Abstract: We show how the renormalized force correlator $Î(u)$, 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 $Î(u)$ 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: $Î(u)$ exhibits a cusp and differs from the statics. We propose that the FRG functions be measured in wetting and magnetic interfaces experiments.
Abstract: We show how the renormalized force correlator $Î(u)$, 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: $Î(u)$ exhibits a cusp and differs from the statics. We propose that the FRG functions be measured in wetting and magnetic interfaces experiments.
Abstract: 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 $Râ(u)$ 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.
Abstract: 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.
Abstract: 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.
Abstract: The Lässig-Wiese (LW) field theory for the freezing transition of random RNA secondary structures is generalized to the situation of an external force. We find a second-order phase transition at a critical applied force$f=f_c$. For $f<f_c$ forces are irrelevant. For $f>f_c$, the extension $\cal L$ as a function of pulling force $f$ scales as $\cal L(f) \sim (f-f_c)^1/γ-1$. The exponent γ is calculated in an ε-expansion: At 1-loop order $γ = 1/2$, equivalent to the disorder-free case. At 2-loop order $γ = 0.6$. Using a locking argument, we speculate that this result extends to the strong-disorder phase.
Abstract: In this note, we clarify the stability of the large-$N$ functional RG fixed points of the order/disorder transition in the random-field (RF) and random-anisotropy (RA) O($N$) models. We carefully distinguish between infinite $N$, and large but finite $N$. For infinite $N$, the Schwarz-Soffer inequality does not give a useful bound, and all fixed points found in cond-mat/0510344 (Phys. Rev. Lett. 96, 197202 (2006)) correspond to physical disorder. For large but finite $N$ (i.e. to first order in $1/N$) the non-analytic RF fixed point becomes unstable, and the disorder flows to an analytic fixed point characterized by dimensional reduction. However, for random anisotropy the fixed point remains non-analytic (i.e. exhibits a cusp) and is stable in the $1/N$ expansion, while the corresponding dimensional-reduction fixed point is unstable. In this case the Schwarz-Soffer inequality does not constrain the 2-point spin correlation. We compute the critical exponents of this new fixed point in a series in $1/N$ and to 2-loop order.
Abstract: 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 $ζ=1/3$ (one loop), $ζ=0.388\pm 0.002$ (numerics) while experiments give $ζ \approx 0.5$. 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 $ζ \approx 0.45$ (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 θ near $Ï/2$, where cubic terms in the energy vanish, is suggested.
Abstract: 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.
Abstract: 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.
Abstract: 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 $d=4+ε$ 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 $d=d_\mathrmlc$ plunges below $d_\mathrmlc=4$: we find two fixed points, one describing the quasi-ordered phase, the other is novel and describes the F/D transition. $d_\mathrmlc$ can be obtained in an $(N_c-N)$-expansion. The theory is also analyzed at large $N$ and a glassy regime is found.
Abstract: We give a pedagogical introduction into the functional renormalization group treatment of disordered systems. After a review of its 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 renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how the non-analyticity can be measured in a simulation or experiment. We then construct a renormalizable field theory beyond leading order. We discuss 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. We further consider random field magnets, and supersymmetry. We finally discuss depinning, both isotropic and anisotropic, and universal scaling function.
Abstract: We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects correlated as $\sim r^-a $ for large separation $r$. We derive the functional renormalization group equations to one-loop order which allow to describe the universal properties of the system in equilibrium and at the depinning transition. Using a double $\varepsilon=4-d$ and $δ=4-a$ expansion we compute the fixed points characterizing different universality classes and analyze their regions of stability. The long-range disorder-correlator remains analytic but generates short-range disorder whose correlator exhibits the usual cusp. The critical exponents and universal amplitudes are computed to first order in $\varepsilon$ and δ at the fixed points. At depinning a velocity-versus-force exponent β larger than unity can occur. We discuss possible realizations using extended defects.
Abstract: The field theory of self-avoiding tethered membranes still poses major challenges. In this article, we report progress on the toy-model of a manifold repelled by a single point. Our approach allows to sum the perturbation expansion in the strength $g_0$ of the interaction exactly in the limit of internal dimension $D \to 2$, yielding an analytic solution for the strong-coupling limit. This analytic solution is the starting point for an expansion in $2-D$, which aims at connecting to the well studied case of polymers ($D=1$). We give results to fourth order in $2-D$, where the dependence on $g_0$ is again summed exactly. As an application, we discuss plaquette density functions, and propose a Monte-Carlo experiment to test our results. These methods should also allow to shed light on the more complex problem of self-avoiding manifolds.
