Delocalization in continuous disordered systems

Hilke, Michael; Flores, J. C.

doi:10.1103/physrevb.55.10625

Cited by 48 publications

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“…In the case of IQHE, a continuous quantum phase transition means algebraic divergence of the longitudinal Hallresistivity slope in temperature T at the transition point. However, recent experiments [5] showed that such slopes remain finite when they are extrapolated to T 0. This implies a nonscaling behavior around a transition point, contradicting the expectation of continuous quantum phase transitions suggested by the theories.…”

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confidence: 99%

“…As a function of the magnetic field, the Hall conductance jumps from one QH plateau to another when the Fermi energy crosses an extended-state level. Many previous studies [1][2][3][4][5] have been focused on how such a transition occurs.One overlooked issue regarding IQHE is the nature of a transition from one QH plateau to another. All existing theories assume it to be a continuous quantum phase transition.…”

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confidence: 99%

“…-f, 71.30. +h, 73.20.Jc Recently there has been a great renewal of interest on transitions from integer quantum Hall effect (IQHE) states to an insulator [1][2][3][4][5]. According to the scaling theory of localization [6], all electrons in a disordered two-dimensional system are localized in the absence of a magnetic field.…”

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Metallic Phase in Quantum Hall Systems due to Inter-Landau-Band Mixing

et al. 2001

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The electronic eigenstates of a quantum Hall (QH) system are chiral states. Strong inter-Landau-band mixings among these states can occur when the bandwidth is comparable to the spacing of two adjacent Landau bands. We show that mixing of localized states with opposite chirality can delocalize electronic states. Based on numerical results, we propose the existence of a metallic phase between two adjacent QH phases and between a QH phase and the insulating phase. This result is consistent with nonscaling behaviors observed in recent experiments on a quantum Hall liquid-to-insulator transition. DOI: 10.1103/PhysRevLett.87.216802 PACS numbers: 73.43. -f, 71.30. +h, 73.20.Jc Recently there has been a great renewal of interest on transitions from integer quantum Hall effect (IQHE) states to an insulator [1][2][3][4][5]. According to the scaling theory of localization [6], all electrons in a disordered two-dimensional system are localized in the absence of a magnetic field. In the presence of a strong magnetic field, a series of disorder-broadened Landau bands (LBs) will appear, and extended states reside at the centers of these bands while states at other energies are localized. The integrally quantized Hall (QH) plateaus are observed when the Fermi level lies in the localized states, with the value of the Hall conductance, s xy ne 2 ͞h, related to the number of filled LBs ͑n͒. As a function of the magnetic field, the Hall conductance jumps from one QH plateau to another when the Fermi energy crosses an extended-state level. Many previous studies [1][2][3][4][5] have been focused on how such a transition occurs.One overlooked issue regarding IQHE is the nature of a transition from one QH plateau to another. All existing theories assume it to be a continuous quantum phase transition. The fingerprint of a continuous phase transition is scaling laws around the transition point, and this assumption is mainly due to the early scaling experiments [7]. In the case of IQHE, a continuous quantum phase transition means algebraic divergence of the longitudinal Hallresistivity slope in temperature T at the transition point. However, recent experiments [5] showed that such slopes remain finite when they are extrapolated to T 0. This implies a nonscaling behavior around a transition point, contradicting the expectation of continuous quantum phase transitions suggested by the theories. It also means that one should reexamine the nature of plateau transitions.In this Letter we show that a narrow metallic phase may exist between two adjacent IQHE phases and between an IQHE phase and an insulating phase when the effect of interband mixing of opposite chirality is taken into account. Thus, it corresponds to two consecutive quantum phase transitions instead of one when the Hall conductance jumps from one plateau to another, consistent with the nonscaling behavior observed in experiments.According to the semiclassical theory [8], an electronic state in a strong magnetic field and in a smooth potential can be decomposed into a rapid c...

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Metallic Phase in Quantum Hall Systems due to Inter-Landau-Band Mixing

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“…[12]. In this case the random potential is V (x, y) = n,m V n,m (y)δ(na−x)δ(ma−y), where V n,m are random.…”

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Noninteracting Electrons and the Metal-Insulator Transition in Two Dimensions with Correlated Impurities

Hilke

2003

Phys. Rev. Lett.

