Randomly Pinned Systems: Interfaces and Luttinger Liquids

The present thesis adresses three different topics that are all in direct connection to the statistical physics of disordered systems. The first chapter, which is the main part of this work, is devoted to driven interfaces in random media. The main focus is on the case of an ac driving. Our investigations start with the associated mean field theory. At the beginning we consider the case of a constant driving force and discuss the critical behaviour close to the depinning transition. Subsequently, we examine the velocity hysteresis curve for small frequencies by use of analytical and numerical methods. It turns out that all observables under consideration exhibit scaling behaviour on approaching the adiabatic case. Next, we analyse perturbation theory for ac-driven interfaces. Tt turns out, that perturbation theory can only work for sufficiently high interface dimensionalities D>4. The failure of perturbation theory in low dimensions is discussed. The second part of this thesis is concerned with the influence of electroelastic coupling on the behaviour of domain walls in ferroelectrics. For the case of electrostrictive coupling in the high temperature phase we find, that merely the interface tension coefficients are changed in an anisotropic fashion. On the contrary, a piezo effect in the paraphase entails a long range term. After deriving the interface hamiltonian in the harmonic approximation, we determine the roughness of domain walls due to random field disorder using an Imry-Ma type argument. On large length scales, domain walls turn out to be flat. In the last chapter, we consider one-dimensional electronic systems, so-called Luttinger liquids. The subject of the first part is the competition of two phases of one-dimensional disordered fermionic systems, namely the Mott and the Anderson insulator. By relating both, the compressibility as well as a finite optical conductivity to a vanishing mass of charged excitations, the existence of an intermediate Mott glass phase can be excluded. Combining informations about the renormalisation group flow and the energy of topological excitations, as well as using simple scaling arguments, we construct the phase diagram. In the second part we study the replica trick to describe the quantum creep and the linear conductivity at finite temperatures for Luttinger liquids with relevant disorder. We provide approximate solutions to the Euler-Lagrange equations for the replica action. The replica limit turns out to be problematic. At the example of a toy model we illustrate a method for the replica limit and demonstrate the emergent difficulties

Mean field theory for driven domain walls in disordered environments

We study the mean field equation of motion for driven domain walls in random media. We discuss the two cases of an external constant as well as an oscillating driving force. Our main focus lies on the critical dynamics close to the depinning transition, which we study by analytical and numerical methods. We find power-law scaling for the velocity as well as the hysteresis loop area.Comment: 16 pages, 19 figures, submitted to Phys. Rev.

The Effect of Randomness on the Mott State

We reinvestigate the competition between the Mott and the Anderson insulator state in a one-dimensional disordered fermionic system by a combination of instanton and renormalization group methods. Tracing back both the compressibility and the ac-conductivity to a vanishing kink energy of the electronic displacement field we do not find any indication for the existence of an intermediate (Mott glass) phase.Comment: 4 page

Perturbation theory for ac-driven interfaces in random media

We study $D$-dimensional elastic manifolds driven by ac-forces in a disordered environment using a perturbation expansion in the disorder strength and the mean-field approximation. We find, that for $D\le 4$ perturbation theory produces non-regular terms that grow unboundedly in time. The origin of these non-regular terms is explained. By using a graphical representation we argue that the perturbation expansion is regular to all orders for $D>4$. Moreover, for the corresponding mean-field problem we prove that ill-behaved diagrams can be resummed in a way, that their unbounded parts mutually cancel. Our analytical results are supported by numerical investigations. Furthermore, we conjecture the scaling of the Fourier coefficients of the mean velocity with the amplitude of the driving force $h$.Comment: 23 pages, substantial changes, replaced with the published versio