Abstract: 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 $ε=4-d$, 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 $d_\mathrmu\approx 2.5$.
Abstract: We compute the normalisation factor for the large order asymptotics of perturbation theory for the self-avoiding manifold (SAM) model describing flexible tethered ($D$-dimensional) membranes in $d$-dimensional space, and the ε-expansion for this problem. For that purpose, we develop the methods inspired from instanton calculus, that we introduced in a previous publication (Nucl.\ Phys.\ B 534 (1998) 555), and we compute the functional determinant of the fluctuations around the instanton configuration. This determinant has UV divergences and we show that the renormalized action used to make perturbation theory finite also renders the contribution of the instanton UV-finite. To compute this determinant, we develop a systematic large-$d$ expansion. For the renormalized theory, we point out problems in the interplay between the limits $ε\to 0$ and $d\to\infty$, as well as IR divergences when $ε=0$. We show that many cancellations between IR divergences occur, and argue that the remaining IR-singular term is associated to amenable non-analytic contributions in the large-$d$ limit when $ε=0$. The consistency with the standard instanton-calculus results for the self-avoiding walk is checked for $D=1$.
Abstract: 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. \noindent Dedicated to Lothar Schäfer at the occasion of his 60th birthday.
Abstract: 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 $N\to \infty$. 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.
Abstract: We study elastic systems such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of âdimensional reductionâ, 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 âambiguousâ. 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 β-function differs from previous work and the one at depinning by novel âanomalous termsâ. For interfaces and random bond disorder we find a roughness exponent $ζ=0.20829804 ε + 0.006858 ε^2$, $ε=4-d$. For random field disorder we find $ζ=ε/3$ and compute universal amplitudes to order $O(ε^2)$. 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 ε.
Abstract: Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the KPZ-term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. This mechanism has two steps: First a non-analytic disorder-distribution is generated under renormalization beyond the Larkin-length. This non-analyticity then generates the KPZ-term. We compute the β-function to one loop taking properly into account the non-analyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the non-renormalization of the KPZ-coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow-equations describe a variety of physical situations: We study manifolds in periodic disorder, relevant for charge density waves, as well as in non-periodic disorder. Further the elasticity of the manifold can either be short-range (SR) or long-range (LR). A careful analysis of the flow yields several non-trivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost-mode. For LR elasticity, relevant for contact line depinning, we show that there are two phases depending on the strength of the KPZ coupling. For SR elasticity, using the Cole-Hopf transformed theory we identify a non-trivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau-mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.
Abstract: In this letter, we report progress on the field theory of polymerized tethered membranes. For the toy-model of a manifold repelled by a single point, we are able to sum the perturbation expansion in the strength $g_0$ of the interaction exactly in the limit of internal dimension $D \to 2$. This exact solution is the starting point for an expansion in $2-D$, which aims at connecting to the well studied case of polymers ($D = 1$). We here give results to order $(2-D)^4$, where again all orders in $g_0$ are resummed. This is a first step towards a more complete solution of the self-avoiding manifold problem, which might also prove valuable for polymers.
Abstract: We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.
Abstract: 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 $N\to\infty$. 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.
Abstract: We study the two-terminal transport properties of a metallic single-walled carbon nanotube with good contacts to electrodes, which have recently been shown [W. Liang et al, Nature 441, 665-669 (2001)] to conduct ballistically with weak backscattering occurring mainly at the two contacts. The measured conductance, as a function of bias and gate voltages, shows an oscillating pattern of quantum interference. We show how such patterns can be understood and calculated, taking into account Luttinger liquid effects resulting from strong Coulomb interactions in the nanotube. We treat back-scattering in the contacts perturbatively and use the Keldysh formalism to treat non-equilibrium effects due to the non-zero bias voltage. Going beyond current experiments, we include the effects of possible ferromagnetic polarization of the leads to describe spin transport in carbon nanotubes. We thereby describe both incoherent spin injection and coherent resonant spin transport between the two leads. Spin currents can be produced in both ways, but only the latter allow this spin current to be controlled using an external gate. In all cases, the spin currents, charge currents, and magnetization of the nanotube exhibit components varying quasiperiodically with bias voltage, approximately as a superposition of periodic interference oscillations of spin- and charge-carrying âquasiparticlesâ in the nanotube, each with its own period. The amplitude of the higher-period signal is largest in single-mode quantum wires, and is somewhat suppressed in metallic nanotubes due to their sub-band degeneracy.
