Statistics of rare events in disordered conductors

We determine the statistical properties of wave functions in disordered quantum systems by exact diagonalization of one-, two-and quasi-one dimensional tight-binding Hamiltonians. In the quasi-one dimensional case we find that the tails of the distribution of wave-function amplitudes are described by the non-linear σ-model. In two dimensions, the tails of the distribution function are consistent with a recent prediction based on a direct optimal fluctuation method. 72.15.Rn,71.23.An, Typeset using REVT E X 1 It is well established that disordered quantum systems in the metallic regime (i.e., in the limit of weak disorder) and highly excited classically chaotic quantum systems exhibit universal quantum fluctuations that can be described by random matrix theory (RMT): statistical properties, on the scale of the mean level spacing, of eigenvalues, eigenfunctions, and matrix elements are universal, i.e., they do not depend on the microscopic details of the systems under consideration [1][2][3][4][5].However, in ballistic, classically chaotic quantum systems, non-hyperbolic phase-space structures may lead to deviations from universal RMT statistics [3]. Similarly fluctuations in disordered, classically diffusive quantum systems may deviate considerably from the RMT predictions due to increased localization. This effect is naturally very significant in the tails of distribution functions [6] (corresponding to rare events) of wave-function amplitudes [7][8][9][10][11][12][13], of the local density of states [7,12], of inverse participation ratios [12,13] and of NMR line shapes [7]. In all of these cases (with the exception of Ref. [7] which deals with one-dimensional (1D) systems), the distribution functions have been calculated using the non-linear σ-model (NLSM). Very recently, this approach has been extended to ballistic systems [14,15] (see also [16][17][18] We use the Anderson model of localization [23] which is a tight-binding model on aHere r = (x, y, . . .) denotes sites on the lattice, c † r and c r are the usual creation and annihilation operators, the hopping amplitudes are t rr ′ = 1 for nearest neighbour sites and zero otherwise. The on-site potential υ r is taken to be uncorrelated white noise, with zero mean and variance υ r υ r ′ = δ rr ′ W 2 /12. The parameter W characterizes the disorder strength.As is well-known (see for instance [24,25] We refer to these two cases by assigning, as usual, the parameter β = 1 to the former and β = 2 to the latter. The metallic regime is characterized by g ≫ 1 whereis the dimensionless conductance (we take = 1). Here · · · W denotes an average over disorder realisations. The wave functions are normalizedis a window function of width η, centered around E = 0 and normalized to unity. In the following we describe the results of our calculations and compare them to the predictions of Refs. [8][9][10][11][12][13]19].1D case. The eigenstates in a disordered chain are localized with localization length ξ = 4ℓ. According to Ref.[7] the distribution of wave-function amp...

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“…In the following we describe the results of our calculations and compare them to the predictions of Refs. [8][9][10][11][12][13]19]. …”

Section: Introductionmentioning

confidence: 99%

Exact diagonalization study of rare events in disordered conductors

Röemer¹,

Uski²,

Schreiber³

et al. 2000

Phys. Rev. B

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“…As was pointed out in Ref. 8, these may modify the predictions of the DNLSM (see also Refs. 9 and 10).…”

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confidence: 74%

“…In 3D, by contrast, results corresponding to (8-10) are not known 16 , but it is expected that the background intensity of ALS is characteristically different from that in Q1D samples: ALS in 3D systems are expected 6,8 to exhibit a very narrow maximum near the localisation centre (the width of this peak is a matter of current debate 6,8 ). The background, however, is expected 6,16 to decay very quickly to V |ψ(r)| 2 t ≃ 1.…”

Section: The Anderson Model Is Defined By the Tight-bindingmentioning

confidence: 97%

“…Such a situation could occur if ALS were created not through semiclassical diffusion, but through local potential wells trapping the electrons 8,15 . The mechanism giving rise to ALS is expected 6,8,16 to crucially depend on the dimensionality of the system and may determine their spatial structure.Finally, within the DNLSM, it is possible to obtain the wave-function statistics directly in Q1D, using a transfermatrix technique. In 2D and 3D, on the other hand, a further saddle-point approximation is necessary.…”

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confidence: 99%

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Spatial structure of anomalously localized states in disordered conductors

