Correlated random band matrices: Localization-delocalization transitions

Janssen, Martin; Pracz, Krystian

doi:10.1103/physreve.61.6278

Cited by 11 publications

(8 citation statements)

References 37 publications

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“…The Hamiltonian can be regarded as a random matrix which can be studied in the framework of the random matrix theory (RMT). The RMT has already been applied to IQHE and disordered superconductors [23][24][25]. We expect some insights into the present problem from RMT, which we leave to future work.…”

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confidence: 92%

Topological invariant in three-dimensional band insulators with disorder

Guo

2010

Phys. Rev. B

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Topological insulators in three dimensions are characterized by a Z2-valued topological invariant, which consists of a strong index and three weak indices. In the presence of disorder, only the strong index survives. This paper studies the topological invariant in disordered three-dimensional system by viewing it as a super-cell of an infinite periodic system. As an application of this method we show that the strong index becomes non-trivial when strong enough disorder is introduced into a trivial insulator with spin-orbit coupling, realizing a strong topological Anderson insulator. We also numerically extract the gap range and determine the phase boundaries of this topological phase, which fits well with those obtained from self-consistent Born approximation (SCBA) and the transport calculations.PACS numbers: 72.25.Hg, Time reversal invariant band insulators of noninteracting electrons are basically divided into two classes: the ordinary insulator and the topological insulator [1][2][3][4]. The latter is a novel phase of quantum matter. It has an insulating bulk gap and gapless edge or surface states. These gapless states are topologically protected and are immune to non-magnetic disorder. Recently another kind of non-trivial quantum phase termed topological Anderson insulator (TAI) has been predicted to exist in two dimensions (2D) [5][6][7] and three dimensions (3D) [8], which makes the situation more interesting. In the TAI phase, remarkably, the topologically protected gapless states emerge due to disorder.For systems without disorder, the topological phases can be characterized by studying the gapless states as obtained e.g. from diagonalizing the Hamiltonian in a geometry with edges or surfaces. They can also be characterized by the topological invariants calculated from the bulk Hamiltonian, which have been well studied in recent literature [9,10]. However in the disordered systems, gapless modes alone cannot unambiguously identify the topological phases because they may be localized in space. Instead, the transport properties are usually used to find these extended topologically protected modes. Similarly we may also use the topological invariant to characterize the topological phases induced by disorder. A question naturally arises: how to calculate the topological invariant in the presence of disorder.At first glance, it is not obvious how to generalize the present methods from translation invariant band insulators to the disordered systems. Let us first recall the integer quantum Hall effect (IQHE) where the generalization to the case with disorder is well understood. The topological quantum number in IQHE, which characterizes the quantized Hall conductivity, is known as the first Chern number [also refered to as the Thouless-KohmotoNightingale-den Nijs (TKNN) integer] and is closely related to Berry's phase. In the presence of disorder, the TKNN integers defined for a clean system can be generalized. By introducing generalized periodic boundary conditions and averaging over different boundary condi...

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confidence: 92%

Topological invariant in three-dimensional band insulators with disorder

Guo

2010

Phys. Rev. B

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“…In particular, the direct computation of eigenvalues and eigenvectors combined with finite-size scaling techniques can answer questions about Anderson localization due to disorder [1], quantum chaos in the energy levels or wave functions [2,3], etc. The difficulties become immense when one wants to treat many-body problems in the same framework since the dimension of the Hilbert space increases dramatically (exponentially) with the number of electrons.…”

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confidence: 99%

Level-spacing distribution of a fractal matrix

Katsanos

Evangelou

2001

Physics Letters A

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We diagonalize numerically a Fibonacci matrix with fractal Hilbert space structure of dimension d f = 1.8316... We show that the density of states is logarithmically normal while the corresponding level-statistics can be described as critical since the nearest-neighbor distribution function approaches the intermediate semi-Poisson curve. We find that the eigenvector amplitudes of this matrix are also critical lying between extended and localized.Pacs numbers: 71.30.+h, 71.23.Fb The numerical diagonalization of one-electron Hamiltonians in discrete space tight-binding lattices has been proved a very useful tool to treat the effect of disorder on the motion of quantum particles, alternatively to field theoretic methods. In particular, the direct computation of eigenvalues and eigenvectors combined with finite-size scaling techniques can answer questions about Anderson localization due to disorder [1], quantum chaos in the energy levels or wave functions [2,3], etc. The difficulties become immense when one wants to treat many-body problems in the same framework since the dimension of the Hilbert space increases dramatically (exponentially) with the number of electrons. However, also in this case the nearest-neighbor hopping bounded kinetic energy and the two-body character of the interaction guarantee that the Hamiltonian matrix structure is very sparse, often described as multifractal in the adopted Hilbert space. For example, in ref.[4] multifractal exponents D q were computed to characterize the Hilbert space structure of interacting many-body Hamiltonians.We shall treat a much simpler, albeit interesting problem, with a given Hamiltonian which consists of zeroes and ones, neither periodic nor random but with a simple fractal structure. We construct and diagonalize a Fibonacci matrix of order n of size N n × N n , with N n = N n−1 + N n−2 , N 0 = N −1 = 1. This matrix represents a self-similar fractal object itself and was introduced in ref. [5] in order to obtain the ground state of the square Ising antiferromagnet in a maximum critical field. The Fibonacci matrix is a transfer matrix connecting the possible ground states of the n × m and n × (m + 1) square lattices. For size N n it has the block form

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“…This definition is, however, not appropriate for precise and systematic calculations, because we cannot determine which wavefunctions are more multifractal if localization lengths of two wavefunctions are both close to the system size. We show that the correlation function of box-measures of multifractal critical wavefunctions 8,26,27 works quite well for the definition of ALS. In order to demonstrate how ALS influence critical properties of the Anderson transition, we examine the distribution function of the correlation dimension D 2 of critical wavefunctions.…”

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confidence: 79%

Anomalously localized states and multifractal correlations of critical wave functions in two-dimensional electron systems with spin-orbital interactions

Obuse

Yakubo

2004

Phys. Rev. B

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Anomalously localized states (ALS) at the critical point of the Anderson transition are studied for the SU(2) model belonging to the two-dimensional symplectic class. Giving a quantitative definition of ALS to clarify statistical properties of them, the system-size dependence of a probability to find ALS at criticality is presented. It is found that the probability increases with the system size and ALS exist with a finite probability even in an infinite critical system, though the typical critical states are kept to be multifractal. This fact implies that ALS should be eliminated from an ensemble of critical states when studying critical properties from distributions of critical quantities. As a demonstration of the effect of ALS to critical properties, we show that the distribution function of the correlation dimension D2 of critical wavefunctions becomes a delta function in the thermodynamic limit only if ALS are eliminated.

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Correlated random band matrices: Localization-delocalization transitions

Cited by 11 publications

References 37 publications

Topological invariant in three-dimensional band insulators with disorder

Topological invariant in three-dimensional band insulators with disorder

Level-spacing distribution of a fractal matrix

Anomalously localized states and multifractal correlations of critical wave functions in two-dimensional electron systems with spin-orbital interactions

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