Thouless Formula Research Articles

A useful formula for periodic Jacobi matrices on trees

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its "half-tree" restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees.

Winding Number, Density of States, and Acceleration

Abstract Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy, respectively. We will prove that the winding number is in fact Avila’s acceleration, it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila’s global theory. Communicating Editor: Prof Svetlana Jitomirskaya

Sets of Non-Lyapunov Behaviour for Scalar and Matrix Schrödinger Cocycles

Abstract We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_{n}$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_{n}^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $\rho (t)$ such that $\rho (t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.

Localization for One-Dimensional Anderson–Dirac Models

We prove spectral and dynamical localization for a one-dimensional Dirac operator to which is added an ergodic random potential, with a discussion on the different types of potential. We use scattering properties to prove the positivity of the Lyapunov exponent through Fürstenberg theorem. We get then the Hölder regularity of the integrated density of states through a new version of Thouless formula, and thus the Wegner estimate necessary for the multiscale analysis.

Probing band-center anomaly with the Kernel polynomial method

We investigate the anomalous behavior of localization length of a non-interacting one-dimensional Anderson model at zero temperature. We report numerical calculations of the Thouless expression of localization length, based on the Kernel polynomial method (KPM), which has an computational complexity, where N is the system size. The KPM results show excellent agreement with perturbative results in a large system size limit, confirming the validity of the Thouless formula. In the perturbative regime, we show that the KPM approximation of the Thouless expression produces the correct localization length at the band center in the thermodynamic limit. The Thouless expression relates localization length in terms of density of states in a one-dimensional disordered system. By calculating the KPM estimates of the density of states, we find a cusp-like behavior around the band center in the perturbative regime. This cusp-like singularity can not be obtained by approximate analytical calculations within the second-order approximations, reflects the band-center anomaly.

Probing the Global Delocalization Transition in the de Moura-Lyra Model with the Kernel Polynomial Method

In this paper, we report numerical calculations of the localization length in a non-interacting one-dimensional tight-binding model at zero tem¬perature, holding a correlated disorder model with an algebraic power-spectrum (de Moura-Lyra model). Our calculations were based on a Kernel Polynomial implementation of the Thouless formula for the inverse localization length of a general nearest-neighbor 1D tight-binding model with open boundaries. Our results confirm the delocalization of all eigenstates in de Moura-Lyra model for α > 1 and a localization length which diverges as ξ ∝ (1 – α)–1 for α → 1–, at all energies in the weak disorder limit (as previously seen in [12]).

Anomalies in the 1D Anderson model: Beyond the band-centre and band-edge cases

We consider the one-dimensional Anderson model with weak disorder. Using the Hamiltonian map approach, we analyse the validity of the random-phase approximation for resonant values of the energy, E=2cos(πr), with r a rational number. We expand the invariant measure of the phase variable in powers of the disorder strength and we show that, contrary to what happens at the centre and at the edges of the band, for all other resonant energies the leading term of the invariant measure is uniform. When higher-order terms are taken into account, a modulation of the invariant measure appears for all resonant values of the energy. This implies that, when the localisation length is computed within the second-order approximation in the disorder strength, the Thouless formula is valid everywhere except at the band centre and at the band edges.

Thouless bandwidth formula in the Hofstadter model

We generalize the Thouless bandwidth formula to its nth moment. We obtain a closed expression in terms of polygamma, zeta and Euler numbers.

Landauer–Büttiker and Thouless Conductance

In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample S connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer-B\"uttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of S. In this spirit, we define Landauer-B\"uttiker crystalline currents by extending the Landauer-B\"uttiker formula to a setup where the sample S is replaced by a periodic structure whose unit cell is S. We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.

A Thouless formula and Aubry duality for long-range Schrödinger skew-products

In this paper, we study the dynamical properties of a class of ergodic linear skew-products which includes the linear skew-products defined by quasi-periodic Schrödinger operators and their duals, in Aubry sense, when the potential is a trigonometric polynomial. Notably, these linear skew-products preserve an adapted complex-symplectic structure. We prove a Thouless formula relating the sum of the positive Lyapunov exponents and the logarithmic potential associated with the density of states of the corresponding operator. In particular, for quasi-periodic Schrödinger operators and their duals, we prove an identity for the upper Lyapunov exponent of the skew-product and the sum of the positive Lyapunov exponents of their dual, which generalizes the well-known formula for the Almost Mathieu. We illustrate these identities with some numerical illustrations.

