arXiv subjects

Kay Joerg Wiese

Publications and source records attributed to Kay Joerg Wiese.

At least 73 records · Page 4Linked to original sources

arXiv2006

How to measure Functional RG fixed-point functions for dynamics and at depinning

We show how the renormalized force correlator Delta(u), the function computed in the functional RG (FRG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in presence of pinning disorder. For equilibrium dynamics we recover the relation obtained recently in the statics between Delta(u) and a physical observable. Its extension to depinning reveals interesting relations to stick-slip models of avalanches used in dry friction and earthquake dynamics. The particle limit (d=0) is solved for illustration: Delta(u) exhibits a cusp and differs from the statics. We propose that the FRG functions be measured in wetting and magnetic interfaces experiments.

cond-mat.dis-nn↗

arXiv2006

Systematic Field Theory of the RNA Glass Transition

We prove that the Laessig-Wiese (LW) field theory for the freezing transition of the secondary structure of random RNA is renormalizable to all orders in perturbation theory. The proof relies on a formulation of the model in terms of random walks and on the use of the multilocal operator product expansion. Renormalizability allows us to work in the simpler scheme of open polymers, and to obtain the critical exponents at 2-loop order. It also allows to prove some exact exponent identities, conjectured in LW.

q-bio.BM↗

arXiv2006

Universal distribution of threshold forces at the depinning transition

We study the distribution of threshold forces at the depinning transition for an elastic system of finite size, driven by an external force in a disordered medium at zero temperature. Using the functional renormalization group (FRG) technique, we compute the distribution of pinning forces in the quasi-static limit. This distribution is universal up to two parameters, the average critical force, and its width. We discuss possible definitions for threshold forces in finite-size samples. We show how our results compare to the distribution of the latter computed recently within a numerical simulation of the so-called critical configuration.

cond-mat.dis-nn↗

arXiv2006

Wetting and Minimal Surfaces

We study minimal surfaces which arise in wetting and capillarity phenomena. Using conformal coordinates, we reduce the problem to a set of coupled boundary equations for the contact line of the fluid surface, and then derive simple diagrammatic rules to calculate the non-linear corrections to the Joanny-de Gennes energy. We argue that perturbation theory is quasi-local, i.e. that all geometric length scales of the fluid container decouple from the short-wavelength deformations of the contact line. This is illustrated by a calculation of the linearized interaction between contact lines on two opposite parallel walls. We present a simple algorithm to compute the minimal surface and its energy based on these ideas. We also point out the intriguing singularities that arise in the Legendre transformation from the pure Dirichlet to the mixed Dirichlet-Neumann problem.

hep-th↗

arXiv2006

Measuring functional renormalization group fixed-point functions for pinned manifolds

Exact numerical minimization of interface energies is used to test the functional renormalization group (FRG) analysis for interfaces pinned by quenched disorder. The fixed-point function R(u) (the correlator of the coarse-grained disorder) is computed. In dimensions D=d+1, a linear cusp in R''(u) is confirmed for random bond (d=1,2,3), random field (d=0,2,3), and periodic (d=2,3) disorders. The functional shocks that lead to this cusp are seen. Small, but significant, deviations from 1-loop FRG results are compared to 2-loop corrections. The cross-correlation for two copies of disorder is compared with a recent FRG study of chaos.

cond-mat.dis-nn↗

arXiv2006

The Freezing of Random RNA

We study secondary structures of random RNA molecules by means of a renormalized field theory based on an expansion in the sequence disorder. We show that there is a continuous phase transition from a molten phase at higher temperatures to a low-temperature glass phase. The primary freezing occurs above the critical temperature, with local islands of stable folds forming within the molten phase. The size of these islands defines the correlation length of the transition. Our results include critical exponents at the transition and in the glass phase.

q-bio.BM↗

arXiv2005

Why one needs a functional renormalization group to survive in a disordered world

In these proceedings, we discuss why functional renormalization is an essential tool to treat strongly disordered systems. More specifically, we treat elastic manifolds in a disordered environment. These are governed by a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss how a renormalizable field theory can be constructed even beyond 2-loop order. We then consider an elastic manifold embedded in N dimensions, and give the exact solution for N to infinity. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. Finally, the effective action at order 1/N is reported.

cond-mat.dis-nn↗

arXiv2005

Random field spin models beyond one loop: a mechanism for decreasing the lower critical dimension

