This PhD thesis has the following structure: Chapter 1 - General introduction; Chapter 2 - Preliminaries; Chapter 3 - The Replicated Transfer Matrix; Chapter 4 - Finite Size Corrections On Random Graphs; Chapter 5 - The Random Field Ising Model; Chapter 6 - The Euclidean Assignment Problem; Chapter 7 - The Euclidean Matching Problem; Chapter 8 - The Large M Expansion.
We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density.
We derive the analytical expression for the first finite size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree.
We evaluate the electron-electron interaction corrections to the electronic thermal conductivity in a disordered conductor in the diffusive regime. We use a diagrammatic many-body method analogous to that of Altshuler and Aronov for the electrical conductivity. We derive results in one, two and three dimensions for both the singlet and triplet channels, and in all cases find that the Wiedemann-Franz law is violated.
Using the mapping of the partition function of the two-dimensional Ising model onto a pfaffian we evaluate the domain wall free energy difference for the pure and disordered Ising model close to the pure fixed point. Using this method very large lattices can be studied exactly and we confirm that disorder even including frustrating interactions indeed are irrelevant close to the pure fixed point. The finite-size renormalization group flow shows a power-law behavior modified by a logarithmic term that dominates for small lattice sizes.
Anderson localization has been studied extensively for more than half a century. However, while our understanding has been greatly enhanced by calculations based on a small epsilon expansion in d = 2 + epsilon dimensions in the framework of non-linear sigma models, those results can not be safely extrapolated to d = 3. Here we calculate the leading scale-dependent correction to the frequency-dependent conductivity sigma(omega) in dimensions d <= 3. At d = 3 we find a leading correction Re{sigma(omega)} ~ |omega|, which at low frequency is much larger than the omega^2 correction deriving from t…
The electrical {\em dc}-conductivity of disordered, non-interacting electrons is calculated in the asymptotic limit of high lattice dimensions $d\to \infty$. To go beyond the lowest-order contribution in the expansion parameter $1/d$ of the single bubble diagram, vertex corrections are calculated from an asymptotic expression for the two-particle vertex. A mean-field approximation for the dc-conductivity containing the leading high-dimensional vertex corrections is proposed which is free of spurious non-analyticities, i.e. the conductivity is non-negative and shows no unphysical behavior in $d…
The mean free path is an essential characteristic length in disordered systems. In microscopic calculations, it is usually approximated by the classical value of the elastic mean free path. It corresponds to the Boltzmann mean free path when only isotropic scattering is considered, but it is different for anisotropic scattering. In this paper, we work out the corrections to the so called Boltzmann mean free path due to multiple scattering effects on finite size scatterers, in the s-wave approximation, ie. when the elastic mean free path is equivalent to the Boltzmann mean free path. The main r…
The electron-electron interaction corrections to the transport coefficients are calculated for a two-dimensional disordered metal in a parallel magnetic field via the quantum kinetic equation approach. For the thermal transport, three regimes (diffusive, quasiballistic and truly ballistic) can be identified as the temperature increases. For the diffusive and quasiballistic regimes, the Lorentz number dependence on the temperature and on the magnetic field is studied. The electron-electron interactions induce deviations from the Wiedemann-Franz law, whose sign depend on the temperature: at low …
We study the behavior of a quantum dipolar Bose condensate with the Lee-Huang-Yang quantum corrections at zero temperature in the presence of weak disorder potential. We solve the underlying nonlocal Gross-Pitaevskii equation using a perturbative theory. The role of both the Lee-Huang-Yang term and the disorder potential on the condensed fraction, the equation of state, the compressibility and the superfluid density is deeply analyzed. Surprisingly, we find that the Lee-Huang-Yang quantum corrections not only arrest the dipolar implosion but also lead to reduce the condensate fluctuations due …
Mean-field theory of non-interacting disordered electron systems is widely and successfully used to describe equilibrium properties of alloys in the whole range of disorder strengths. It, however, fails to take into account effects of quantum coherence and localizing back-scattering effects when applied to transport phenomena. We present an approximate scheme extending the mean-field theory for one-electron properties in that it offers a formula for the two-particle vertex and the electrical conductivity non-perturbatively including the leading-order vertex corrections in a way that the approx…
The nonlinear behavior of the Hall resistivity at low magnetic fields in single quantum well GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures with degenerated electron gas is studied. It has been found that this anomaly is accompanied by the weaker temperature dependence of the conductivity as compared with that predicted by the first-order theory of the quantum corrections to the conductivity. We show that both effects in strongly disordered systems stem from the second order quantum correction caused by the effect of weak localization on the interaction correction and vice versa. This correctio…
For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: ν_{FS}<= 2/d. It is believed that the true critical exponent νof a disorder induced phase transition satisfies the same bound. We argue that in disordered systems the standard averaging introduces a noise, and a corresponding new diverging length scale, characterized by ν_{FS}=2/d. This length scale, however, is independent of the system's own correlation length ξ. Therefore νcan be less than 2/d. We illustrate these ideas on two exact examples, with ν< 2/d. We propose…
We investigate the deviation of the level-correlation functions from the universal form for the chiral symmetric classes. Using the supersymmetric nonlinear sigma model we formulate the perturbation theory. The large energy behavior is compared with the result of the diagrammatic perturbation theory. We have the diffuson and cooperon contributions even in the average density of states. For the unitary and orthogonal classes we get the small energy behavior that suggests a weakening of the level repulsion. For the symplectic case we get a result with opposite tendency.
