We numerically investigate the vibrational spectra of single-component clusters in two-dimensions. Stable configurations of clusters at local energy minima are obtained, and for each the hessian matrix is evaluated and diagonalized to obtain eigenvalues as well as eigenvectors. We study the density of states so obtained as a function of the width of the potential well describing the two-body interaction. As the width is reduced, as in three dimensions, we find that the density of states approaches a common form, but the two-peak behavior survives. Further, calculations of the participation ratio show that most states are extended, although a smaller fraction of the degrees of freedom are involved in these modes, compared to three dimensions. We show that the fluctuation properties of these modes converges to those of the Gaussian orthogonal ensemble of random matrices, in common with previous results on three dimensional amorphous clusters and molecular liquids
We investigate the high-frequency behavior of the density of vibrational states in three-dimensional elasticity theory with spatially fluctuating elastic moduli. At frequencies well above the mobility edge, instanton solutions yield an exponentially decaying density of states. The instanton solutions describe excitations, which become localized due to the disorder-induced fluctuations, which lower the sound velocity in a finite region compared to its average value. The exponentially decaying density of states (known in electronic systems as the Lifshitz tail) is governed by the statistics of a…
We consider the localization properties of a lattice of coupled masses and springs with random mass and spring constant values. We establish the full phase diagrams of the system for pure mass and pure spring disorder. The phase diagrams exhibit regions of stable as well as unstable wave modes. The latter are of interest for the instantaneous-normal-mode spectra of liquids and the nascent field of acoustic metamaterials. We show the existence of delocalization-localization transitions throughout the phase diagram and establish, by high-precision numerical studies, that the universality of thes…
Disordered solids exhibit unusual properties of their vibrational states and thermal conductivities. Recent progresses have well established the concept of "elastic heterogeneity", i.e., disordered materials show spatially inhomogeneous elastic moduli. In this study, by using molecular-dynamics simulations, we gradually introduce "disorder" into a numerical system to control its modulus heterogeneity. The system starts from a perfect crystalline state, progressively transforms into an increasingly disordered crystalline state, and finally undergoes structural amorphisation. We monitor independ…
In crystals, molecules thermally vibrate around the periodic lattice sites. Vibrational motions are well understood in terms of phonons, which carry heat and control heat transport. The situation is notably different in disordered solids, where vibrational excitations are not phonons and can be even localized. Recent numerical work has established the concept of elastic heterogeneity: disordered solids show inhomogeneous local mechanical response. Clearly, the heterogeneous nature of elastic properties strongly influences vibrational and thermal properties, and it is expected to be the origin …
The low temperature universal properties in disordered and amorphous solids are considered. We introduce a model that includes two types of two level systems (TLSs), which, based on their local symmetry, interact weakly or strongly with the phonon field. This accounts well for the experimental results, and addresses some long-standing questions: the nature of the TLSs; the smallness and universality of the phonon attenuation, and the energy scale of $3$K below which universality is observed. Our model describes disordered lattices; we also discuss its application to amorphous solids.
We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $\mathcal{D}(ω)$ of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness $κ$ drawn from a distribution $p(κ)$, subjected to a constant field $h$ and interacting bilinearly with a coupling of strength $J$. We investigate the vibrational properties of its ground state at zero temperature. When $p(κ)$ is gapped, the emergent $\mathcal{D}(ω)$ is also gapped, for small $J$. Upon increasing $J$, the gap vanishes on…
In this article we present numerical results of conduction in a disordered quasi-1D wire in the possible presence of magnetic impurities. Our analysis leads us to the study of universal properties in different conduction regimes such as the localized and metallic ones. In particular, we analyse the cross-over between universality classes occurring when the strength of magnetic disorder is increased. For this purpose, we use a numerical Landauer approach, and derive the scattering matrix of the wire from electron's Green's function.
The low-frequency vibrations of glasses are markedly different from those of crystals. These vibrations have recently been categorized into two types: spatially extended vibrations, whose vibrational density of states (vDOS) follows a non-Debye quadratic law, and quasilocalized vibrations (QLVs), whose vDOS follows a quartic law. The former are explained by elasticity theory with quenched disorder and microscopic replica theory as being a consequence of elastic instability, but the origin of the latter is still debated. Here, we show that the latter can also be directly derived from elasticity…
Interface fluctuations in disordered systems: Universality and non-Gaussian statistics
我们采用了一个功能重整化群来研究在$d=4-ε$尺寸的钉扎电位的存在下的接口。与以前的方法相反[D. S. Fisher，Phys. Rev. Lett. { \bf 56}，1964（1986）]我们使用软截止方案。用这里开发的方法，我们确认的值的粗糙度指数$ζ\approx 0.2083 ε$顺序$ε$。超越以前的工作，我们证明了这个指数是普遍的。此外，我们分析了在无序分布和温度作为一个危险的无关变量的作用，产生更高的累积量。
Disordered 2D chiral fermions provide an effective description of several materials including graphene and topological insulators. While previous analysis considered delta correlated disorder and no ultraviolet cut-offs, we consider here the effect of short range correlated disorder and the presence of a momentum cut-off, providing a more realistic description of condensed matter models. We show that the density of states is anomalous with a critical exponent function of the disorder and that conductivity is universal only when the ultraviolet cut-off is removed, as consequence of the supersym…
We develop a new method to isolate localized defects from extended vibrational modes in disordered solids. This method augments particle interactions with an artificial potential that acts as a high-pass filter: it preserves small-scale structures while pushing extended vibrational modes to higher frequencies. The low-frequency modes that remain are "bare" defects; they are exponentially localized without the quadrupolar tails associated with elastic interactions. We demonstrate that these localized excitations are excellent predictors of plastic rearrangements in the solid. We characterize se…