The functional RG for the random field and random anisotropy O(N) sigma-models is studied to two loop. The ferromagnetic/disordered (F/D) transition fixed point is found to next order in d=4+epsilon for N > N_c (N_c=2.8347408 for random field, N_c=9.44121 for random anisotropy). For N < N_c the lower critical dimension plunges below d=4: we find two fixed points, one describing the quasi-ordered phase, the other is novel and describes the F/D transition. The lower critical dimension can be obtained in an (N_c-N)-expansion. The theory is also analyzed at large N and a glassy regime is found.

cond-mat.dis-nn↗

arXiv2005

Supersymmetry Breaking in Disordered Systems and Relation to Functional Renormalization and Replica-Symmetry Breaking

In this article, we study an elastic manifold in quenched disorder in the limit of zero temperature. Naively it is equivalent to a free theory with elasticity in Fourier-space proportional to k^4 instead of k^2, i.e. a model without disorder in two space-dimensions less. This phenomenon, called dimensional reduction, is most elegantly obtained using supersymmetry. However, scaling arguments suggest, and functional renormalization shows that dimensional reduction breaks down beyond the Larkin length. Thus one equivalently expects a break-down of supersymmetry. Using methods of functional renormalization, we show how supersymmetry is broken. We also discuss the relation to replica-symmetry breaking, and how our formulation can be put into work to lift apparent ambiguities in standard functional renormalization group calculations.

cond-mat.dis-nn↗

arXiv2005

2-loop Functional Renormalization for elastic manifolds pinned by disorder in N dimensions

We study elastic manifolds in a N-dimensional random potential using functional RG. We extend to N>1 our previous construction of a field theory renormalizable to two loops. For isotropic disorder with O(N) symmetry we obtain the fixed point and roughness exponent to next order in epsilon=4-d, where d is the internal dimension of the manifold. Extrapolation to the directed polymer limit d=1 allows some handle on the strong coupling phase of the equivalent N-dimensional KPZ growth equation, and eventually suggests an upper critical dimension of about 2.5.

cond-mat.dis-nn↗

arXiv2004

Can non-linear elasticity explain contact-line roughness at depinning?

We examine whether cubic non-linearities, allowed by symmetry in the elastic energy of a contact line, may result in a different universality class at depinning. Standard linear elasticity predicts a roughness exponent zeta=1/3 (one loop), zeta=0.39 (numerics) while experiments give zeta of about 0.5. Within functional RG we find that a non-local KPZ-type term is generated at depinning and grows under coarse graining. A fixed point with zeta=0.45 (one loop) is identified, showing that large enough cubic terms increase the roughness. This fixed point is unstable, revealing a rough strong-coupling phase. Experimental study of contact angles theta near pi/2, where cubic terms vanish in the energy, is suggested.

cond-mat.soft↗

arXiv2004

Derivation of the Functional Renormalization Group Beta-Function at order 1/N for Manifolds Pinned by Disorder

In an earlier publication, we have introduced a method to obtain, at large N, the effective action for d-dimensional manifolds in a N-dimensional disordered environment. This allowed to obtain the Functional Renormalization Group (FRG) equation for N=infinity and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mezard-Parisi solution. Here we compute the corrections at order 1/N. We introduce two novel complementary methods, a diagrammatic and an algebraic one, to perform the complicated resummation of an infinite number of loops, and derive the beta-function of the theory to order 1/N. We present both the effective action and the corresponding functional renormalization group equations. The aim is to explain the conceptual basis and give a detailed account of the novel aspects of such calculations. The analysis of the FRG flow, comparison with other studies, and applications, e.g. to the strong-coupling phase of the Kardar-Parisi-Zhang equation are examined in a subsequent publication.

cond-mat.dis-nn↗

arXiv2003

Functional Renormalization Group at Large N for Disordered Elastic Systems, and Relation to Replica Symmetry Breaking

We study the replica field theory which describes the pinning of elastic manifolds of arbitrary internal dimension d in a random potential, with the aim of bridging the gap between mean field and renormalization theory. The full effective action is computed exactly in the limit of large embedding space dimension N. The second cumulant of the renormalized disorder obeys a closed self-consistent equation. It is used to derive a Functional Renormalization Group (FRG) equation valid in any dimension d, which correctly matches the Balents-Fisher result to first order in epsilon=4-d. We analyze in detail the solutions of the large-N FRG for both long-range and short-range disorder, at zero and finite temperature. We find consistent agreement with the results of Mezard Parisi (MP) from the Gaussian variational method (GVM) in the case where full replica symmetry breaking (RSB) holds there. We prove that the cusplike non-analyticity in the large N FRG appears at a finite scale, corresponding to the instability of the replica symmetric solution of MP. We show that the FRG exactly reproduces, for any disorder correlator and with no need to invoke Parisi's spontaneous RSB, the non-trivial result of the GVM for small overlap. A formula is found yielding the complete RSB solution for all overlaps. Since our saddle-point equations for the effective action contain both the MP equations and the FRG, it can be used to describe the crossover from FRG to RSB. A qualitative analysis of this crossover is given, as well as a comparison with previous attempts to relate FRG to GVM. Finally, we discuss applications to other problems and new perspectives.

