Final Activity Report Summary - FUNRG (Functional renormalization group approach to disordered systems)
The functional renormalisation group (FRG) is a promising new method which can handle systems that are complicated by the presence of multiple minima, rare events, dimensional reduction, etc. The project concentrated on a class of disordered systems which were of importance for a vast number of experimental applications but were unsatisfactorily handled by the standard methods.
Elastic objects in disordered media model the physics of diverse systems such as domain walls in magnets, charge density waves in solids, vortices in type-II superconductors, etc. Using FRG, we computed for the first time the universal distributions of threshold forces and avalanches at the depinning transition. Remarkably the threshold force distribution was almost Gaussian in the quasi-static limit, while it was governed by extreme value statistics and was given by the Gumbel distribution in the critical configuration.
Exploring the connections between the depinning problem and sandpiles we proposed a field theory for the loop-erased random walk (LERW), which received significant attention because of potential applications in combinatorics, self-organised criticality and, more recently, in conformal field theory and SLE. It was surprising that contrary to the self-avoiding walk, which was described by the O(N=0) model there was no field-theoretic approach up to now to compute the LERW fractal dimension using an expansion of around d = 4.
We also studied the influence of transverse fluctuations and disorder correlation on the statics and dynamics of elastic systems. It was known that columnar defects which were produced in superconductors by heavy ion irradiation significantly enhanced vortex pinning which prevented dissipation, and thus had a great importance for applications. We found, surprisingly, that for isotropic correlated disorder, in contrast to uncorrelated disorder, at the depinning transition a velocity-versus-force exponent could be larger than unity as is usually observed in experiments. We showed for the first time that in isotropic correlated disorder vortices could form a new 'strong' Bragg-glass phase with vanishing linear tilt modulus. Elastic systems with columnar disorder exhibited a transverse Meissner effect. Disorder generated the critical field below which there was no response to a tilt and above which the tilt angle behaved as a power-law with a universal exponent. We showed that vortices in the presence of columnar disorder could form a weakly pinned Bose-glass which shared features of the Bragg-glass, such as power-law decay of the translation order and the strong Bose-glass, namely transverse Meissner effect. Such a glassy phase was recently observed in numerical simulations.
The obtained results were used to study the surface grow which was described by the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. FRG could also be used to describe Burgers turbulence. We proposed a new method to study decaying and forced Navier-Stokes turbulence using FRG. We found that, in contrast to the Burgers turbulence, the FRG equations for the fully developed turbulence were highly non-local. The reason was that, although Burgers equation looked similar to the Navier-Stokes equation, there was a crucial difference. The velocity field was not incompressible but potentially led to an essentially different spectrum of elementary excitations.
We furthermore studied the long distance properties of the O(N) model in the presence of correlated random fields and random anisotropies. Using FRG, we obtained the phase diagram and computed the critical exponents. We found that below the lower critical dimension two different types of quasi-long-range-order with zero order-parameter but infinite correlation length could exist. This could explain the existence of two different magnetic states of He-3A in aerogel which was recently observed in experiments.
Elastic objects in disordered media model the physics of diverse systems such as domain walls in magnets, charge density waves in solids, vortices in type-II superconductors, etc. Using FRG, we computed for the first time the universal distributions of threshold forces and avalanches at the depinning transition. Remarkably the threshold force distribution was almost Gaussian in the quasi-static limit, while it was governed by extreme value statistics and was given by the Gumbel distribution in the critical configuration.
Exploring the connections between the depinning problem and sandpiles we proposed a field theory for the loop-erased random walk (LERW), which received significant attention because of potential applications in combinatorics, self-organised criticality and, more recently, in conformal field theory and SLE. It was surprising that contrary to the self-avoiding walk, which was described by the O(N=0) model there was no field-theoretic approach up to now to compute the LERW fractal dimension using an expansion of around d = 4.
We also studied the influence of transverse fluctuations and disorder correlation on the statics and dynamics of elastic systems. It was known that columnar defects which were produced in superconductors by heavy ion irradiation significantly enhanced vortex pinning which prevented dissipation, and thus had a great importance for applications. We found, surprisingly, that for isotropic correlated disorder, in contrast to uncorrelated disorder, at the depinning transition a velocity-versus-force exponent could be larger than unity as is usually observed in experiments. We showed for the first time that in isotropic correlated disorder vortices could form a new 'strong' Bragg-glass phase with vanishing linear tilt modulus. Elastic systems with columnar disorder exhibited a transverse Meissner effect. Disorder generated the critical field below which there was no response to a tilt and above which the tilt angle behaved as a power-law with a universal exponent. We showed that vortices in the presence of columnar disorder could form a weakly pinned Bose-glass which shared features of the Bragg-glass, such as power-law decay of the translation order and the strong Bose-glass, namely transverse Meissner effect. Such a glassy phase was recently observed in numerical simulations.
The obtained results were used to study the surface grow which was described by the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. FRG could also be used to describe Burgers turbulence. We proposed a new method to study decaying and forced Navier-Stokes turbulence using FRG. We found that, in contrast to the Burgers turbulence, the FRG equations for the fully developed turbulence were highly non-local. The reason was that, although Burgers equation looked similar to the Navier-Stokes equation, there was a crucial difference. The velocity field was not incompressible but potentially led to an essentially different spectrum of elementary excitations.
We furthermore studied the long distance properties of the O(N) model in the presence of correlated random fields and random anisotropies. Using FRG, we obtained the phase diagram and computed the critical exponents. We found that below the lower critical dimension two different types of quasi-long-range-order with zero order-parameter but infinite correlation length could exist. This could explain the existence of two different magnetic states of He-3A in aerogel which was recently observed in experiments.