Divergence Of Localization Length Research Articles

Metal-insulator crossover in monolayer MoS2

We report on transport measurements in monolayer MoS2 devices, close to the bottom of the conduction band edge. These devices were annealed in situ before electrical measurements. This allows us to obtain good ohmic contacts at low temperatures, and to measure precisely the conductivity and mobility via four-probe measurements. The measured effective mobility up to μ eff = 180 cm2 V−1 s−1 is among the largest obtained in CVD-grown MoS2 monolayer devices. These measurements show that electronic transport is of the insulating type for σ ≤ 1.4e 2/h and n ≤ 1.7 × 1012 cm−2, and a crossover to a metallic regime is observed above those values. In the insulating regime, thermally activated transport dominates at high temperature (T > 120 K). At lower temperatures, conductivity is driven by Efros–Schklovkii variable range hopping in all measured devices, with a universal and constant hopping prefactor, that is a clear indication that hopping is not phonon-mediated. At higher carrier density, and high temperature, the conductivity is well modeled by the Boltzmann equation for a non-interacting Fermi gas, taking into account both phonon and impurity scatterings. Finally, even if this apparent metal-insulator transition can be explained by phonon-related phenomena at high temperature, the possibility of a genuine 2D MIT cannot be ruled out, as we can observe a clear power-law diverging localization length close to the transition, and a one-parameter scaling can be realized.

Flat band induced metal-insulator transitions for weak magnetic flux and spin-orbit disorder

We consider manifolds of tunable all-bands-flat (ABF) lattices in dimensions $d=1,2$, parameterized by a manifold angle parameter $\ensuremath{\theta}$. We study localization properties of eigenstates in the presence of weak magnetic flux disorder and weak spin-orbit disorder. We demonstrate that weakly disordered ABF lattices are described by effective scale-free models, where the disorder strength is scaled out. For weak magnetic flux disorder, we observe subexponential localization at flatband energies in $d=1$, which differs from the usual Anderson localization. We also find diverging localization length at flatband energies for weak flux values in $d=2$; however, the character of the eigenstates at these energies is less clear. For weak spin-orbit coupling disorder in $d=2$ we identify a tunable metal-insulator transition with mobility edges. We also consider the case of mixed spin-orbit and diagonal disorder and obtain the metal-insulator transition driven by the manifold parameter $\ensuremath{\theta}$.

Topological quantum criticality of the disordered Chern insulator

We consider the two-dimensional topological Chern insulator in the presence of static disorder. Generic quantum states in this system are Anderson localized. However, topology requires the presence of a subset of critical states, with diverging localization length (the Chern insulator analog of the ‘center of the Landau band states’ of the quantum Hall insulator.) We discuss geometric criteria for the identification of these states at weak disorder, and their extension into the regime of strong disorder by analytical methods. In this way, we chart a critical surface embedded in a phase space spanned by energy, topological control parameter, and disorder strength. Our analytical predictions are supplemented by a numerical analysis of the position of the critical states, and their multifractal properties.

Proposal for realizing anomalous Floquet insulators via Chern band annihilation

Two-dimensional periodically driven systems can host an unconventional topological phase unattainable for equilibrium systems, termed the Anomalous Floquet-Anderson insulator (AFAI). The AFAI features a quasi-energy spectrum with chiral edge modes and a fully localized bulk, leading to non-adiabatic but quantized charge pumping. Here, we show how such a Floquet phase can be realized in a driven, disordered Quantum Anomalous Hall insulator, which is assumed to have two critical energies where the localization length diverges, carrying states with opposite Chern numbers. Driving the system at a frequency close to resonance between these two energies localizes the critical states and annihilates the Chern bands, giving rise to an AFAI phase. We exemplify this principle by studying a model for a driven, magnetically doped topological insulator film, where the annihilation of the Chern bands and the formation of the AFAI phase is demonstrated using the rotating wave approximation. This is complemented by a scaling analysis of the localization length for two copies of a quantum Hall network model with a tunable coupling between them. We find that by tuning the frequency of the driving close to resonance, the driving strength required to stabilize the AFAI phase can be made arbitrarily small.

