result is generalized to higher-order multicritical points, yielding exact values for all their critical exponents and exact forms for the associated scaling functions.

cond-mat.stat-mech↗

arXiv2001

Unusual corrections to scaling in the 3-state Potts antiferromagnet on a square lattice

At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2; and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as β\equiv J/kT \to \infty according to ξ\sim A e^{2β} (1 + bβe^{-β} + ...), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal.

cond-mat.stat-mech↗

arXiv2001

Conformal Invariance and Percolation

These lectures give an introduction to the methods of conformal field theory as applied to deriving certain results in two-dimensional critical percolation: namely the probability that there exists at least one cluster connecting two disjoint segments of the boundary of a simply connected region; and the mean number of such clusters. No previous familiarity with conformal field theory is assumed, but in the course of the argument many of its important concepts are introduced in as simple a manner as possible. A brief account is also given of some recent alternative approaches to deriving these kinds of result.

math-ph↗

arXiv2000

Linking numbers for self-avoiding walks and percolation: application to the spin quantum Hall transition

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point. This yields many results, for example the distribution of the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. These twist operators correspond to (1,2) in the Kac classification of conformal field theory, so that their higher-point correlations, which describe linking numbers around multiple points, may be computed exactly. As an application we compute the exact value \sqrt 3/2 for the dimensionless conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.

cond-mat.stat-mech↗

arXiv2000

The Field Theory of the q->4+ Potts Model

The q-state Potts model in two dimensions exhibits a first-order transition for q>4. As q->4+ the correlation length at this transition diverges. We argue that this limit defines a massive integrable quantum field theory whose lowest excitations are kinks connecting 4+1 degenerate ground states. We construct the S-matrix of this theory and the two-particle form factors, and hence estimate a number of universal amplitude ratios. These are in very good agreement with the results of extrapolated series in q^(-1/2) as well as Monte Carlo results for q=5.

hep-th↗

arXiv1999

Correlated correlation functions in random-bond ferromagnets

The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2+epsilon) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator is presented.

cond-mat.stat-mech↗

arXiv1999

Logarithmic Correlations in Quenched Random Magnets and Polymers

It is argued that logarithmic factors multiplying power law behavior are to be expected at or near non-mean field critical points of systems with short-range interactions described theoretically by any kind of n -> 0 limit, in which the effective free energy vanishes. Explicit examples are given for quenched random ferromagnets, polymer statistics and percolation, but the phenomenon is quite general.

cond-mat.stat-mech↗

arXiv1999

Susceptibility amplitude ratios in the two-dimensional Potts model and percolation

The high-temperature susceptibility of the $q$-state Potts model behaves as $Γ|T-T_c|^{-γ}$ as $T\to T_c+$, while for $T\to T_c-$ one may define both longitudinal and transverse susceptibilities, with the same power law but different amplitudes $Γ_L$ and $Γ_T$. We extend a previous analytic calculation of the universal ratio $Γ/Γ_L$ in two dimensions to the low-temperature ratio $Γ_T/Γ_L$, and test both predictions with Monte Carlo simulations for $q=3$ and 4. The data for $q=4$ are inconclusive owing to large corrections to scaling, while for $q=3$ they appear consistent with the prediction for $Γ_T/Γ_L$, but not with that for $Γ/Γ_L$. A simple extrapolation of our analytic results to $q\to1$ indicates a similar discrepancy with the corresponding measured quantities in percolation. We point out that stronger assumptions were made in the derivation of the ratio $Γ/Γ_L$, and our work suggests that these may be unjustified.

cond-mat.stat-mech↗

arXiv1999

Renormalisation Group Theory of Branching Potts Interfaces

We develop a field-theoretic representation for the configurations of an interface between two ordered phases of a q-state Potts model in two dimensions, in the solid-on-solid approximation. The model resembles the field theory of directed percolation and may be analysed using similar renormalisation group methods. In the one-loop approximation these reveal a simple mechanism for the emergence of a critical value q_c, such that for q q_c the interfacial width diverges at a finite value of the tension, indicating a first-order transition. The value of the Widom exponent for q minus infinity limit of our model corresponds to directed percolation and that the values for the exponents in the one-loop approximation are in reasonable agreement with accepted values.

cond-mat.stat-mech↗

arXiv1999

Lattice field theories with an energy current

We investigate a lattice scalar field theory in the presence of a bias favouring the establishment of an energy current, as a model for stationary nonequilibrium processes at low temperature in a non-integrable system. There is a transition at a finite value of the bias to a gapless modulated phase which carries a classical current; however, unlike in similar, integrable, models, quantum effects also allow for a non-zero current at arbitrarily small bias. The transition is second order in the magnetically disordered phase, but is pre-empted by a first-order transition in the ferromagnetic case, at least at the mean-field level.

