Non-Hermitian β-ensemble with real eigenvalues

By removing the Hermitian condition of the so-called β-ensemble of tridiagonal matrices, an ensemble of non-Hermitian random matrices is constructed whose eigenvalues are all real. It is shown that they belong to the class of pseudo-Hermitian operators. Its statistical properties are investigated.


I. INTRODUCTION
About a decade ago, evidences have been gathered that complex Hamiltonian may have real spectrum. 1It has also been identified that this property is a consequence of the invariance of the Hamiltonian under the combined parity and time-reversal transformations.This understanding lead to an extension of quantum mechanics to include this special class of non-Hermitian Hamiltonians with unbroken PT symmetry 2 (see 3 for a review).From a more general approach, the main property of this class of non-Hermitian operators is to be connected to their respective adjoints by the similarity transformation (1) which defines the so-called pseudo-Hermitian operators and means that operator and its adjoint have the same set of eigenvalues. 4Concomitantly, there has been attempts to find, in the context of random matrix theory (RMT), ensembles of matrices satisfying the above relation. 5,6 is effort produced, so far, an ensemble of 2×2 matrices, the case of arbitrary size matrices remaining open.Non-Hermitian random matrices were introduced by Ginibre 7 few years after Wigner's proposal of the Hermitian ones by the end of the 50's. 8He just removed the Hermitian condition from the three classes of matrices of the Gaussian ensemble, namely the orthogonal (GOE), the unitary (GUE), and the symplectic (GSE) with real, complex, and quaternion elements respectively.Denoted by Dyson index β = 1, 2, 4 which gives the degree of freedom of the number, these classes constitute what Dyson called the "three-fold way". 9The β-ensemble generalized this "three-fold way" by constructing a β dependent ensemble of tridiagonal Hermitian matrices in which β is a parameter that can assume any positive real value. 10,11 r the integer 1, 2, 4 values of β, the statistical properties of the Gaussian ensemble 12 are reproduced.
Ginibre's non-Hermitian classes of matrices have found applications in the study of physical open systems 13 and have been matter of recent investigation. 14However, they do not constitute good candidates to the construction of a pseudo-Hermitian RMT.It is unlikely that by imposing some restriction on their matrices, they can be made to satisfy Eq. (1).It is our purpose to show that, on the other hand, by removing the Hermitian condition of the tridiagonal matrices, the β-ensemble naturally extends RMT to include the pseudo-Hermiticity condition.This is shown below in the third section, after the presentation of the β-ensemble in the next section.

II. THE HERMITIAN β-ENSEMBLE
For completeness and future comparison we give in this section a summary of some basic results regarding the β-ensemble.A matrix of this ensemble is 11 where the diagonal elements are normally distributed and the sub-diagonal elements χ ν / √ 2 are distributed according to the distribution derived from the χ 2 -distribution.From this definition, it is found that the joint density distribution of the eigenvalues is given by 11 Although some results have already been obtained, 15,16 analytical derivations of exact expressions describing eigenvalue statistical properties for arbitrary values of β are still lacking.Results, however, can be inferred in the two extreme limits β → 0 and β → ∞ and, also, in the asymptotic regime of large matrices.For instance, since H 0 becomes diagonal with eigenvalues behaving independently.Therefore, in this limit, the βensemble becomes a Poissonian ensemble of uncorrelated levels.Regarding the infinite limit, as the two first moments of the distribution f ν (y) are and it follows that for large β, that is large ν, the average y diverges while the variance vanishes.Under this circunstance, the diagonal elements of H β becomes negligible compared to the sub-diagonals ones and the fluctuations are progressively supressed turning rigid the ensemble.Asymptotically, as the size n of the matrices increases, the eigenvalues of H β distribute themselves on the real axis with the same density the Gaussian ensemble. 11This means they occupy, as proved below, a compact support defined by Wigner semi-circle law 12 Finally, we consider the case of matrix of size n = 2 which analytically can be fully worked out.The important measure is the density distribution of the distance s between the couple of eigenvalues.It is given by which, after a simple calculation, leads to After the rescaling, s = t/ t , of the variable such that s = 1, the distribution P(s) with that generalizes the so-called Wigner surmise is obtained.In agreement with the above discussion, P(s) has the limits when β → 0 and when β → ∞ The distribution P(s) fit reasonably well the spacing between eigenvalues of large matrices.