Abstract: Recently, B. Gerganov, A. LeClair and M. Moriconi [Phys. Rev. Lett. 86 (2001) 4753] have proposed an âexactâ (all orders) β-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 β-function to 4-loop order, for the classical groups $G = \mboxSU(N)$, $\mboxSO(N)$ and $\mboxSP(N)$. We find a contribution at 4-loop order, proportional to a higher-order group-theoretical invariant, which is incompatible with the proposed β-function in all possible regularization schemes.
Abstract: 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 $ε=4-d$. 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 Parisiâs 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.
Abstract: 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 $G(q)=1/q^d+2 ζ$; ζ, 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.
Abstract: 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 $d=4-ε$ predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator $G(q)=1/q^d+2 ζ$, ζ 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 $ømega=-ε$ and not $-ε/3$ 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.
Abstract: In this article we study the effect of a $δ $-interaction on a polymerized membrane of arbitrary internal dimension $D$. Depending on the dimensionality of membrane and embedding space, different physical scenarios are observed. We emphasize on the difference of polymers from membranes. For the latter, non-trivial contributions appear at the 2-loop level. We also exploit a âmassive schemeâ inspired by calculations in fixed dimensions for scalar field theories. Despite the fact that these calculations are only amenable numerically, we found that in the limit of $D\to 2$ each diagram can be evaluated analytically. This property extends in fact to any order in perturbation theory, allowing for a summation of all orders. This is a novel and quite surprising result. Finally, an attempt to go beyond $D=2$ is presented. Applications to the case of self-avoiding membranes are mentioned.
Abstract: We introduce a method, based on an exact calculation of the effective action at large $N$, which aims to bridge the gap between mean field theory and renormalization in complex systems. We apply it to a $d$-dimensional manifold in a random potential for large embedding space dimension $N$. This yields a functional renormalization group equation valid for any $d$, which contains both the $O(ε)$ results of Balents-Fisher and some of the non-trivial results of the Mezard-Parisi solution thus shedding light on both. Corrections are computed at order $O(1/N)$. Applications to the problems of KPZ, random field and mode coupling in glasses are mentioned.
Abstract: In this article, we review basic facts about disordered systems, especially the existence of many metastable states and and the resulting failure of dimensional reduction. Besides techniques based on the Gaussian variational method and replica-symmetry breaking (RSB), the functional renormalization group (FRG) is the only general method capable of attacking strongly disordered systems. We explain the basic ideas of the latter method and why it is difficult to implement. We finally review current progress for elastic manifolds in disorder.
Abstract: We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the non-analytic form of the dynamical action. This cures the inability of the 1-loop flow-equations to distinguish between statics and quasi-static depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the 2-loop β-function and show the generation of âirreversibleâ anomalous terms, originating from the non-analytic nature of the theory, which cause the statics and driven dynamics to differ at 2-loop order. We obtain the roughness exponent ζ and dynamical exponent $z$ to order $ε^2$. This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that random-field disorder does indeed attract all disorder of shorter range. It also shows that the conjecture $ζ=ε/3$ is incorrect, and allows to compute the violations, as $ζ=\fracε3(1 + 0.14331 ε)$, $ε=4-d$. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields $ζ=\fracε3(1 + 0.39735 ε)$, $ε=2-d$ (vs.\ the standard prediction $ ζ=1/3$ for $d=1$), in reasonable agreement with the most recent simulations. The high value of $ζ\approx 0.5$ found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.
Abstract: We introduce a method, based on an exact calculation of the effective action at large $N$, which aims to bridge the gap between mean field theory and renormalization in complex systems. We apply it to a $d$-dimensional manifold in a random potential for large embedding space dimension $N$. This yields a functional renormalization group equation valid for any $d$, which contains both the $O(ε)$ results of Balents-Fisher and some of the non-trivial results of the Mezard-Parisi solution thus shedding light on both. Corrections are computed at order $O(1/N)$. Applications to the problems of KPZ, random field and mode coupling in glasses are mentioned.