cond-mat.stat-mech↗

arXiv1999

Directed Percolation and Generalized Friendly Walkers

We show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of m directed random walkers with rather general attractive interactions, when suitably continued to m=0. In 1+1 dimensions, this is dual to a model of interacting steps on a vicinal surface. A similar correspondence with interacting self-avoiding walks is constructed for isotropic percolation.

cond-mat.stat-mech↗

arXiv1999

The Reaction Process A+A->O in Sinai Disorder

The single-species reaction-diffusion process $A+A\to O$ is examined in the presence of an uncorrelated, quenched random velocity field. Utilising a field-theoretic approach, we find that in two dimensions and below the density decay is altered from the case of purely diffusing reactants. In two-dimensions the density amplitude is reduced in the presence of weak disorder, yielding the interesting result that Sinai disorder can cause reactions to occur at an {\it increased} rate. This is in contrast to the case of long-range correlated disorder, where it was shown that the reaction becomes sub-diffusion limited. However, when written in terms of the microscopic diffusion constant it is seen that increasing the disorder has the effect of reducing the rate of the reaction. Below two dimensions, the effect of Sinai disorder is much more severe and the reaction is shown to become sub-diffusion limited. Although there is no universal amplitude for the time-dependence of the density, it is universal when expressed in terms of the disorder-averaged diffusion length. The appropriate amplitude is calculated to one-loop order.

cond-mat.stat-mech↗

arXiv1999

Critical Exponents near a Random Fractal Boundary

The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary is a random fractal, specifically a self-avoiding walk or the frontier of a Brownian walk, in two dimensions, and show that the boundary scaling behaviour of the correlation function is characterised by a set of multifractal boundary exponents, given exactly by conformal invariance arguments to be $λ_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1)$. This result may be interpreted in terms of a scale-dependent distribution of opening angles $α$ of the fractal boundary: on short distance scales these are sharply peaked around $α=π/3$. Similar arguments give the multifractal exponents for the case of coupling to a quenched random bulk geometry.

cond-mat.stat-mech↗

arXiv1998

Quenched Randomness at First-Order Transitions

A rigorous theorem due to Aizenman and Wehr asserts that there can be no latent heat heat in a two-dimensional system with quenched random impurities. We examine this result, and its possible extensions to higher dimensions, in the context of several models. For systems whose pure versions undergo a strong first-order transition, we show that there is an asymptotically exact mapping to the random field Ising model, at the level of the interface between the ordered and disordered phases. This provides a physical explanation for the above result and also implies a correspondence between the problems in higher dimensions, including scaling relations between their exponents. The particular example of the q-state Potts model in two dimensions has been considered in detail by various authors and we review the numerical results obtained for this case. Turning to weak, fluctutation-driven first-order transitions, we describe analytic renormalisation group calculations which show how the continuous nature of the transition is restored by randomness in two dimensions.

cond-mat.stat-mech↗

arXiv1998

Asymptotic form of the approach to equilibrium in reversible recombination reactions

The reversible reactions A+A<=>C and A+B<=>C are investigated. From the exact Langevin equations describing our model, we set up a systematic approximation scheme to compute the approach of the density of C particles to its equilibrium value. We show that for sufficiently long time t, this approach takes the form of a power law At^(-d/2), for any dimension d. The amplitude A is also computed exactly, but is expected to be model dependent. For uncorrelated initial conditions, the C density turns out to be a monotonic time function. The cases of correlated initial conditions and unequal diffusion constants are investigated as well. In the former, correlations may break the monotonicity of the density or in some special cases they may change the long time behavior. For the latter, the power law remains valid, only the amplitude changes, even in the extreme case of immobile C particles. We also consider the case of segregated initial condition for which a reaction front is observed, and confirm that its width is governed by mean-field exponent in any dimension.

cond-mat.stat-mech↗

arXiv1997

Critical Behaviour of Random Bond Potts Models: A Transfer Matrix Study

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q>4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent beta/nu varies continuously with q, assuming non-Ising values for q>4, whereas the correlation length exponent nu is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour.

cond-mat.stat-mech↗

arXiv1997

Extraordinary transition in the two-dimensional O(n) model

The extraordinary transition which occurs in the two-dimensional O(n) model for $n<1$ at sufficiently enhanced surface couplings is studied by conformal perturbation theory about infinite coupling and by finite-size scaling of the spectrum of the transfer matrix of a simple lattice model. Unlike the case of $n\geq1$ in higher dimensions, the surface critical behaviour differs from that occurring when fixed boundary conditions are imposed. In fact, all the surface scaling dimensions are equal to those already found for the ordinary transition, with, however, an interesting reshuffling of the corresponding eigenvalues between different sectors of the transfer matrix.

cond-mat.stat-mech↗