III. THE PSEUDO-HERMITIAN β-ENSEMBLE
Considering the general case of a non-Hermitian tridiagonal matrix, H, with diagonal a = (a n , ..., a 1 ), upper sub-diagonal b = (b n − 1 , ..., b 1 ), and lower sub-diagonal b = (c n − 1 , ..., c 1 ), in which sub-diagonal elements are different from zero but otherwise matrix elements can assume any real value, then two theorems are easily proved: Theorem 1.The matrix H is pseudo-Hermitian.
Proof.Define the invertible diagonal matrix η whose elements are given by diag then, it is immediately verified that H and its adjoint H † satisfy Eq. (1) belonging therefore to the class of pseudo-Hermitian matrices.

Theorem 2. If the products b i c i are positive then all eigenvalues of H are real.
Proof.Define the invertible diagonal matrix η 1 2 whose elements are obtained by taking the square roots of the elements of η, that is then, the matrix is Hermitian with diagonal a and sub-diagonal Let us now introduce the random non-Hermitian matrix Ĥβ obtained from H β by allowing the elements of the two sub-diagonals to be different though sorted from the same f (n−i)β (y) distribution.In this case, the probability of matrices with zero elements is negligible, and, from the above theorems, Ĥβ is a pseudo-Hermitian matrix and its eigenvalues are real.Moreover, the eigenvalues of Ĥβ are the same of the matrix where κ ν are distributed according to the K-distribution where K 0 (x) is the modified Bessel function.This distribution has the limit such that Ĥ0 coincides with H 0 when β → 0. In the other extreme, the moments and implies that Ĥβ also becomes rigid when β → ∞.Therefore, the non-Hermitian and the Hermitian beta ensembles coincide in the two extreme limits.For large matrix sizes, the eigenvalues of the matrices K β , and a fortiori those of Ĥβ , are asymptotically distributed on the real axis as those of H β , namely according to the Wigner semicircle, Eq. ( 8).To prove this, we resort to the fact that the characteristic polynomials of the general tridiagonal matrix H satisfy, with P −1 = 0 and P 0 = 1, the recurrence relation In particular, applying to H β and K β , the independence of their elements can be used to define an average characteristic polynomial P n constructed by taking the average of Eq. ( 22).Considering first the Hermitian case, H β , a n = 0 and which replaced in Eq. ( 23) gives Comparing this relation with the recurrence relation of the Hermite polynomials, the identification follows. 11Therefore, eigenvalues of H β fluctuate around the zeros of the Hermite polynomials which are known to be distributed according to Eq. ( 8). 17 Considering now the non-Hermitian case, for large n we have implying the same eigenvalue asymptotic density distribution of the Hermitian case, namely Eq. (8).Therefore, although differences are to be expected in their fluctuations properties, the eigenvalues of the tridiagonal matrices of the non-Hermitian β-ensemble, asymptotically occupy, on the real axis, the same compact interval of the Hermitian ones.Though the above argument mathematically is not rigorous, the correctness of its prediction is shown in Fig. 1, where the eigenvalue density distribution generated by an ensemble of matrices of size N = 50 is compared with the semi-circle law.The histogram for β = 5 shows that at this relatively high value of the parameter, the eigenvalues place themselves in a crystal lattice structure fluctuating weakly around average positions.
Considering the 2×2 case, replacing in Eq. ( 9) the distribution f β (y) by g β (y), the spacing distribution between the eigenvalues is obtained.As before, a new variable s is defined by the rescaling which ensures that s = 1.Replacing in Eq. ( 28) the Bessel function by its behavior −log (x) at the origin we find that, at small separations, the eigenvalues repel one another as We remark that logarithmic dependent repulsion of pseudo-Hermitian matrices has already been reported. 18On the other hand, for large values of s, making in integral the change of variable v = 2u/t 2 we find that P(s) decays exponentially as without the power factor of the Hermitian case, Eq. (10).By comparing Eqs. ( 10) and (29), we conclude that the repulsion between eigenvalues becomes larger when the Hermitian condition is removed and, at the same time, the absence of the power factor in Eq. (30) shows that larger separations become less probable.In Fig. 2, a set of spacing distributions calculated with Eqs. ( 10) and (28) illustrates this discussion.They show the evolution of the spacings towards the Gaussian distribution when β decreases and the tendency for a more localized distribution as beta reaches higher values.Fig. 2 shows that the non-Hermitian ensemble moves faster in this direction explaining the structure in the eigenvalue density for the value β = 5 in Fig. 1.
In conclusion, we have shown that by an appropriate removal of the Hermitian condition, the β-ensemble of tridiagonal matrices becomes a model of pseudo-Hermitian matrices with real eigenvalues.This extension of the β-ensemble parallels in RMT the extension of quantum mechanics to incorporate complex Hamiltonian with real eigenvalues.

5 FIG. 1 .
FIG.1.For β = 0.5 and β = 5, the density distribution of eigenvalues of an ensemble of non-Hermitian matrices of size N = 50 is compared with the semi-circle law prediction.