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Disordered elastic systems: Fluctuations and AC-dynamics

Andreas Glatz

Published Jan 1, 2004• 1 citation
OpenAlex

Abstract

Disordered elastic systems are studied in the quantum, classical, and dynamic regime in the present thesis. The thesis consists of two main parts: In the first part we study the low temperature phase diagram of one-dimensional weakly disordered quantum systems like charge or spin density waves and Luttinger liquids by a full finite temperature renormalization group (RG) calculation. In the classical region, for vanishing quantum fluctuations those results are supplemented by an exact solution of the model in the case of strong disorder, described by the ground state and the correlation function. Furthermore, by a mapping of the problem onto a Burgers equation with noise, in the case of weak disorder, we can derive an expression for the correlation length. At zero temperature we reproduce the (quantum) phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wave length of the phason excitations. The existence of a zero temperature transition is reflected in a rich cross-over phase diagram determined by the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential, and for the case of density waves we discuss the influence of quantum phase-slips, which might lead to a new scenario for the unpinning (delocalization) transition at zero temperature. Additionally, we analyze the current noise in a classical one-dimensional charge density wave system in the weak pinning regime by solving the overdamped equation of motion numerically. At low temperatures and just above the zero temperature depinning threshold, the power spectrum of the current noise reveals the existence of 1/f or flicker noise. Our results are in agreement with experimental measurements in quasi-one-dimensional charge density wave systems. In part two, we examine the viscous motion of an interface driven by a periodically oscillating external field in a random medium. The velocity exhibits a smeared depinning transition showing a pronounced hysteresis, which is absent in the adiabatic case. Its width is determined by a new length scale, introduced by the frequency of the external drive and - in the low frequency regime - by the critical exponents of the zero frequency depinning transition. Scaling arguments and an approximate renormalization group treatment predict a power law behavior of the velocity as a function of frequency and field amplitude at the dc depinning threshold, which is confirmed numerically. Thermal fluctuations lead to an additional smearing of the transition. If the amplitude of the external driving force is smaller than the depinning field, the motion of the interface by avalanches has to be taken into account in order to describe the velocity hysteresis. To check the validity of our model with respect to experimental results, we calculate the complex susceptibilities using an adiabatic and non-adiabatic approach numerically for finite systems and compare it to measurements of a superferromagnetic granular multilayer system and find that the model of an ac driven interface in a disordered environment can describe the main features of the experimental results. Additionally, the influence of a strong surface potential on the critical depinning of an elastic system, driven in a random medium is considered. If the surface potential prevents depinning completely, the curvature of the displacement profile at zero temperature exhibits a pronounced rhombic hysteresis curve with width of two times the (bulk) depinning field. The hysteresis disappears at non-zero temperatures if the driving force is changed adiabatically. If the surface depinns by the applied force or thermal creep, the curvature is reduced with increasing velocity. The results apply, e.g., to driven magnetic domain walls, flux-line lattices, and charge-density waves. As addendum we examine low dimensional interacting, but clean Bose systems at low temperatures. The interaction leads to scattering events of the over-condensate particles for which we calculate the scattering times using Fermi's golden rule. With that we derive the thermal conductivity and the related weak localization corrections. This is done for short and long-range interactions.

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Disordered elastic systems are studied in the quantum, classical, and dynamic regime in the present thesis. The thesis consists of two main parts: In the first part we study the low temperature phase diagram of one-dimensional weakly disordered quantum systems like charge or spin density waves and Luttinger liquids by a full finite temperature renormalization group (RG) calculation. In the classical region, for vanishing quantum fluctuations those results are supplemented by an exact solution of the model in the case of strong disorder, described by the ground state and the correlation function. Furthermore, by a mapping of the problem onto a Burgers equation with noise, in the case of weak disorder, we can derive an expression for the correlation length. At zero temperature we reproduce the (quantum) phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wave length of the phason excitations. The existence of a zero temperature transition is reflected in a rich cross-over phase diagram determined by the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential, and for the case of density waves we discuss the influence of quantum phase-slips, which might lead to a new scenario for the unpinning (delocalization) transition at zero temperature. Additionally, we analyze the current noise in a classical one-dimensional charge density wave system in the weak pinning regime by solving the overdamped equation of motion numerically. At low temperatures and just above the zero temperature depinning threshold, the power spectrum of the current noise reveals the existence of 1/f or flicker noise. Our results are in agreement with experimental measurements in quasi-one-dimensional charge density wave systems. In part two, we examine the viscous motion of an interface driven by a periodically oscillating external field in a random medium. The velocity exhibits a smeared depinning transition showing a pronounced hysteresis, which is absent in the adiabatic case. Its width is determined by a new length scale, introduced by the frequency of the external drive and - in the low frequency regime - by the critical exponents of the zero frequency depinning transition. Scaling arguments and an approximate renormalization group treatment predict a power law behavior of the velocity as a function of frequency and field amplitude at the dc depinning threshold, which is confirmed numerically. Thermal fluctuations lead to an additional smearing of the transition. If the amplitude of the external driving force is smaller than the depinning field, the motion of the interface by avalanches has to be taken into account in order to describe the velocity hysteresis. To check the validity of our model with respect to experimental results, we calculate the complex susceptibilities using an adiabatic and non-adiabatic approach numerically for finite systems and compare it to measurements of a superferromagnetic granular multilayer system and find that the model of an ac driven interface in a disordered environment can describe the main features of the experimental results. Additionally, the influence of a strong surface potential on the critical depinning of an elastic system, driven in a random medium is considered. If the surface potential prevents depinning completely, the curvature of the displacement profile at zero temperature exhibits a pronounced rhombic hysteresis curve with width of two times the (bulk) depinning field. The hysteresis disappears at non-zero temperatures if the driving force is changed adiabatically. If the surface depinns by the applied force or thermal creep, the curvature is reduced with increasing velocity. The results apply, e.g., to driven magnetic domain walls, flux-line lattices, and charge-density waves. As addendum we examine low dimensional interacting, but clean Bose systems at low temperatures. The interaction leads to scattering events of the over-condensate particles for which we calculate the scattering times using Fermi's golden rule. With that we derive the thermal conductivity and the related weak localization corrections. This is done for short and long-range interactions.
The low-temperature phase diagram of one-dimensional weakly disordered quantum systems such as charge or spin density waves and Luttinger liquids is studied by a full finite temperature renormalization-group (RG) calculation. For vanishing quantum fluctuations this approach is amended by an exact solution in the case of strong disorder and by a mapping onto the Burgers equation with noise in the case of weak disorder, respectively. At zero temperature we reproduce the quantum phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wavelength of the phason excitations. The existence of a zero-temperature transition is reflected in a rich crossover phase diagram of the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential. Applications to related systems are discussed as well.
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.
The low temperature phase diagram of 1D disordered quantum systems such as charge or spin density waves, superfluids, and related systems is considered by a full finite- T renormalization group approach for the first time. At zero temperature the consideration of quantum phase slips leads to a new scenario for the unpinning (delocalization) transition. In the strong pinning limit the model is solved exactly. At finite T a rich crossover diagram with various scaling regions is found which reflects the zero temperature quantum critical behavior.
The low temperature phase diagram of 1D disordered quantum systems like charge or spin density waves, superfluids, and related systems is considered by a full finite T renormalization group approach for the first time. At zero temperature the consideration of quantum phase slips leads to a new scenario for the unpinning (delocalization) transition. In the strong pinning limit the model is solved exactly. At finite T a rich crossover diagram with various scaling regions is found which reflects the zero temperature quantum critical behavior.
Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ −1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.
In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.