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While standard scaling arguments show that a system of non-interacting electrons in two dimensions and in the presence of uncorrelated disorder is insulating, in this work we discuss the case where inter-impurity correlations are included. We find that for point-like impurities and an infinite inter-impurity correlation length a mobility edge exists in 2D even if the individual impurity potentials are random. In the uncorrelated system we recover the scaling results, while in the intermediate regime for length scales comparable to the correlation length, the system behaves like a metal but with increasing fluctuations, before strong localization eventually takes over for length scales much larger than the correlation length. In the intermediate regime, the relevant length scale is not given by the elastic scattering length but by the inter-impurity correlation length, with important consequences for high mobility systems.It is generally believed that a non-interacting two dimensional (2D) system in the presence of disorder is always insulating. This result is based on extensive work on scaling theory pioneered by the 'gang of four ' [1]. It has also received considerable amount of numerical support [2]. The general statement can be summarized as follows. In scaling theory the localization length in 2D is given by L c ≃ λe πkF λ/2 [3], where λ is the elastic scattering length. Hence, as soon as λ is finite, the localization length is finite. However, because λ appears in the exponent, when k F λ >> 1 the localization length can be extremely large and difficult to probe experimentally. In the standard Born approximation λ −1 ∼ n I , where n I is the two-dimensional impurity concentration, hence, a non-zero n I would lead to a finite L c . At zero temperature a finite localization length implies that the resistance diverges exponentially for a system size exceeding the localization length. Equivalently, for an infinite system, the resistance diverges when the temperature, T tends to zero because the phase coherence length l φ is infinite at zero T .When considering discrete models based on Anderson's disordered tight binding model [4] very similar results are obtained. All states are localized for any strength of disorder. These results apply only if all sites are uncorrelated. Indeed, even in 1D there exist special long and short range correlations in the disorder, which can lead to the existence of extended states in these systems [5]. Similarly special systems can also be found in 2D [6]. The exact conditions under which these localization conditions apply has recently gained considerable interest because 2D electronic systems confined in a variety of semiconducting structures, such as Si-MOSFETs and p-type GaAs/AlGaAs have shown a strong metallic-like T -dependence of the resistance [7,8,9] down to the lowest experimental T . In metallic, we understand a positive or vanishing derivative of the resistivity as a function of the temperature, i.e., ∂ρ/∂T ≥ 0.Because in these systems disorder is always present, whic...

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“…5 ͑iii͒ The number of extended states is proportional to ͱN,which means that the relative number of extended states tends to zero as 1/ͱN; therefore the delocalization properties are important only in small systems. 6 In conclusion, there are no periodic wave functions in the RDM, but close to the critical energies there exist periodic envelopes. These envelopes are given by the nondisordered properties of the system.…”

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Comment on “Periodic wave functions and number of extended states in random dimer systems”

1998

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In a recent report, Huang, Wu, and Gong ͓Phys. Rev. B 55, 11 018 ͑1997͔͒ studied numerically the electronic properties of a random dimer model ͑RDM͒ and found extended periodic wave functions near the critical energies. We argue that there are no periodic wave functions in the RDM but close to the critical energies there exist periodic envelopes. These envelopes are given by the nondisordered properties of the system. ͓S0163-1829͑98͒10337-5͔In a recent paper, 1 Huang, Wu, and Gong studied numerically the electronic properties of a random dimer model ͑RDM͒ and found extended periodic wave functions near the critical energies. They also claimed that the number of extended states is proportional to ͱN, where N is the size of the system, which is consistent with previous results.͑i͒ There are no periodic wave functions in the RDM. Indeed, even at the critical energies it was shown previously 2 that the wave functions have a constant norm but the phases are random. Essentially the wave function is i ϭe i i , where i is random. This does, however, lead to a constant envelope. We define the envelope as the ensemble of curves which include all possible values of the random wave function. In this point we present the analytical derivation against the main result presented in Ref. 1 and show that the envelope ͑and not the wave function itself͒ is periodic near the critical energies as long as the localization length exceeds the size of the system. The RDM can be analyzed in terms of the product of the following two transfer matrices 3They reduce to the negative identity matrix when EϭV A or EϭV B ͑critical energies͒, which means that T A and T B commute. Close to the critical energy, i.e., when ⌬EϭEϪV A is small, the commutator ͓T A ,T B ͔ϭO(⌬E), as opposed to the case without the dimer condition, where the commutator is always of order 1. The total system is described by a random as long as N is smaller than 1/⌬E. Our main result is that the envelope of Eq. ͑2͒ reproduces all the numerical figures of Ref. 1. Indeed Huang, Wu, and Gong studied numerically two cases:͑a͒ V A ϭϪ1, V B ϭ1, and EϭϪ1ϩ⌬E. As the envelope can be obtained from the solution of T B m , the corresponding wave functions can be written as i ϳcos(ki), where EϪV B ϭ2 cos(k). To first order in ⌬E we obtain kӍ Ϫͱ⌬E/2.

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Delocalization in continuous disordered systems

Cited by 48 publications

References 33 publications

Metallic Phase in Quantum Hall Systems due to Inter-Landau-Band Mixing

Metallic Phase in Quantum Hall Systems due to Inter-Landau-Band Mixing

Noninteracting Electrons and the Metal-Insulator Transition in Two Dimensions with Correlated Impurities

Comment on “Periodic wave functions and number of extended states in random dimer systems”

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