Uski¹,

Mehlig²,

Schreiber³

2002

Phys. Rev. B

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We study the spatial structure of wave functions with exceptionally high local amplitudes in the Anderson model of localisation. By means of exact diagonalisations of finite systems, we obtain and analyse images of these wave functions: we compare the spatial structure of such anomalously localised states in quasi-one-dimensional samples to that in three-dimensional samples. In both cases the average wave-function intensity exhibits a very narrow peak. The background intensity, however, is found to be very different in these two cases: in three dimensions, it is constant, independent of the distance to the localisation centre (as expected for extended states). In quasi-one dimensional samples, on the other hand, it is redistributed towards the localisation centre and approaches a characteristic form predicted in [A. D. Mirlin, Phys. Rep. 326, 249 (2000)].Statistical properties of physical observables in disordered electronic quantum systems have attracted considerable interest in the last decade. In such systems, quantum interference may cause the (non-interacting) conduction electrons to localise 1 . In three dimensions (3D), this Anderson localisation occurs when the disorder strength exceeds a critical value. Beyond this value (which depends on the Fermi energy and the symmetries of the system), electron wave functions are confined to a limited spatial region of the sample.In the metallic regime, on the contrary, wave functions typically spread over the whole sample, and they contribute to electron transport. However, some wave functions show localised behaviour even in this weakly disordered regime (these are so-called anomalously localised states, abbreviated as ALS in the following). This leads to a non-zero, albeit small, probability of observing exceptionally large wave-function amplitudes, often in the form of a log-normal distribution function of wavefunction intensities. ALS in electronic conductors have been studied intensively in recent years, using the socalled diffusive non-linear sigma model (DNLSM) 2,3,4 . An overview of the main results and predictions based on the DNLSM is given in Ref. 6. Moreover, possible complications due to non-diffusive, so-called ballistic effects on length scales smaller than the mean free path are discussed. As was pointed out in Ref. 8, these may modify the predictions of the DNLSM (see also Refs. 9 and 10). ALS are expected to occur in lower-dimensional disordered systems, too, when the disorder is weak.These interesting analytical results have motivated a number of numerical studies: in Ref. 7, for example, log-normal statistics of wave-function amplitudes in two-dimensional (2D) conductors near the delocalisationlocalisation transition was observed. In Ref. 11 it was confirmed that as the disorder is reduced to reach the weakly disordered regime, the distribution function remains log-normal. It has, however, not been possible to resolve a discrepancy (between the DNLSM 5,6 and the so-called direct optimal fluctuation method 8 ) in the prediction of the parameters ...

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“…They are anomalously rare in weakly disordered conductors, i.e., the tail of f E (t) is not generated by a large number of more or less equal-weight impurity configurations. 11 The effect of spectral averaging is not a priori obvious. 20 Namely, the short scale physics to be discussed below can correlate eigenstates with neighboring eigenvalues and therefore produce artifacts of the numerical procedure employed to get f E (t) in a finite-size system with discrete spectrum, when the original definition (1) in terms of delta functions is abandoned.…”

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confidence: 99%

Quest for rare events in mesoscopic disordered metals

Nikolić

2001

Phys. Rev. B

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The study reports on the first large statistics numerical experiment searching for rare eigenstates of anomalously high amplitudes in three-dimensional diffusive metallic conductors. Only a small fraction of a huge number of investigated eigenfunctions generates the far asymptotic tail of their amplitude distribution function. The relevance of the relationship between disorder and spectral averaging, as well as of the quantum transport properties of the investigated mesoscopic samples, for the numerical exploration of eigenstate statistics is divulged. The quest provides exact results to serve as a reference point in understanding the limits of approximations employed in different analytical predictions, and thereby the physics (quantum vs semiclassical) behind large deviations from the universal predictions of random matrix theory.PACS numbers: 72.15.Rn, 05.45.Mt, Despite years of extensive studies, initiated by the seminal work of Anderson, 1 the problem of quantum particle in a random potential ("disorder") still poses new challenges. The biggest shift in approach and intuition came with the advent of mesoscopic quantum physics 2 -from the critical phenomena like description 3 of the localization-delocalization (LD) transition, to realization that complete understanding of quantum interference effects, generating LD (zero-temperature) transition for strong enough disorder, requires to study the full distribution functions 4 of relevant physical quantities (such as conductance, local density of states, current relaxation times, etc.5 ) in finite-size disordered electronic systems. Even in the diffusive metallic conductors, the tails of such distribution functions show large deviations from the ubiquitous Gaussian distributions, expected in the limit of infinite dimensionless zero-temperature conductance g = G/(e 2 /π ). These asymptotic tails are putative precursors of the developing Anderson localization, occurring at 3 g ∼ 1. A conjecture about unusual eigenstates, being microscopically responsible for the asymptotic tails 4 of various distribution functions related to transport, was put forward early in the development of the mesoscopic program.5 However, it is only recently 6 that eigenstate statistics have come into the focus of mesoscopic community. Similar investigation in the guise of quantum chaos 2,7 started a decade earlier (leading to the concept of scarring 8 in quantum chaotic wave functions). Thus, the notion of "prelocalized" states has emerged from the studies of quantum disordered systems 5,9,10 of finite g. In three-dimensional (3D) conductors these states have sharp amplitude peaks on the top of a homogeneous background. Their appearance, even in good metals, has been viewed as a sign of an incipient localization.5 However, the non-universality of the prelocalized states tempers this view, 11 nonetheless, pointing out to the importance of non-semiclassical effects 11 in systems whose typical transport properties are properly described by semiclassical theories [including the perturbative q...

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Statistics of rare events in disordered conductors

Cited by 57 publications

References 14 publications

Exact diagonalization study of rare events in disordered conductors

Exact diagonalization study of rare events in disordered conductors

Spatial structure of anomalously localized states in disordered conductors

Quest for rare events in mesoscopic disordered metals

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