For disordered elastic manifolds in the ground state (equilibrium) we obtain the critical exponents for the roughness and the correction-to-scaling up to 3-loop order, i.e. third order in $ε=4-d$, where $d$ is the internal dimension $d$. We also give the full 2-point function up to order $ε^{2}$, i.e. at 2-loop order.
We consider the singular electron-electron interaction corrections to the transport coefficients in disordered metals to test the validity of the Wiedemann-Franz law. We develop a local, quantum kinetic equation approach in which the charge and energy conservation laws are explicitly obeyed. To obtain the local description, we introduce bosonic distribution functions for the neutral, low-energy collective modes (electron-hole pairs). The resulting system of kinetic equations enables us to distinguish between the different physical processes involved in the charge and energy transport: elastic …
We study the finite size fluctuations at the depinning transition for a one-dimensional elastic interface of size $L$ displacing in a disordered medium of transverse size $M=k L^ζ$ with periodic boundary conditions, where $ζ$ is the depinning roughness exponent and $k$ is a finite aspect ratio parameter. We focus on the crossover from the infinitely narrow ($k\to 0$) to the infinitely wide ($k\to \infty$) medium. We find that at the thermodynamic limit both the value of the critical force and the precise behavior of the velocity-force characteristics are {\it unique} and $k$-independent. We al…
A Bose-Einstein condensate in an external potential consisting of a superposition of a harmonic and a random potential is considered theoretically. From a semi-quantitative analysis we find the size, shape and excitation energy as a function of the disorder strength. For positive scattering length and sufficiently strong disorder the condensate decays into fragments each of the size of the Larkin length ${\cal L}$. This state is stable over a large range of particle numbers. The frequency of the breathing mode scales as $1/{\cal L}^2$. For negative scattering length a condensate of size ${\cal…
This paper is intended to demonstrate that there is no need to revise the existing theory of the transport properties of disordered conductors in the so-called weak localization regime. In particular, we demonstrate explicitly that recent attempts to justify theoretically that the dephasing rate (extracted from the magnetoresistance) remains finite at zero temperature are based on the profoundly incorrect calculation. This demonstration is based on a straightforward evaluation of the effect of the electron-electron interaction on the weak localization correction to the conductivity of disorder…
We study the effects of quenched disorder on the first-order phase transition in the two-dimensional three-color Ashkin-Teller model by means of large-scale Monte Carlo simulations. We demonstrate that the first-order phase transition is rounded by the disorder and turns into a continuous one. Using a careful finite-size-scaling analysis, we provide strong evidence for the emerging critical behavior of the disordered Ashkin-Teller model to be in the clean two-dimensional Ising universality class, accompanied by universal logarithmic corrections. This agrees with perturbative renormalization-gr…
The diagrammatic approach is applied to study quasiparticle transport properties in two-dimensional d-wave superconductors with dilute nonmagnetic impurities both in Born and in unitary limits. It is found that a novel quantum interference process gives rise to a weak-antilocalization correction to the spin conductivity, indicating the existence of extended low-energy quasiparticle states. With comimg close to unitarity and the nesting, this correction is suppressed and eventually vanishes due to the global particle-hole symmetry.
The DC conductivity of monolayer and bilayer graphene is studied perturbatively for different types of disorder. In the case of monolayer, an exact cancellation of logarithmic divergences occurs for all disorder types. The total conductivity correction for a random vector potential is zero, while for a random scalar potential and a random gap it acquires finite corrections. We identify the diagrams which are responsible for these corrections and extrapolate the finite contributions to higher orders which gives us general expressions for the conductivity of weakly disordered monolayer graphene.…
We present a microscopic derivation of self-consistent equations of Anderson localization in a disordered medium of finite size. The derivation leads to a renormalized, position-dependent diffusion coefficient. The position dependence of the latter is due to the position dependence of return probability in a bounded medium.