Fractional corner magnetization of collinear antiferromagnets

Recent studies revealed that the electric multipole moments of insulators result in fractional electric charges localized to the hinges and corners of the sample. We here explore the magnetic analog of this relation. We show that a collinear antiferromagnet with spin $S$ defined on a $d$-dimensional cubic lattice features fractionally quantized magnetization ${M}_{\text{c}}^{z}=S/{2}^{d}$ at the corners. We find that the quantization is robust even in the presence of gapless excitations originating from the spontaneous formation of the N\'eel order, although the localization length diverges, suggesting a power-law localization of the corner magnetization. When the spin rotational symmetry about the $z$ axis is explicitly broken, the corner magnetization is no longer sharply quantized. Even in this case, we numerically find that the deviation from the quantized value is negligibly small based on quantum Monte Carlo simulations.

Marginally localized edges of time-reversal symmetric topological superconductors

We demonstrate that the one-dimensional helical Majorana edges of two-dimensional time-reversal symmetric topological superconductors (class DIII) can become gapless and insulating by a combination of random edge velocity and interaction. Such a gapless insulating edge breaks time-reversal symmetry inhomogeneously, and the local symmetry broken regions can be regarded as static mass potentials or dynamical Ising spins. In both limits, we find that such gapless insulating Majorana edges are generically exponentially localized and trap Majorana zero modes. Interestingly, for a statistically time-reversal symmetric edge (symmetry is broken locally, but the symmetry breaking order parameter is zero on average), the low-energy theory can be mapped to a Dyson model at zero energy, manifesting a diverging density of states and exhibiting marginal localization (i.e., a diverging localization length). Although the ballistic edge state transport is absent, the localized Majorana zero modes reflect the nontrivial topology in the bulk. Experimental signatures are also discussed.

Extracting many-body localization lengths with an imaginary vector potential

One challenge of studying the many-body localization transition is defining the length scale that diverges upon the transition to the ergodic phase. In this manuscript we explore the localization properties of a ring with onsite disorder subject to an imaginary magnetic flux. We connect the imaginary flux which delocalizes single-particle orbitals of an Anderson-localized ring with the localization length of an open chain. We thus identify the delocalizing imaginary flux per site with an inverse localization length characterizing the transport properties of the open chain. We put this intuition to use by exploring the phase diagram of a disordered interacting chain, and we find that the inverse imaginary flux per bond provides an accessible description of the transition and its diverging localization length.

Topological phase transitions in glassy quantum matter

Amorphous systems have rapidly gained promise as novel platforms for topological matter. In this work we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover unique critical properties of Chern and $\mathbb{Z}_2$ glass transitions. Even for short-range hopping models the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behaviour of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of new research directions.

Thermal conductivity in 1d: Disorder-induced transition from anomalous to normal scaling

It is well known that the contribution of harmonic phonons to the thermal conductivity of 1D systems diverges with the harmonic chain length L. Within various one-dimensional models containing disorder it was shown that the thermal conductivity scales as under certain boundary conditions. Here we show that when the chain is weakly coupled to the heat reservoirs and there is strong disorder this scaling can be violated. We find a weaker power-law dependence on L, and show that for sufficiently strong disorder the thermal conductivity ceases to be anomalous – it does not depend on L and hence obeys Fourier's law. This is despite both density of states and the diverging localization length scaling anomalously. Surprisingly, in this strong disorder regime two anomalously scaling quantities cancel each other to recover Fourier's law of heat transport.