Abstract: We examine, beyond one loop, the candidate field theories for equilibrium and driven dynamics of elastic systems pinned by disorder. To escape dimensional reduction, the action is non-analytic at T=0. We show two-loop renormalizability of (quasi-static) depinning and compute roughness ζ and dynamical exponents $z$ for periodic systems and interfaces. We prove that random field disorder attracts shorter range disorder and find a $\cal O(ε^2)$ violation of the conjecture $ζ=ε/3$, in agreement with simulations. We then discuss the issues arising in the statics. Depinning and static β-functions differ at two loop and contain novel anomalous terms due to the non-analytic nature of the theory
Abstract: In this article, we introduce a generalization of the diffusive motion of point-particles in a turbulent convective flow with given correlations to a polymer or membrane. In analogy to the passive scalar problem we call this the passive polymer or membrane problem. We shall focus on the expansion about the marginal limit of velocity-velocity correlations which are uncorrelated in time and grow with the distance $x$ as $|x|^ε$, and ε small. This relation gets modified for polymers and membranes (the marginal advecting flow has correlations which are shorter ranged.) The construction is done in three steps: First, we reconsider the treatment of the passive scalar problem using the most convenient treatment via field theory and renormalization group. We explicitly show why IR-divergences and thus the system-size appear in physical observables, which is rather unusual in the context of ordinary field-theories, like the $Ï^4$-model. We also discuss, why the renormalization group can nevertheless be used to sum these divergences and leads to anomalous scaling of 2$n$-point correlation functions as e.g.\ $S^2n(x) := \left<[ Î(x,t) - Î(0,t) ]^2n \right>$. In a second step, we reformulate the problem in terms of a Langevin equation. This is interesting in its own, since it allows for a distinction between single-particle and multi-particle contributions, which is not obvious in the Focker-Planck treatment. It also gives an efficient algorithm to determine $S^2n$ numerically, by measuring the diffusion of particles in a random velocity field. In a third and final step, we generalize the Langevin treatment of a particle to polymers and membranes, or more generally to an elastic object of inner dimension $D$ with $0\le D \le 2$. These objects can intersect each other. We also analyze what happens when self-intersections are no longer allowed.
Abstract: We study the dynamics of a polymer or a $D$-dimensional elastic manifold diffusing and convected in a non-potential static random flow (the ârandomly driven polymer modelâ). We find that short-range (SR) disorder is relevant for $d < 4$ for directed polymers (each monomer sees a different flow) and for $d < 6$ for isotropic polymers (each monomer sees the same flow) and more generally for $d<d_c(D)$ in the case of a manifold. This leads to new large scale behavior, which we analyze using field theoretical methods. We show that all divergences can be absorbed in multilocal counter-terms which we compute to one loop order. We obtain the non trivial roughness ζ, dynamical $z$ and transport exponents Ï in a dimensional expansion. For directed polymers we find ζ about 0.63 ($d=3$), ζ about 0.8 ($d=2$) and for isotropic polymers ζ about 0.8 ($d=3$). In all cases $z>2$ and the velocity versus applied force characteristics is sublinear, i.e.\ at small forces $v(f)\sim f^Ï$ with $Ï > 1$. It indicates that this new state is glassy, with dynamically generated barriers leading to trapping, even by a divergenceless (transversal) flow. For random flows with long-range (LR) correlations, we find continuously varying exponents with the ratio $g_L/g_T$ of potential to transversal disorder, and interesting crossover phenomena between LR and SR behavior. For isotropic polymers new effects (e.g.\ a sign change of $ζ -ζ_0$) result from the competition between localization and stretching by the flow. In contrast to purely potential disorder, where the dynamics gets frozen, here the dynamical exponent $z$ is not much larger than 2, making it easily accessible by simulations. The phenomenon of pinning by transversal disorder is further demonstrated using a two monomer âdumbbellâ toy model.
