Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

Nishigaki, Shinsuke M.

doi:10.1103/physreve.59.2853

Cited by 65 publications

(47 citation statements)

References 59 publications

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“…An exact expression for the two level critical spectral function was conjectured in [16] for orthogonal and symplectic ensembles. In the context of the Anderson model, a similar result was conjectured by Nishigaki [10].…”

Section: Introductionsupporting

confidence: 70%

“…Recently, new random matrix ensembles [1,2,3,4,5,6,7,8] depending on additional parameters have been proposed to describe spectral correlations in this critical case. These new models for critical statistics have been successfully utilized to describe the spectral correlations of a disordered system at the Anderson transition in three dimensions [9,10], two dimensional Dirac fermions in a random potential [11], the quantum Hall transition [12] and of the QCD Dirac operator in a liquid of instantons [13,7].…”

Section: Introductionmentioning

confidence: 99%

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Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the ∆ 3 -statistic is accurately described by our analytical results.

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Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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“…(4.3) is a natural extension of the Painlevé V type equation [13] derived for the resolvent of the sine kernel (2.5) at a = 0. One of the authors (SMN) then confirmed that the LSDs at the mobility edge are well fitted, with a single tunable parameter a, to these analytic formulas from the deformed kernel (blue curves in Fig.2) for three symmetry classes [36]. The χ 2 /dof of the fitting in the range 0 ≤ s ≤ 5 (with bin-size 0.05) is as small as 0.23 (orthogonal) and 0.…”

Section: Pos(lattice 2013)018mentioning

confidence: 66%

“…Entrusting that these universality among three different ensembles [34] to be an indication of uniqueness of multifractal deformation of the classical random matrices, one of the authors (SMN) computed the LSDs of Ensemble I in three symmetry classes [35,36] (Fig.3). Ensemble I is…”

Section: Pos(lattice 2013)018mentioning

confidence: 99%

Critical statistics at the mobility edge of QCD Dirac spectra

Nishigaki¹,

Giordano²,

Kovács³

et al. 2014

Proceedings of 31st International Symposium on Lattice Field Theory LATTICE 2013 — PoS(LATTICE 2013)

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We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear σ models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scaleinvariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices.

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“…(25-27) with the kernel Eq. (28) give the same leading terms as Eqs. (13-15) with G(s, 0) given by Eq.…”

mentioning

confidence: 89%

Energy level dynamics in systems with weakly multifractal eigenstates: Equivalence to one-dimensional correlated fermions at low temperatures

Kravtsov

Tsvelik

2000

Phys. Rev. B

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It is shown that the parametric spectral statistics in the critical random matrix ensemble with multifractal eigenvector statistics are identical to the statistics of correlated 1D fermions at finite temperatures. For weak multifractality the effective temperature of fictitious 1D fermions is proportional to T ef f ∝ (1 − dn)/n ≪ 1, where dn is the fractal dimension found from the n-th moment of inverse participation ratio. For large energy and parameter separations the fictitious fermions are described by the Luttinger liquid model which follows from the Calogero-Sutherland model. The low-temperature asymptotic form of the two-point equal-parameter spectral correlation function is found for all energy separations and its relevance for the low temperature equal-time density correlations in the Calogero-Sutherland model is conjectured. PACS number(s): 72.15. Rn, 72.70.+m, 72.20.Ht, The spectral statistics in complex quantum systems are signatures of the underlying dynamics of the corresponding classical counterpart. The spectral statistics in chaotic and disordered systems in the limit of infinite dimensionless conductance g is described by the classical random matrix theory of Wigner and Dyson [1] (WD statistics). The WD statistics possess a remarkable property of universality: it depends only on the symmetry class with respect to the time-reversal transformation T . The three symmetry classes correspond to the lack of T -invariance (the unitary ensemble, β = 2); the Tinvariant systems with T 2 = 1 (the orthogonal ensemble, β = 1), and the T -invariant systems with T 2 = −1 (the symplectic ensemble, β = 4), respectively. The physical ground behind this universality is the structureless eigenfunctions in the ergodic regime which implies the invariance of the eigenfunction statistics with respect to a unitary transformation of the basis.In real disordered metals the eigenfunctions are not basis-invariant. The basis-preference reaches its extreme form for the strongly impure metals where all eigenfunctions are localized in the coordinate space but delocalized in the momentum space. In this case the spectral statistics is Poissonian in the thermodynamic (TD) limit.For low-dimensional systems d = 1, 2, where all states are localized in the TD limit, one can observe the smooth crossover from the WD to the Poisson spectral statistics as a function of the parameter ξ/L, where ξ is the localization radius and L is the system size. The dependence of the spectral correlation functions on the energy variable s = E/∆ (∆ is the mean level separation) is non-universal for finite L/ξ but all of them tend to the Poisson limit as L/ξ → ∞.In systems of higher dimensionality d > 2 the situation is different because of the presence of the Anderson localization transition at a critical disorder W = W c . In the metal phase W < W c the dimensional conductance g(L) → ∞ as L → ∞ and one obtains the WD spectral statistics in the TD limit. In the insulator state g(L) → 0 at L → ∞ and the limiting statistics is Poissonian. However, there i...

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Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

Cited by 65 publications

References 59 publications

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Critical statistics at the mobility edge of QCD Dirac spectra

Energy level dynamics in systems with weakly multifractal eigenstates: Equivalence to one-dimensional correlated fermions at low temperatures

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