Localization Properties of the Chalker–Coddington Model

The Chalker Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove firstly that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly that this implies spectral localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov exponent which is independent of M.

On Information Rates of the Fading Wyner Cellular Model via the Thouless Formula for the Strip

In this paper, we apply the theory of random Schrödinger operators to the analysis of multiusers communication channels similar to the Wyner model, which are characterized by short-range intercell interference. With <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> the channel transfer matrix, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HH</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sup> is a narrow band matrix, a fact that does not permit the use of classical random matrices theory. On the other hand, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HH</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sup> is in many aspects similar to a random Schrödinger operator. We relate the per-cell sum-rate capacity of the channel to the integrated density of states of a random Schrödinger operator; the latter is then related to the top Lyapunov exponent of a random sequence of matrices via a version of the Thouless formula. We also derive several bounds on the limiting per-cell sum-rate capacity, some based on the theory of random Schrödinger operators, and some derived from information theoretical considerations. Finally, we get explicit results in the high-signal-to-noise ratio (SNR) regime for some particular cases.

Disk-annulus transition and localization in random non-Hermitian tridiagonal matrices

Eigenvalues and localization of eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano–Nelson deformation. The support of the spectrum undergoes a disk to annulus transition, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are structures that correspond to the two arcs and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the argument principle. This disk-annulus transition is reminiscent of Feinberg and Zee's transition observed in full complex random matrices.

A Matrix-Valued Point Interactions Model

We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that, away from a discrete set, its Lyapunov exponents do not vanish. For this we use a criterion by Gol’dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.

HÖLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR MATRIX-VALUED ANDERSON MODELS

We study a class of continuous matrix-valued Anderson models acting on L2(ℝd) ⊗ ℂN. We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N, we prove the Hölder continuity of the Integrated Density of States under some assumption on the group GμE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GμE is verified. Therefore, the general results developed here can be applied to this model.

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive nonlinear Schrödinger equation with a Gaussian random pulse as an initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of scattering data of the Zakharov-Shabat operator that determine the asymptotics of the eigenfunctions. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of information transmission through nonlinear optical fibers.

The Thouless formula for random non-Hermitian Jacobi matrices

Random non-Hermitian Jacobi matricesJ n of increasing dimensionn are considered. We prove that the normalized eigenvalue counting measure ofJ n converges weakly to a limiting measure μ asn→∞. We also extend to the non-Hermitian case the Thouless formula relating μ and the Lyapunov exponent of the second-order difference equation associated with the sequenceJ n . The measure μ is shown to be log-Hölder continuous. Our proofs make use of (i) the theory of products of random matrices in the form first offered by H. Furstenberg and H. Kesten in 1960 [8], and (ii) some potential theory arguments.

The stability of the renormalization group as a diagnostic tool for localization andits application to the Fibonacci case

A discussion of the investigation of electronic localization at the thermodynamicallimit is given; we show that the usual methods involve practical difficulties,especially for quasicrystals. A method based on the scaling of bands of asupercrystal is proposed. Then, the nature of the localization is related to thestability of the renormalization group around the fixed points of a trace map. Forquasiperiodic systems, the scaling exponents are obtained using modifiedLyapunov exponents and a Thouless formula. The case of the Fibonacci chain isconsidered as an example, and analytical expressions for the scaling exponents arefound.

Density of States and Thouless Formula for Random Unitary Band Matrices

We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can also be considered as a unitary version of the Anderson model. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.

Effects of Correlated Noise in Random-Mass Dirac Fermions

In the previous paper, we studied the random-mass Dirac fermion in one dimension by using the transfer-matrix methods. We furthermore employed the imaginary vector potential methods for calculating the localization lengths. Especially we investigated effects of the nonlocal but short-range correlations of the random mass. In this paper, we shall study effects of the long-range correlations of the random mass especially on the delocalization transition and singular behaviours at the band center. We calculate localization lengths and density of states for various nonlocally correlated random mass. We show that there occurs a "phase transition" as the correlation length of the random Dirac mass is varied. The Thouless formula, which relates the density of states and the localization lengths, plays an important role in our investigation.