Abstract: We present a simple argument to show that the beta-function of the $d$-dimensional KPZ-equation ($d\ge 2$) is to all orders in perturbation theory given by $β(g) = (d-2) g - 2(8 Ï)^-d/2 Î(2-d/2) g^2$ . Neither the dynamical exponent $z$ nor the roughness-exponent ζ have any correction in any order of perturbation theory. This shows that standard perturbation theory cannot attain the strong-coupling regime and in addition breaks down at $d=4$. We also calculate a class of correlation-functions exactly.
Abstract: We study the dynamics of polymers and elastic manifolds in non potential static random flows. We find that barriers are generated from combined effects of elasticity, disorder and thermal fluctuations. This leads to glassy trapping even in pure barrier-free divergenceless flows $v \stackrelf \to 0\sim f^Ï$ ($Ï > 1$). The physics is described by a new RG fixed point at finite temperature. We compute the anomalous roughness $R \sim L^ζ$ and dynamical $t\sim L^z$ exponents for directed and isotropic manifolds.
Abstract: We discuss an elementary problem in electrostatics: What does the charge distribution look like for a free charge on a strictly one-dimensional wire of finite length? To the best of our knowledge this question has so far not been discussed anywhere. One notices that a solution of this problem is not as simple as it might appear at first sight.
Abstract: We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The $n$-th order is shown to have factorial growth as $(-\mboxcst)^n (n!)^1-ε/D$, where $D$ is the âinternalâ dimension of the membrane and ε the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension $d$ of bulk space. This is the starting point of a systematic $1/d$ expansion. As a consequence, the epsilon-expansion of self-avoiding membranes has a factorial growth, like the epsilon-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.
Abstract: We introduce a generalization of the $O(N)$ field theory to $N$-colored membranes of arbitrary inner dimension $D$. The $O(N)$ model is obtained for $D\to 1$, while $N\to 0$ leads to self-avoiding tethered membranes (as the $O(N)$ model reduces to self-avoiding polymers). The model is studied perturbatively by a 1-loop renormalization group analysis, and exactly as $N\to\infty$. Freedom to choose the expansion point $D$, leads to precise estimates of critical exponents of the $O(N)$ model. Insights gained from this generalization include a conjecture on the nature of droplets dominating the 3d-Ising model at criticality; and the fixed point governing the random bond Ising model.
Abstract: The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension $D$ are studied using model A dynamics. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent $z$ is given by $z=2+D/ν$. This result applies especially to membranes ($D=2$) but also to polymers ($D=1$), for which this scaling relation had been suggested but not proven.
Abstract: The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension $D$ embedded into $d$ dimensions are studied including hydrodynamical interactions. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent $z$ is given by $z=d$. The crossover to the region, where the membrane is crumpled swollen but the hydrodynamic interaction irrelevant is discussed. The results apply as well to polymers ($D=1$) as to membranes ($D=2$).
Abstract: We introduce a geometric generalization of the $O(N)$-field theory that describes $N$-colored membranes with arbitrary dimension $D$. As the $O(N)$-model reduces in the limit $N\to 0$ to self-avoiding polymers, the $N$-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension $D\to 1$, the manifold model reduces to the $O(N)$-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large $N$ limit and Hartree approximation, can also be applied to this model. By comparison of low and high temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d-Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in $d=3$. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.
Abstract: The scaling properties of self-avoiding polymerized 2-dimensional membranes are studied via renormalization group methods based on a multilocal operator product expansion. The renormalization group functions are calculated to second order. This yields the scaling exponent ν to order $ε^2$. Our extrapolations for ν agree with the Gaussian variational estimate for large space dimension $d$ and are close to the Flory estimate for $d=3$. The interplay between self-avoidance and rigidity at small $d$ is briefly discussed.
Abstract: In this article, we perform a careful analysis of the renormalization procedure used in existing calculations to derive critical exponents for the KPZ-equation at 2-loop order. This analysis explains the discrepancies between the results of the different groups. The correct critical exponents in $d=2+ε$ dimensions at the crossover between weak- and strong-coupling regime are $ζ=O(ε^3)$ and $z=2+O(ε^3)$. No strong-coupling fixed point exists at 2-loop order.
Abstract: The scaling properties of selfavoiding polymerized membranes are studied using renormalization group methods. The scaling exponent ν is calculated for the first time at two loop order. ν is found to agree with the Gaussian variational estimate for large space dimension $d$ and to be close to the Flory estimate for $d=3$.