It is well known that conductivity of disordered metals is suppressed in the limit of low frequencies and temperatures by quantum corrections. Although predicted by theory to exist up to much higher energies, such corrections have so far been experimentally proven only for $\lesssim$80 meV. Here, by a combination of transport and optical studies, we demonstrate that the quantum corrections are present in strongly disordered conductor MoC up to at least $\sim$4 eV, thereby extending the experimental window where such corrections were found by a factor of 50. The knowledge of both, the real and …
We calculate the first-order perturbative correction to the transition temperature $T_c$ in a superconductor with both non-magnetic and magnetic impurities. We do this by first evaluating the correction to the effective potential, $Ω(Δ)$, and then obtain the first-order correction to the order parameter, $Δ$, by finding the minimum of $Ω(Δ)$. Setting $Δ=0$ finally allows $T_c$ to be evaluated. $T_c$ is now a function of both the resistance per square, $R_\square$, a measure of the non-magnetic disorder, and the spin-flip scattering rate, $1/τ_s$, a measure of the magnetic disorder. We find tha…
We study the density of states (DOS) in band gaps of ideal and disordered three-dimensional photonic crystals of finite size. The ideal crystal is a diamond lattice of resonant point scatterers (atoms) whereas the disordered one is obtained from it by displacing the scatterers by random distances in random directions. We find that DOS inside a band gap of the ideal crystal decreases as the inverse of the crystal size. Disorder narrows the band gap and DOS exhibits enhanced fluctuations near the new band edges. However, the average DOS still exhibits the same scaling with the crystal size withi…
We calculate the corrections to the conductivity and compressibility of a disordered metal when the mean free path is smaller than the screening length. Such a condition is shown to be realized for low densities and large disorder. Analysis of the stability of the metallic state reveals a transition to the insulating state in two-dimensions.
We study the problem of disordered interacting bosons within grand-canonical thermodynamics and Bogoliubov theory. We compute the fractions of condensed and non-condensed particles and corrections to the compressibility and the speed of sound due to interaction and disorder. There are two small parameters, the disorder strength compared to the chemical potential and the dilute-gas parameter.
We simulated site dilute Ising models in $d=3$ dimensions for several lattice sizes $L$. For each $L$ singular thermodynamic quantities $X$ were measured at criticality and their distributions $P(X)$ were determined, for ensembles of several thousand random samples. For $L \to \infty$ the width of $P(X)$ tends to a universal constant, i.e. there is no self averaging. The width of the distribution of the sample dependent pseudocritical temperatures $T_c(i,L)$ scales as $δT_c(L) \sim L^{-1/ν}$ and NOT as $\sim L^{-d/2}$. Finite size scaling holds; the sample dependence of $X_i(T_c)$ enters predo…
The localization in a disordered system of $N$ interacting spins coupled by the long-range anisotropic interaction $1/R^α$ is investigated using a finite size scaling in a $d=1$ -dimensional system for $N=8, 10, 12, 14$. The results supports the absence of localization in the infinite system at $α<2d$ and a scaling of a critical energy disordering $W_{c} \propto N^{2d-α}$ in agreement with the analytical theory suggesting the energy delocalization in the subset of interacting resonant pairs of spins as a precursor of the many-body delocalization.The spin relaxation rate $k$ dependence on disor…
We investigate the growth of the entanglement entropy $S_{\textrm{ent}}$ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case we show that an intermediate weak localization regime exists in which $S_{\textrm{ent}}(t)$ grows logarithmically in time $t$ before Anderson localization sets in. For the case of binary bond disorder near the percolation transition we find additive logarithmic corrections to area and volume laws as well as a scaling at long times which is consistent with an infinite randomness fixed point.
