{'fraction_non_alphanumeric': 0.04508320726172466, 'fraction_numerical': 0.00827029752899647, 'mean_word_length': 4.191623036649215, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Spectral Curves and Localization in Random Non-Hermitian Tridiagonal Matrices

Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The deformed spectrum is annular-shaped, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arc 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.


I. INTRODUCTION
Hermitian tridiagonal random matrices are studied in great detail, and many results are available on spectral properties such as density, statistics and localization of eigenvectors. They appear in several models of physics, as Dyson's random chains, Anderson's models for transport in disordered potentials, Ising spin models with random couplings, β-ensembles of tridiagonal random matrices. Hatano and Nelson 1 introduced a beautiful method to study the localization of eigenvectors, by forcing an asymmetry of upper and lower nondiagonal elements. Then the eigenvalues are driven from the real axis to curves in the complex plane, in patterns that measure the localization length of the corresponding eigenvectors.
Tridiagonal random matrices that are non-Hermitian from the start are less studied. They model systems with asymmetric hopping amplitudes 2,3,4 , describe properties of 1D random walks 5,6 or the evolution of population biology 7 . Their spectrum is complex. In this work we study how the Hatano-Nelson deformation modifies it, the occurrence of spectral curves and the connection with the localization of eigenvectors.
Let us then consider complex tridiagonal matrices with corners where all matrix elements are i.i.d. complex random variables. Here we use the uniform distribution in the unitary disk of the complex plane. This implies that the eigenvalue density of the ensemble is only a function of the modulus of the eigenvalue. The eigenvalues of a sample matrix of size n = 800 are shown in Fig.1 (left).
We next consider two deformations of the matrix M , by a complex parameter z = e ξ+iϕ : The two matrices are similar, M b (z) = SM (z n )S −1 , through a diagonal matrix with entries S ii = z i . The balanced matrix M b (z) is more convenient for numerical work. Since the matrices share the same set of eigenvalues, a rotation of z by 2π/n does not change the eigenvalues of M b (z). The eigenvalues of M b (z) are shown in Fig.1 (right). The distribution looks remarkable: a "circle" centered in the origin bounds an outer annular halo where the eigenvalues appear exactly in the same positions as those in the left figure. The inner region is void: all the eigenvalues that were there before deformation (ξ = 0) have moved to the boundary circle. As |z| becomes larger, see Fig.2, the circle enlarges as well, but the eigenvalues in  the annular halo do not move, until they are swept by the circle. For large ξ only the circle remains. This is not surprising: in the limit of large |z| the matrix M b (z) simplifies to bidiagonal. The eigenvalue equation can be solved explicitly and gives E n = z n b 1 · · · b n . Then the eigenvalues E k = |z|e ln |b| exp i(θ + 2πk/n) are equally spaced and lie on a circle of radius r such that log r = ξ+ log |b| . Eigenvalues on the circle and in the halo respond differently to the phase ϕ = arg z. As ϕ sweeps the Brillouin zone from 0 to 2π/n, only the eigenvalues sitting on the circle move (and remain therein), while the outer ones do not have measurable changes at all. This is illustrated in Fig.3, which also shows that an eigenvalue on the circle moves to the position of a neighboring one as ϕ is increased by 2π/n. By increasing the size n of the matrices, the "circle" is seen to become independent of the sample and more regular, (Fig.4).
The phenomenon described is analogous to what Hatano and Nelson 1 discovered for random tridiagonal Hermitian matrices (a k real, c k+1 = b * k ) where the undeformed eigenvalues (ξ = 0) are real. The deformation forces them to move into the complex plane and distribute along a two-arc loop, with possible external wings of unaffected eigenvalues in the real axis (Fig.5). The two-arc loop and wings of the Hermitian model correspond to the circle and annular halo of the non-Hermitian model discussed here.  In Section II and III we study the spectral density of the undeformed ensemble and the localization of eigenvectors, measured by the Lyapunov exponent or by the variance. In Section IV we explain the observed spectral features of the deformed ensemble by means of the Argument Principle of complex analysis and a spectral duality between the eigenvalues of M (z n ) and those of the transfer matrix.

II. THE SPECTRUM OF M
Since the matrix entries of M are chosen to be uniformly distributed in the unit complex disk, the average eigenvalue density of the ensemble, ρ 0 (E) = 1 n i δ 2 (E − E i ) , depends on |E|. The eigenvalue equation written for the component u k with highest absolute value, implies the inequality |E| ≤ |a k |+|b k |+|c k | ≤ 3. Thus the disk that supports the density has a radius not exceeding 3; the numerical evidence is that it has length 2.

