Exponent Of Localization Length Research Articles

Metal-insulator transition in a two-dimensional system of chiral unitary class

We perform a numerical investigation of Anderson metal-insulator transition (MIT) in a two-dimensional system of chiral symmetry class AIII by combining finite-size scaling, transport, density of states, and multifractality studies. The results are in agreement with the $\ensuremath{\sigma}$-model renormalization-group theory where MIT is driven by proliferation of vortices. We determine the phase diagram and find an apparent nonuniversality of several parameters on the critical line of MIT, which is consistent with the analytically predicted slow renormalization towards the ultimate fixed point of the MIT. The localization-length exponent $\ensuremath{\nu}$ is estimated as $\ensuremath{\nu}=1.55\ifmmode\pm\else\textpm\fi{}0.1$.

Edge State, Localization Length, and Critical Exponent from Survival Probability in Topological Waveguides.

Edge states in topological phase transitions have been observed in various platforms. To date, verification of the edge states and the associated topological invariant are mostly studied, and yet a quantitative measurement of topological phase transitions is still lacking. Here, we show the direct measurement of edge states and their localization lengths from survival probability. We employ photonic waveguide arrays to demonstrate the topological phase transitions based on the Su-Schrieffer-Heeger model. By measuring the survival probability at the lattice boundary, we show that in the long-time limit, the survival probability is P=(1-e^{-2/ξ_{loc}})^{2}, where ξ_{loc} is the localization length. This length derived from the survival probability is compared with the distance from the transition point, yielding a critical exponent of ν=0.94±0.04 at the phase boundary. Our experiment provides an alternative route to characterizing topological phase transitions and extracting their key physical quantities.

Scaling Collapse of Longitudinal Conductance near the Integer Quantum Hall Transition.

Within the mature field of Anderson transitions, the critical properties of the integer quantum Hall transition still pose a significant challenge. Numerical studies of the transition suffer from strong corrections to scaling for most observables. In this Letter, we suggest to overcome this problem by using the longitudinal conductance g of the network model as the scaling observable, which we compute for system sizes nearly 2 orders of magnitude larger than in previous studies. We show numerically that the sizable corrections to scaling of g can be accounted for in a remarkably simple form, which leads to an excellent scaling collapse. Surprisingly, the scaling function turns out to be indistinguishable from a Gaussian. We propose a cost-function-based approach and estimate ν=2.609(7) for the localization length exponent, consistent with previous results, but considerably more precise than in most works on this problem. Extending previous approaches for Hamiltonian models, we also confirm our finding using integrated conductance as a scaling variable.

Fractional Chern insulators with a non-Landau level continuum limit

Recent developments in fractional quantum Hall (FQH) physics highlight the importance of studying FQH phases of particles partially occupying energy bands that are not Landau levels. FQH phases in the regime of strong lattice effects, called fractional Chern insulators, provide one setting for such studies. As the strength of lattice effects vanishes, the bands of generic lattice models asymptotically approach Landau levels. In this paper, we construct lattice models for single-particle bands that are distinct from Landau levels even in this continuum limit. We describe how the distinction between such bands and Landau levels is quantified by band geometry over the magnetic Brillouin zone and reflected in the electromagnetic response. We analyze the localization-delocalization transition in one such model and compute a localization length exponent of 2.57(3). Moreover, we study interactions projected to these bands and find signatures of bosonic and fermionic Laughlin states. Most pertinently, our models allow us to isolate conditions for optimal band geometry and gain further insight into the stability of FQH phases on lattices.

Criticality of Two-Dimensional Disordered Dirac Fermions in the Unitary Class and Universality of the Integer Quantum Hall Transition.

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 etal. [Phys. Rev. B 50, 7526 (1994)PRBMDO0163-182910.1103/PhysRevB.50.7526] 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 critical properties. Given the far-reaching implications for the emerging field of the quantum anomalous Hall effect, modern condensed matter physics, and our general understanding of disordered critical points, it is surprising that this conjecture has never been tested numerically. Here, we report the results of extensive numerics on the phase diagram and criticality of 2D DDF in the unitary class. We find a critical line at m=0, with an energy-dependent localization length exponent. At large energies, our results for the DDF are consistent with state-of-the-art numerical results ν_{IQH}=2.56-2.62 from models of the IQHT. At E=0, however, we obtain ν_{0}=2.30-2.36 incompatible with ν_{IQH}. This result challenges conjectured relations between different models of the IQHT, and several interpretations are discussed.

Mobility edges and critical exponents in the disordered double chains with long-range correlation

Abstract In this paper, we investigate the localization-delocalization transitions of two types of disordered double chain models with long-range correlation. Based on the empirical methods, we find the exact positions of the two pairs of mobility edges between the localized states and the extended states, which are perfectly in agreement with the numerical results obtained by the transfer matrix method. In the case with interchain correlation in on-site energy (a linear relationship between the corresponding site energies of the two chains), a second-order quantum phase transition has been identified, which is manifested as the jump of critical exponent of the localization length. While for the case without interchain correlation in on-site energy, i.e., the disordered on-site energies of the two chains are independently, the critical exponent is determined by one of the two chains, of which the degree of long-range correlation is smaller.

Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map.

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys. Rev. E 97, 012210 (2018)2470-004510.1103/PhysRevE.97.012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena on the number M of the harmonic perturbations, where M+1=d corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization for our systems, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT by largely increasing M. The numerically obtained index of critical subdiffusion t^{α} (t:time) agrees well with the prediction of one-parameter scaling theory α=2/(M+1), but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: If M is fixed, then the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is 1/(M-1) times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

Random network models with variable disorder of geometry

Recently it was shown (I.A.Gruzberg, A. Kl\"umper, W. Nuding and A. Sedrakyan, Phys.Rev.B 95, 125414 (2017)) that taking into account random positions of scattering nodes in the network model with $U(1)$ phase disorder yields a localization length exponent $2.37 \pm 0.011$ for plateau transitions in the integer quantum Hall effect. This is in striking agreement with the experimental value of $2.38 \pm 0.06$. Randomness of the network was modeled by replacing standard scattering nodes of a regular network by pure tunneling resp.reflection with probability $p$ where the particular value $p=1/3$ was chosen. Here we investigate the role played by the strength of the geometric disorder, i.e. the value of $p$. We consider random networks with arbitrary probability $0 <p<1/2$ for extreme cases and show the presence of a line of critical points with varying localization length indices having a minimum located at $p=1/3$.

Integer quantum Hall transition on a tight-binding lattice

Even though the integer quantum Hall transition has been investigated for nearly four decades its critical behavior remains a puzzle. The best theoretical and experimental results for the localization length exponent $\nu$ differ significantly from each other, casting doubt on our fundamental understanding. While this discrepancy is often attributed to long-range Coulomb interactions, Gruzberg et al. [Phys. Rev. B 95, 125414 (2017)] recently suggested that the semiclassical Chalker-Coddington model, widely employed in numerical simulations, is incomplete, questioning the established central theoretical results. To shed light on the controversy, we perform a high-accuracy study of the integer quantum Hall transition for a microscopic model of disordered electrons. We find a localization length exponent $\nu=2.58(3)$ validating the result of the Chalker-Coddington network.

Localization-length exponent in two models of quantum Hall plateau transitions

Motivated by the recent numerical studies on the Chalker-Coddington network model that found a larger-than-expected critical exponent of the localization length characterizing the integer quantum Hall plateau transitions, we revisited the exponent calculation in the continuum model and in the lattice model, both projected to the lowest Landau level or subband. Combining scaling results with or without the corrections of an irrelevant length scale, we obtain $\nu = 2.48 \pm 0.02$, which is larger but still consistent with the earlier results in the two models, unlike what was found recently in the network model. The scaling of the total number of conducting states, as determined by the Chern number calculation, is accompanied by an effective irrelevant length scale exponent $y = 4.3$ in the lattice model, indicating that the irrelevant perturbations are insignificant in the topology number calculation.

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.

Many-body localization in Ising models with random long-range interactions

We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, $V_{ij}\propto\left|i-j\right|^{-\alpha}$, where the exponent of the interaction range $\alpha$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\alpha$, the critical exponent experiences a sharp increase at about $\alpha=1$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\alpha<1$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\alpha>1$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.

Quantum Hall effect and hopping conductivity in n-InGaAs/InAlAs nanoheterostructures

The longitudinal and Hall magnetoresistances are measured in the quantum Hall effect regime in the n-InGaAs/InAlAs heterostructures at temperatures of T = (1.8–30) K in magnetic fields up to B = 9 T. Temperature-induced transport in the region of the longitudinal resistance minima, corresponding to the plateau regions at Hall resistance, is investigated within the framework of the concept of hopping conductivity in a strongly localized electron system. The analysis of variable-range hopping conductivity in the region of the second, third, and fourth plateau of the quantum Hall effect provides the possibility of determining the localization length exponent.

Localization of directed polymers in random media due to a columnar defect

We investigate the localization of directed polymers in random media due to the presence of an attractive columnar defect at the center of a two-dimensional substrate. If the defect’s strength is too weak to affect the polymers, the localization length of the polymers exhibits a power-law behavior as a function of the polymer length, as in the case of no defect. When the defect’s strength is greater than a critical value, the localization length approaches a finite value, thereby yielding a localization length exponent (or liberation exponent) ν ⊥ = 1.8004(31). The correlation length, which is perpendicular to the localization length, is defined as the distance over which the polymer is localized by the defect. The correlation length exponent ν ‖ = 2.881(5) is estimated from the data collapse via the scaling relation ζ = ν ⊥/ν ‖, where ζ = 5/8 represents the wandering exponent. In addition, we measure the number of times that the optimal path passes through the defect, because that number increases as the defect’s strength increases. This measurement yields a new critical exponent λ = 0.59.