We studied lasing modes in a disordered photonic crystal. The scaling of the lasing threshold with the system size depends on the strength of disorder. For sufficiently large size, the minimum of the lasing threshold occurs at some finite value of disorder strength. The highest random cavity quality factor was comparable to that of an intentionally introduced single defect. At the minimum, the lasing threshold showed a super-exponential decrease with the size of the system. We explain it through a migration of the lasing mode frequencies toward the photonic bandgap center, where the localizati…
Weak localization leads to the same correction to both the conductivity and the McMillan's electron-phonon coupling constant $λ$ (and $λ_{tr}$). Consequently the temperature dependence of the thermal electrical resistivity is decreasing as the conductivity is decreasing due to weak localization, which results in the decrease of the temperature coefficient of resistivity (TCR) with increasing the residual resistivity. When $λ$ and $λ_{tr}$ are approaching zero, only residual resistivity part remains and gives rise to the negative TCR. Accordingly, the Mooij rule is a manifestation of weak local…
Capillary fingering is a displacement process that can occur when a non-wetting fluid displaces a wetting fluid from a homogeneous disordered porous medium. Here, we investigate how this process is influenced by a pore size gradient. Using microfluidic experiments and computational pore-network models, we show that the non-wetting fluid displacement behavior depends sensitively on the direction and the magnitude of the gradient. The fluid displacement depends on the competition between a pore size gradient and pore-scale disorder; indeed, a sufficiently large gradient can completely suppress c…
Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/R^α$ is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins $N$. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for $d$-dimensional system including the absence of localization in the infinite system at $α<2d$ and a universal scaling of a critical energy disordering $W_{c} \propto N^{\frac{2d-α}{d}}$. %The finite size e…
A diagrammatic technique for two-particle vertex functions is used to describe systematically the influence of spatial quantum coherence and backscattering effects on transport properties of noninteracting electrons in a random potential. In analogy with many-body theory we construct parquet equations for topologically distinct {\em nonlocal} irreducible vertex functions into which the {\em local} one-particle propagator and two-particle vertex of the coherent-potential approximation (CPA) enter as input. To complete the two-particle parquet equations we use an integral form of the Ward identi…
We study the transport properties of two electrons in a quasi one-dimensional disordered wire. The electrons are subject to both, a disorder potential and a short range two-body interaction. Using the approach developed by Iida et al. [ Ann. Phys. (N.Y.) 200 (1990) 219 ], the supersymmetry technique, and a suitable truncation of Hilbert space, we work out the two-point correlation function in the framework of a non-linear sigma model. We study the loop corrections to arbitrary order. We obtain a remarkably simple and physically transparent expression for the change of the localization length c…
The transfer matrix method is applied to finite quasi-1D disordered samples attached to perfect leads. The model is described by structured band matrices with random and regular entries. We investigate numerically the level spacing distribution for finite-length Lyapunov exponents as well as the conductance and its fluctuations for different channel numbers and sample sizes. A comparison is made with theoretical predictions and with numerical results recently obtained with the scattering matrix approach. The role of the coupling and finite size effects is also discussed.
We study the conductance of disordered graphene superlattices with short-range structural correlations. The system consists of electron- and hole-doped graphenes of various thicknesses, which fluctuate randomly around their mean value. The effect of the randomness on the probability of transmission through the system of various sizes is studied. We show that in a disordered superlattice the quasiparticle that approaches the barrier interface almost perpendicularly transmits through the system. The conductivity of the finite-size system is computed and shown that the conductance vanishes when t…
Effect of the "random anisotropy" type disorder on orientation of the magnetization in an amorphous magnet is considered. It is shown, that the principal corrections to the free energy of the magnet originate from the fluctuations of the order parameter in the direction of its degeneracy. The types of disorder are found, which lift the continuous degeneracy and suppress the Larkin-Imry-Ma mechanism of disruption of the long-range order in continuously degenerate systems.
We study the critical dynamics of hyper-cubic finite size system in the presence of quenched short-range correlated disorder. By using the random $T_c$ model A for the critical dynamics and the renormalization group method in the vicinity of the upper critical dimension $d=4$, we derive in first order of $ε$ the expression for the relaxation time. Its finite-size scaling behavior is discussed both analytically and numerically in details. This was made possible by analyzing carefully the finite--size effects on the Onsager kinetic coefficient. The obtained results are compared to those reported…
A spin-wave (SW) approach for hard-core bosons is presented to treat the problem of two dimensional boson localization in a random potential. After a short review of the method to compute 1/S-corrected observables, the case of random on-site energy is discussed. Whereas the mean-field solution does not display a Bose glass (BG) phase, 1/S corrections do capture BG physics. In particular, the localization of SW excitations is discussed through the inverse participation ratio.
The low lying excited states of the three-dimensional minimum matching problem are studied numerically. The excitations' energies grow with their size and confirm the droplet picture. However, some low energy, infinite size excitations create multiple valleys in the energy landscape. These states violate the droplet scaling ansatz, and are consistent with mean field predictions. A similar picture may apply to spin glasses whereby the droplet picture describes the physics at small length scales, while mean field describes that at large length scales.