III. THE LYAPUNOV EXPONENT
The numerical evaluation of the eigenvectors of M shows that they are strongly localized for all eigenvalues; indeed, for large matrix size, they decay exponentially (Anderson localization). The rate of decay is measured by the Lyapunov exponent, an asymptotic property of transfer matrices.
The transfer matrix of a realization M is the product of 2 × 2 random matrices: Its eigenvalues can be written as z n ± = e n(ξ±+iϕ±) . For large n the exponents ξ ± (E) become opposite: ξ + + ξ − = 1 n log | det t(E)| = 1 n n k=1 log |c k | − log |b k | → 0. According to the theory of random matrix products, for large n the positive exponent ξ + becomes independent of n and the realization of randomness, and converges to the Lyapunov exponent of the matrix ensemble. The Lyapunov exponent can be evaluated by an extension of Thouless formula to non-Hermitian matrices 2,8 , where ρ 0 is the eigenvalue density of the ensemble of matrices M . Note that for complex spectra, the equation for γ implies the Poisson equation ∇ 2 γ(E) = 2πρ 0 (E). Therefore γ(E) can be understood as the electrostatic potential generated by a charge distribution in the plane with density ρ 0 (E). For a distribution of matrix entries that is uniform in the unit disk, it is log |b| = −1/2 and ρ 0 is rotation invariant. Then the integral can be simplified: The integral 2π 0 log |r − r ′ e iϕ |dϕ = 2π log max(r, r ′ ) was used. N 0 (|E|) is the fraction of the spectrum inside the disk of radius |E|. For |E| larger than the spectral radius it is γ(|E|) = log(|E|) + 1/2, |E| ≥ 2.
The Lyapunov exponent is an increasing function of |E|. Its numerical evaluation is shown in Fig.7.
We checked numerically the exponential decay of eigenvectors with a rate given by γ(|E|). If u is an eigenvector of M , with components {u k } n k=1 , the numbers |u k | 2 provide the probability distribution for the position of a particle in the lattice 1 . . . n. We choose to measure the where k = k k|u k | 2 is the mean position of the particle. Other measures could be used, as the inverse participation ratio or the Shannon entropy; in this case the eigenvectors are peaked on small intervals, and the variance has a clear meaning.
For an ideal state v that is exponentially localized, |v k | 2 = (tanh γ) e −2γ|k| , (γn ≫ 1), the variance is var[ v] = sinh(1/γ). We use the same relation to compute a rate γ a from the numerically evaluated variance of an eigenstate u a . In Fig.8 we plot the numerical pairs (|E a |, γ a ) for the eigenvalues and eigenvectors of a single matrix M of size n = 800, together with the Lyapunov exponent γ(|E|), given by Thouless formula (7). The numerical data are consistent with the picture of exponential localization of eigenvectors.

IV. HOLE, HALO AND LOCALIZATION
When the parameter ξ is switched on, it appears in the corners of the matrix M (z n ) and modifies the boundary conditions in (4). In the transition from M to M (z n ), one expects that the eigenvalues of enough localized eigenstates do not change. An eigenstate u = {u k } of M (z n ) corresponds to an eigenstate S u of M b (z), with components z k u k . If |u k | ≈ e −γ|k| for large k, the factor e kξ delocalizes it if γ < ξ. This simple argument by Hatano and Nelson indicates a threshold value γ(|E|) = ξ at which eigenvalues must be drastically influenced by the deformation.
In Fig.9 we plot the variances (z axis) of the eigenvectors of a matrix M b (z = 2) of size n = 800, and the cor-responding complex eigenvalues (horizontal plane). The boundary of the circular hole is populated by the eigenvectors which are delocalized.
The existence of an empty disk and a halo of fixed eigenvalues for the deformed ensemble thus reflects the localization properties of the eigenvectors of M as a function of |E|, i.e. the function γ(|E|).
Since γ(0) is nonzero, there is a threshold value ξ min ≈ 0.291 below which no hole opens in the spectral support.

V. CONCLUSIONS
The Hatano Nelson deformation opens a hole in the spectrum of non Hermitian tridiagonal random matrices with i.i.d. matrix elements. The eigenvalues that are swept to the boundary of the hole correspond to states that are no longer Anderson localized. This is explained in terms of a spectral duality, stability of the Lyapunov exponent, and the Argument Principle.
Tridiagonal matrices with different strengths of randomness in the three diagonals would also show similar spectral features.
Aknowledgements: L.G.M. wishes to thank prof. I. Goldsheid for having inspired him the present investigation, and for useful comments.