Non-universality of scaling exponents in quantum Hall transitions

We have investigated experimentally the scaling behaviour of quantum Hall transitions in GaAs/AlGaAs heterostructures of a range of mobility, carrier concentration, and spacer layer width. All three critical scaling exponents γ, κ and p were determined independently for each sample. We measure the localization length exponent to be γ ≈ 2.3, in good agreement with expected predictions from scaling theory, but κ and p are found to possess non-universal values. Results obtained for κ range from κ = 0.16 ± 0.02 to κ = 0.67 ± 0.02, and are found to be Landau level (LL) dependent, whereas p is found to decrease with increasing sample mobility. Our results demonstrate the existence of two transport regimes in the LL conductivity peak; universality is found within the quantum coherent transport regime present in the tails of the conductivity peak, but is absent within the classical transport regime found close to the critical point at the centre of the conductivity peak. We explain these results using a percolation model and show that the critical scaling exponent depends on certain important length scales that correspond to the microscopic description of electron transport in the bulk of a two-dimensional electron system.

Finite-Size Effects and Irrelevant Corrections to Scaling Near the Integer Quantum Hall Transition

We present a numerical finite-size scaling study of the localization length in long cylinders near the integer quantum Hall transition employing the Chalker-Coddington network model. Corrections to scaling that decay slowly with increasing system size make this analysis a very challenging numerical problem. In this work we develop a novel method of stability analysis that allows for a better estimate of error bars. Applying the new method we find consistent results when keeping second (or higher) order terms of the leading irrelevant scaling field. The knowledge of the associated (negative) irrelevant exponent y is crucial for a precise determination of other critical exponents, including multifractal spectra of wave functions. We estimate |y|>/~0.4, which is considerably larger than most recently reported values. Within this approach we obtain the localization length exponent 2.62±0.06 confirming recent results. Our stability analysis has broad applicability to other observables at integer quantum Hall transition, as well as other critical points where corrections to scaling are present.

Asymmetric Metal-Insulator Transition in Disordered Ferromagnetic Films

We present experimental data and a theoretical interpretation of the conductance near the metal-insulator transition in thin ferromagnetic Gd films of thickness b ≈ 2-10 nm. A large phase relaxation rate caused by scattering of quasiparticles off spin-wave excitations renders the dephasing length L(ϕ) ≲ b in the range of sheet resistances considered, so that the effective dimension is d = 3. The conductivity data at different stages of disorder obey a fractional power-law temperature dependence and collapse onto two scaling curves for the metallic and insulating regimes, indicating an asymmetric metal-insulator transition with two distinctly different critical exponents; the best fit is obtained for a dynamical exponent z ≈ 2.5 and a correlation (localization) length critical exponent ν- ≈ 1.4 (ν+ ≈ 0.8) on the metallic (insulating) side.

Dissipation and Criticality in the Lowest Landau Level of Graphene

The lowest Landau level of graphene is studied numerically by considering a tight-binding Hamiltonian with disorder. The Hall conductance sigma_{xy} and the longitudinal conductance sigma_{xx} are computed. We demonstrate that bond disorder can produce a plateaulike feature centered at nu=0, while the longitudinal conductance is nonzero in the same region, reflecting a band of extended states between +/-E_{c}, whose magnitude depends on the disorder strength. The critical exponent corresponding to the localization length at the edges of this band is found to be 2.47+/-0.04. When both bond disorder and a finite mass term exist the localization length exponent varies continuously between approximately 1.0 and approximately 7/3.

Crossover effects in the random-exchange spin-12antiferromagnetic chain

The random antiferromagnetic spin-1/2 XX and XXZ chain is studied numerically for varying strength of the disorder, using exact diagonalization and stochastic series expansion methods. The spin-spin correlation function as well as the stiffness display a clear crossover from the pure behavior (no disorder) to the infinite randomness fixed point or random singlet behavior predicted by the the real space renormalization group. The crossover length scale is shown to diverge as $\xi\sim{\mathcal D}^{-\gamma}$, where ${\mathcal D}$ is the variance of the random bonds. Our estimates for the exponent $\gamma$ agrees well within the error bars with the one for the localization length exponent emerging within an analytical bosonization calculation. Exact diagonalization and stochastic series expansion results for the string correlation function are also presented.

Levitation of Extended States in a Random Magnetic Field with a Finite Mean**The project supported by National Natural Science Foundation of China and in part by the Natural Science Foundation of Henan Province of China

We study the localization properties of electrons in a two-dimensional system in a random magnetic field with the average and the amplitude of the magnetic field fluctuations δB. The localization length of the system is calculated by using the finite-size scaling method combined with the transfer-matrix technique. In the case of weak δB, we find that the random magnetic field system is equivalent to the integer quantum Hall effect system, namely, the energy band splits into a series of disorder broadened Landau bands, at the centers of which states are extended with the localization length exponent . With increasing δB, the extended states float up in energy, which is similar to the levitation scenario proposed for the integer quantum Hall effect.