Using Gaussian integral transform techniques borrowed from functional-integral field theory and the replica trick we derive a version of the coherent-potential approximation (CPA) suited for describing ($i$) the diffusive (hopping) motion of classical particles in a random environment and ($ii$) the vibrational properties of materials with spatially fluctuating elastic coefficients in topologically disordered materials. The effective medium in the present version of the CPA is not a lattice but a homogeneous and isotropic medium, representing an amorphous material on a mesoscopic scale. The tr…
Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random K-SAT…
Amorphous solids or glasses are known to exhibit stretched-exponential decay over broad time intervals in several of their macroscopic observables: intermediate scattering function, dielectric relaxation modulus, time-elastic modulus etc. This behaviour is prominent especially near the glass transition. In this Letter we show, on the example of dielectric relaxation, that stretched-exponential relaxation is intimately related to the peculiar lattice dynamics of glasses. By reformulating the Lorentz model of dielectric matter in a more general form, we express the dielectric response as a funct…
This thesis presents several aspects of the physics of disordered elastic systems and of the analytical methods used for their study. On one hand we will be interested in universal properties of avalanche processes in the statics and dynamics (at the depinning transition) of elastic interfaces of arbitrary dimension in disordered media at zero temperature. To study these questions we will use the functional renormalization group. After a review of these aspects we will more particularly present the results obtained during the thesis on (i) the spatial structure of avalanches and (ii) the corre…
The relaxation dynamics and the vibrational spectra of amorphous solids, such as metal alloys, have been intensely investigated as well separated topics in the past. The aim of this review is to summarize recent results in both these areas in an attempt to establish, or unveil, deeper connections between the two phenomena of relaxation and vibration. Theoretical progress in the area of slow relaxation dynamics of liquid and glassy systems and in the area of vibrational spectra of glasses and liquids is reviewed. After laying down a generic modelling framework to connect vibration and relaxatio…
We review the results of our recent numerical investigations on the electronic properties of disordered two dimensional systems with chiral unitary, chiral orthogonal, and chiral symplectic symmetry. Of particular interest is the behavior of the density of states and the logarithmic scaling of the smallest Lyapunov exponents in the vicinity of the chiral quantum critical point in the band center at E=0. The observed peaks or depressions in the density of states, the distribution of the critical conductances, and the possible non-universality of the critical exponents for certain chiral unitary…
Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E=0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m=0. This observation lead Ludwig et al [Phys. Rev. B 50, 7526 (1994)] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal cri…
Disordered non-interacting systems are classified into ten symmetry classes, with the unitary class being the most fundamental. The three and four dimensional unitary universality classes are attracting renewed interest because of their relation to three dimensional Weyl semi-metals and four dimensional topological insulators. Determining the critical exponent of the correlation/localistion length for the Anderson transition in these classes is important both theoretically and experimentally. Using the transfer matrix technique, we report numerical estimations of the critical exponent in a U(1…
We investigate the effect of spatially correlated disorder on the Anderson transition of phonons in three dimensions using a Greens function based approach, namely, the typical medium dynamical cluster approximation (TMDCA), in mass-disordered systems. We numerically demonstrate that correlated disorder with pairwise correlations mitigates the localization of the vibrational modes. A correlation driven localization-delocalization transition can emerge in a three-dimensional disordered system with an increase in the strength of correlations.
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…
It is now possible to study experimentally the combined effect of disorder and interactions in cold atom physics. Motivated by these developments we investigate the dynamics around the metal-insulator transition (MIT) in a one-dimensional (1D) Fermi gas with short-range interactions in a quasiperiodic potential by the time-dependent density-matrix renormalization group (tDMRG) technique. By tuning disorder and interactions we study the MIT from the weakly to the strongly interacting limit. The MIT is not universal as time evolution, well described by a process of anomalous diffusion, depends q…
Quasiperiodic system is an intermediate state between periodic and disordered systems with unique delocalization-localization transition driven by the quasiperiodic potential (QP). One of the intriguing questions is whether the universality class of the Anderson transition (AT) driven by QP is similar to that of the AT driven by the random potential in the same symmetry class. Here, we study the critical behavior of the ATs driven by QP in the three-dimensional (3D) Anderson model, Peierls phase model, and Ando model, which belong to the Wigner-Dyson symmetry classes. The localization length a…
Amorphous solids, as well as many disordered lattices, display remarkable universality in their low temperature acoustic properties. This universality is attributed to the attenuation of phonons by tunneling two-level systems (TLSs), facilitated by the interaction of the TLSs with the phonon field. TLS-phonon interaction also mediates effective TLS-TLS interactions, which dictates the existence of a glassy phase and its low energy properties. Here we consider KBr:CN, the archetypal disordered lattice showing universality. We calculate numerically, using conjugate gradients method, the effectiv…
Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational…
Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensiv…
We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak inter-color coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization gro…
It was recently shown that different simple models of glass formers with binary interactions define a universality class in terms of the density of states of their quasi-localized low-frequency modes. Explicitly, once the hybridization with standard Debye (extended) modes is avoided, a number of such models exhibit a universal density of state, depending on the mode frequencies as $D(ω) \sim ω^4$. It is unknown however how wide is this universality class, and whether it also pertains to more realistic models of glass formers. To address this issue we present analysis of the quasi-localized mod…
A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We ch…