Abstract: A complete classification of the renormalization-group flow is given for impurity-like marginal operators of membranes whose elastic stress scales like $(Î r)^2$ around the external critical dimension $d_c=2$. These operators are classified by characteristic functions on $\mathbbR^2 \times \mathbbR^2$.
Abstract: The scaling properties of self-avoiding tethered membranes at the tricritical point (theta-point) are studied by perturbative renormalization group methods. To treat the 3-body repulsive interaction (known to be relevant for polymers), new analytical and numerical tools are developped and applied to 1-loop calculations. These technics are a prerequisite to higher order calculations for self-avoiding membranes. The cross-over between the 3-body interaction and the modified 2-body interaction, attractive at long range, is studied through a new double epsilon-expansion. It is shown that the latter interaction is relevant for 2-dimensional membranes at the theta-point.
Abstract: The renormalization ζ-function for supersymmetric nonlinear sigma-models is calculated up to three-loop order. For a wide class of models, which includes the $N$-vector model and matrix models, the result can be summarized as follows: If the ζ-function for the bosonic model is $ζ_\mathrmBos(t_R)=at_R +O(t_R^ 2)$, then the ζ-function for the supersymmetric model takes the form $ζ_ \mathrmSUSY(t_R)=at_R+O(t_R^4)$. This is the case for arbitrary harmonic polynomials of the field variables (so called âsoft operatorsâ).
Abstract: Die Renormierungsgruppen ζ-Funktion für supersymmetrischeh nichtlineare Sigma-Modelle wird berechnet bis zur 3-loop Ordnung. For eine groà e Klasse von Modellen, eingeschlossen das $N$-Vector Modell und Matrix-Modelle, kann das Resultat wie folgt zusammengefasst werden: Wenn die ζ-Funktion für das bosonische Modell $ζ_\mathrmBos(t_R)=a t_R +O(t_R^2)$ ist, dann besitzt die ζ-Funktion für das supersymmetrische Modell die Form $ζ_\mathrmSusy= a t_R + O(t_R^4)$. Dieses Resultat ist gültig für beliebige harmonische Polynome des Feldes (sog.\ âWeiche Operatorenâ).
Abstract: In these lecture notes, we give an overview about non-local field-theories and their application to polymerized membranes, i.e.\ membranes with a fixed internal connectivity. The main technical tool is the multi-local operator product expansion (MOPE), generalizing ideas from local field theories to the multi-local situation. Dedicated to Hagen Kleinert at the occasion of his 60th birthday.
Abstract: Membranes are of great technological and biological as well as theoretical interest. Two main classes of membranes can be distinguished: Fluid membranes and polymerized, tethered membranes. Here, we review progress in the theoretical understanding of polymerized membranes, i.e.\ membranes with a fixed internal connectivity. We start by collecting basic physical properties, clarifying the role of bending rigidity and disorder, theoretically and experimentally as well as numerically. We then give a thorough introduction into the theory of self-avoiding membranes, or more generally non-local field theories with δ-like interactions. A couple of tools is developed. Based on a proof of perturbative renormalizability for non-local field-theories, renormalization group calculations can be performed up to 2-loop order, which in 3 dimensions predict a crumpled phase with fractal dimension of about 2.4; this phase is however seemingly unstable towards the inclusion of bending rigidity. The tricritical behavior of membranes is discussed and shown to be quite different from that of polymers. Dynamical properties are studied in the same frame-work. Exact scaling relations, suggested but not demonstrated long time ago by De Gennes for polymers, are established. Along the same lines, disorder can be included leading to interesting applications. We also construct a generalization of the $O(N)$-model, which in the limit $N\to0$ reduces to self-avoiding membranes in analogy with the $O(N)$-model, which in the limit $N\to0$ reduces to self-avoiding polymers. Since perturbation theory is at the basis of the above approach, one has to ensure that the perturbation expansion is not divergent or at least Borel-summable. Using a suitable reformulation of the problem, we obtain the instanton governing the large-order behavior. This suggests that the perturbation expansion is indeed Borel-summable and the presented approach meaningful. Some technical details are relegated to the appendices. A final collection of various topics may also serve as exercises.