A generic phase between disordered Weyl semimetal and diffusive metal

Quantum phase transitions of three-dimensional (3D) Weyl semimetals (WSMs) subject to uncorrelated on-site disorder are investigated through quantum conductance calculations and finite-size scaling of localization length. Contrary to previous claims that a direct transition from a WSM to a diffusive metal (DM) occurs, an intermediate phase of Chern insulator (CI) between the two distinct metallic phases should exist due to internode scattering that is comparable to intranode scattering. The critical exponent of localization length is ν,simeq 1.3 for both the WSM-CI and CI-DM transitions, in the same universality class of 3D Gaussian unitary ensemble of the Anderson localization transition. The CI phase is confirmed by quantized nonzero Hall conductances in the bulk insulating phase established by localization length calculations. The disorder-induced various plateau-plateau transitions in both the WSM and CI phases are observed and explained by the self-consistent Born approximation. Furthermore, we clarify that the occurrence of zero density of states at Weyl nodes is not a good criterion for the disordered WSM, and there is no fundamental principle to support the hypothesis of divergence of localization length at the WSM-DM transition.

Effect of weak disorder on delocalization properties of gapped graphene superlattices

We study the effect of weak disorder on the delocalization properties of gapped graphene superlattice (SL) formed by periodically located rectangular potential barriers. We consider two types of the SLs: the SLs with uniform gap and SLs consisting of alternating layers of gapped and gapless graphene regions. Using the perturbative approach we obtain an analytical expression for the inverse localization length (ILL) derived for the case of randomly fluctuating geometric and energetic parameters. In the first case, when the barrier (well) width fluctuates around its mean value, the corresponding equation for the ILL reveals the presence of the Fabry–Perot resonances, at which the localization length diverges. These resonances are exact, i.e., are stored in any degree of disorder. It has been found that the localization properties manifest stronger for the particles with energies lying in the non-resonant bands where our approach is extremely sensitive to the degree of disorder. For the case of weakly fluctuating both barrier and well widths we analytically obtain ILL taking correlations into account. The main effect of the correlations, which lead to an increase (or decrease) in the localization length, was revealed near the double resonance arising at coincidence of two Fabry–Perot resonances associated with barrier and well widths. The random fluctuations of the potential strength also lead to the delocalization resonances. However, they exist only in a weak-disorder approximation. We found that, for an array composed of alternating strips of gapless and gapped graphene modifications these resonances can appear only for normally incident particles in contrast to the SL with a uniform gap. For such particles, the delocalization resonances occur also in the purely random potential. This means, in particular, that in the one-dimensional case, not all the states of the massive Dirac particles are localized in the presence of weak disorder.

Topological criticality in the chiral-symmetric AIII class at strong disorder.

The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

Spin-directed network model for the surface states of weak three-dimensionalZ2topological insulators

A two-dimensional spin-directed $\mathbb{Z}^{\,}_{2}$ network model is constructed that describes the combined effects of dimerization and disorder for the surface states of a weak three-dimensional $\mathbb{Z}^{\,}_{2}$ topological insulator. The network model consists of helical edge states of two-dimensional layers of $\mathbb{Z}^{\,}_{2}$ topological insulators which are coupled by time-reversal symmetric interlayer tunneling. It is argued that, without dimerization of interlayer couplings, the network model has no insulating phase for any disorder strength. However, a sufficiently strong dimerization induces a transition from a metallic phase to an insulating phase. The critical exponent $\nu$ for the diverging localization length at metal-insulator transition points is obtained by finite-size scaling analysis of numerical data from simulations of this network model. It is shown that the phase transition belongs to the two-dimensional symplectic universality class of Anderson transition.

Metallic Phase of the Quantum Hall Effect in Four-Dimensional Space

We study the phase diagram of the quantum Hall effect in four-dimensional (4D) space. Unlike in 2D, in 4D there exists a metallic as well as an insulating phase, depending on the disorder strength. The critical exponent ν≈1.2 of the diverging localization length at the quantum Hall insulator-to-metal transition differs from the semiclassical value ν=1 of 4D Anderson transitions in the presence of time-reversal symmetry. Our numerical analysis is based on a mapping of the 4D Hamiltonian onto a 1D dynamical system, providing a route towards the experimental realization of the 4D quantum Hall effect.