cond-mat↗

arXiv2003

Functional Renormalization Group and the Field Theory of Disordered Elastic Systems

We study elastic systems such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of ``dimensional reduction'', we use the functional renormalization group. Difficulties arise in the calculation of the renormalization group functions beyond 1-loop order. Even worse, observables such as the 2-point correlation function exhibit the same problem already at 1-loop order. These difficulties are due to the non-analyticity of the renormalized disorder correlator at zero temperature, which is inherent to the physics beyond the Larkin length, characterized by many metastable states. As a result, 2-loop diagrams, which involve derivatives of the disorder correlator at the non-analytic point, are naively "ambiguous''. We examine several routes out of this dilemma, which lead to a unique renormalizable field-theory at 2-loop order. It is also the only theory consistent with the potentiality of the problem. The beta-function differs from previous work and the one at depinning by novel "anomalous terms''. For interfaces and random bond disorder we find a roughness exponent zeta = 0.20829804 epsilon + 0.006858 epsilon^2, epsilon = 4-d. For random field disorder we find zeta = epsilon/3 and compute universal amplitudes to order epsilon^2. For periodic systems we evaluate the universal amplitude of the 2-point function. We also clarify the dependence of universal amplitudes on the boundary conditions at large scale. All predictions are in good agreement with numerical and exact results, and an improvement over one loop. Finally we calculate higher correlation functions, which turn out to be equivalent to those at depinning to leading order in epsilon.

cond-mat↗

arXiv2003

The Functional Renormalization Group Treatment of Disordered Systems: a Review

We review current progress in the functional renormalization group treatment of disordered systems. After an elementary introduction into the phenomenology, we show why in the context of disordered systems a functional renormalization group treatment is necessary, contrary to pure systems, where renormalization of a single coupling constant is sufficient. This leads to a disorder distribution, which after a finite renomalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how a renormalizable field theory can be constructed, even beyond 1-loop order. We then discuss an elastic manifold imbedded in N dimensions, and give the exact solution for N -> oo. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We finally discuss depinning, both isotropic and anisotropic, and the scaling function for the width distribution of an interface.

cond-mat↗

arXiv2003

Universal interface width distributions at the depinning threshold

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independant modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows to compute the small deviations, i.e. a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.

cond-mat↗

arXiv2003

Higher correlations, universal distributions and finite size scaling in the field theory of depinning

Recently we constructed a renormalizable field theory up to two loops for the quasi-static depinning of elastic manifolds in a disordered environment. Here we explore further properties of the theory. We show how higher correlation functions of the displacement field can be computed. Drastic simplifications occur, unveiling much simpler diagrammatic rules than anticipated. This is applied to the universal scaled width-distribution. The expansion in d=4-epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator G(q)=1/q^(d+2 ζ), zeta being the roughness exponent. The deviations from this Gaussian result are small and involve higher correlation functions, which are computed here for different boundary conditions. Other universal quantities are defined and evaluated: We perform a general analysis of the stability of the fixed point. We find that the correction-to-scaling exponent is omega=-epsilon and not -epsilon/3 as used in the analysis of some simulations. A more detailed study of the upper critical dimension is given, where the roughness of interfaces grows as a power of a logarithm instead of a pure power.

cond-mat↗

arXiv2002

The 4-loop beta-function in the 2D Non-Abelian Thirring model, and comparison with its conjectured "exact" form

Recently, B. Gerganov, A. LeClair and M. Moriconi [Phys. Rev. Lett. 86 (2001) 4753] have proposed an "exact" (all orders) beta-function for 2-dimensional conformal field theories with Kac-Moody current-algebra symmetry at any level k, based on a Lie group G, which are perturbed by a current-current interaction. This theory is also known as the Non-Abelian Thirring model. We check this conjecture with an explicit calculation of the beta-function to 4-loop order, for the classical groups G= SU(N), SO(N) and SP(N). We find a contribution at 4-loop order, proportional to a higher-order group-theoretical invariant, which is incompatible with the proposed beta-function in all possible regularization schemes.

cond-mat↗