We demonstrate that a weak external magnetic field can induce negative magneto-thermal-resistance for magnons in a disordered two-dimensional antiferromagnet. We study the main effect of a weak external magnetic field on the longitudinal thermal conductivity, $κ_{xx}$, for a disordered antiferromagnet using the weak-localization theory for magnons. We show that the weak-localization correction term of $κ_{xx}$ positively increases with increasing the magnetic field parallel to the ordered spins. Since this increase corresponds to a decrease of the thermal resistivity, this phenomenon is negati…
The disordered Bose-Hubbard model in two dimensions at non-integer filling admits a superfluid to Bose-glass transition at weak disorder. Less understood are the properties of this system at strong disorder and energy densities corresponding to excited states. In this work we study the Bose-glass transition of the ground state and the related finite energy localization transition, the mobility edge of the quasiparticle spectrum, a critical energy separating extended from localized quasiparticle excitations. To study these the fluctuation operator expansion is used. The level spacing statistics…
The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform disorder is investigated by intensive Monte Carlo simulations. Taking into account finite-time corrections to scaling, simple ageing behaviour is observed in the two-time responses and correlators. The dynamical exponent z and the form of the scaling functions only depend on the ratio eps/T, where eps describes the width of the distribution of the disorder. The agreement of the predictions of local scale-invariance generalised to z<> 2 for the two-time scaling functions of response and correlations with t…
Magnetotransport measurements in combination with molecular dynamics (MD) simulations on two-dimensional disordered Lorentz gases in the classical regime are reported. In quantitative agreement between experiment and simulation, the magnetoconductivity displays a pronounced peak as a function of perpendicular magnetic field $B$ which cannot be explained in the framework of existing kinetic theories. We show that this peak is linked to the onset of a directed motion of the electrons along the contour of the disordered obstacle matrix when the cyclotron radius becomes smaller than the size of th…
We study nonperturbative interaction corrections to the thermodynamic quantities of multichannel disordered wires in the presence of the Coulomb interactions. Within the replica nonlinear $σ$-model (NL$σ$M) formalism, they arise from nonperturbative soliton saddle points of the NL$σ$M action. The problem is reduced to evaluating the partition function of a replicated classical one dimensional Coulomb gas. The state of the latter depends on two parameters: the number of transverse channels in the wire, N_{ch}, and the dimensionless conductance, G(L_T), of a wire segment of length equal to the t…
Weak localization has a strong influence on both the normal and superconducting properties of metals. In particular, since weak localization leads to the decoupling of electrons and phonons, the temperature dependence of resistance decreases with increasing disorder, as manifested by Mooij's empirical rule. In addition, Testardi's universal correlation of Tc and the resistance ratio follows. We show that weak localization leads to the similar correction to other physical quantities, such as electronic thermal resistivity, electronic heat capacity, and ultrasonic attenuation, which are controll…
We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder $Δ_c \approx U/2$. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scalin…
We study the thermal conductivity in disordered $s$-wave superconductors. Expanding on previous works for normal metals, we develop a formalism that tackles particle diffusion as well as the weak localization (WL) and weak anti-localization (WAL) effects. Using a Green's functions diagrammatic technique, which takes into account the superconducting nature of the system by working in Nambu space, we identify the system's low-energy modes, the diffuson and the Cooperon. The time scales that characterize the diffusive regime are energy dependent; this is in contrast with the the normal state, whe…
We provide a vectorial model to simulate second-harmonic generation (SHG) in birefringent, transparent media with an arbitrary configuration of non-linear ($χ^{(2)}$) crystalline grains. We apply this model on disordered assemblies of LiNbO$_3$ and BaTiO$_3$ grains to identify the influence of the birefringence on the random quasi-phase-matching process. We show that in monodispersed assemblies, the birefringence relaxes the grain-size dependence of the SHG efficiency. In polydispersed assemblies with sufficiently large grains, we find that the birefringence introduces an SHG efficiency enhanc…
We investigate the properties of the superfluid phase in the three-dimensional disordered Bose-Hubbard model using Quantum Monte-Carlo simulations. The phase diagram is generated using Gaussian disorder on the on-site potential. Comparisons with box and speckle disorder show qualitative similarities leading to the re-entrant behavior of the superfluid. Quantitative differences that arise are controlled by the specific shape of the disorder. Statistics pertaining to disorder distributions are studied for a range of interaction strengths and system sizes, where strong finite-size effects are obs…