Suppression of Anderson localization of light in one-dimensional disordered photonic superlattices

The localization properties of electromagnetic modes in one-dimensional disordered photonic superlattices are theoretically studied. The multilayered system is considered to be composed of alternating stacks of two different random-thickness slabs, characterized by nondispersive and/or frequency-dependent electric permittivities and magnetic permeabilities. Results for the localization length are evaluated by using an analytical model for weakly disordered systems as well as its general definition through the transmissivity properties of the heterostructure. Good agreement between both results is observed only for small amplitudes of disorder. The critical frequencies at which the localization length diverges are correctly predicted in the whole frequency spectrum by the analytical model and confirmed via the corresponding numerical calculations. The ${\ensuremath{\lambda}}^{2}$ dependence of the localization length, previously observed in disordered heterostructures made of material of positive refractive indexes, are confirmed in the present work. In addition, new ${\ensuremath{\lambda}}^{4}$ and ${\ensuremath{\lambda}}^{\ensuremath{-}4}$ dependencies of the localization length in positive-negative disordered photonic superlattices are obtained, under certain specific conditions, in the long and short wavelength limits, respectively. The asymptotic behavior of the localization length in these limits is essentially determined by the particular frequency dispersion that characterizes the metamaterial used in the left-handed layers. When the effects of absorption are considered, then a divergence of the localization length is still observed, under some conditions, in the short wavelength limit.

Divergence of localization length in short-range correlated superlattices

We study the localization/delocalization property of disordered GaAs–Al x Ga 1− x As semiconductor superlattices with structural short-range correlations (so-called random-dimer model). Our system consists of square quantum barriers and wells with the same thickness and the heights of barriers have two different values. Using transfer matrix method, we calculate localization length of the system and show that there are two different delocalized states in the system. The energy of these extended states depends on the fraction of Al in the barriers. Then, we consider the divergence of the localization length near the extended state.

Particle-hole symmetric localization in optical lattices using time modulated random on-site potentials

The random hopping models exhibit many fascinating features, such as diverging localization length and density of states as energy approaches the bandcenter, due to its particle-hole symmetry. Nevertheless, such models are yet to be realized experimentally because the particle-hole symmetry is easily destroyed by diagonal disorder. Here we propose that a pure random hopping model can be effectively realized in ultracold atoms by modulating a disordered onsite potential in particular frequency ranges. This idea is motivated by the recent development of the phenomena called "dynamical localization" or "coherent destruction of tunneling". Investigating the application of this idea in one dimension, we find that if the oscillation frequency of the disorder potential is gradually increased from zero to infinity, one can tune a non-interacting system from an Anderson insulator to a random hopping model with diverging localization length at the band center, and eventually to a uniform-hopping tight-binding model.

Optical properties of 1D photonic crystals with correlated and uncorrelated disorder

We introduce both correlated disorder and uncorrelated disorder to one-dimensionaldielectric periodic structures and investigate their effects on light transmission, localizationlength, density of photonic states and decay rate of resonant modes. The photonicbandgaps are more robust against uncorrelated disorder due to the preservation oflong-range structural order. While correlated disorder enhances light localization near theband edges, uncorrelated disorder causes a divergence of localization length near the gapedges. The correlated disorder induces a larger fluctuation of decay rates for the bandgapmodes than the pass band modes. In contrast, the resonant modes near the pass bandcenter experience the strongest fluctuation of decay rates in the presence of uncorrelateddisorder.

Electron delocalization in bilayer graphene induced by an electric field

Electronic localization is numerically studied in disordered bilayer graphene with an electric-field-induced energy gap. Bilayer graphene is a zero-gap semiconductor, in which an energy gap can be opened and controlled by an external electric field perpendicular to the layer plane. We found that, in the smooth disorder potential not mixing the states in different valleys ($K$ and ${K}^{\ensuremath{'}}$ points), the gap opening causes a phase transition at which the electronic localization length diverges. We show that this can be interpreted as the integer quantum Hall transition at each single valley, even though the magnetic field is absent.

Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions

Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a ${\mathbb{Z}}_{2}$ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.