Non-Hermitian ß-ensemble with real eigenvalues

O. Bohigas and M. P. Pato

Citation: AIP Advances 3, 032130 (2013); doi: 10.1063/1.4796167 View online: http://dx.doi.org/10.1063/14796167

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Non-Hermitian f-ensemble with real eigenvalues

O. Bohigas1 and M. P. Pato2

1CNRS, Université Paris-Sud, UMR8626, LPTMS, Orsay Cedex, F-91405, France 2Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, 05314-970 Sao Paulo, S.P., Brazil

(Received 4 December 2012; accepted 8 March 2013; published online 15 March 2013)

By removing the Hermitian condition of the so-called i-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. Copyright 2013 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4796167]

I. INTRODUCTION

About a decade ago, evidences have been gathered that complex Hamiltonian may have real spectrum.1 It 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 symmetry2 (see3 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

H = n h n-1 (1)

which defines the so-called pseudo-Hermitian operators and means that operator and its adjoint have the same set of eigenvalues.4 Concomitantly, there has been attempts to find, in the context of random matrix theory (RMT), ensembles of matrices satisfying the above relation.5,6 This effort produced, so far, an ensemble of 2x2 matrices, the case of arbitrary size matrices remaining open.

Non-Hermitian random matrices were introduced by Ginibre7 few years after Wigner's proposal of the Hermitian ones by the end of the 50's.8 He 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 i = 1, 2, 4 which gives the degree of freedom of the number, these classes constitute what Dyson called the "three-fold way".9 The i-ensemble generalized this "three-fold way" by constructing a i dependent ensemble of tridiagonal Hermitian matrices in which i is a parameter that can assume any positive real value.10,11 For the integer 1, 2, 4 values of i, the statistical properties of the Gaussian ensemble12 are reproduced.

Ginibre's non-Hermitian classes of matrices have found applications in the study of physical open systems13 and have been matter of recent investigation.14 However, 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 i-ensemble naturally extends RMT to include the pseudo-Hermiticity condition. This is shown below in the third section, after the presentation of the i-ensemble in the next section.

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2158-3226/2013/3(3)/032130/6 3, 032130-1 © Author(s) 2013

II. THE HERMITIAN 0-ENSEMBLE

For completeness and future comparison we give in this section a summary of some basic results regarding the i -ensemble. A matrix of this ensemble is11

(N(0, 2) X(n-i)p \

X(n-i)i N(0,2) X(n-2)i

xp N(0, 2) )

X2i N(0, 2) Xi

where the diagonal elements are normally distributed and the sub-diagonal elements xv/V2 are distributed according to the distribution

2exp(- y2) yv—1

f„ (y) =

T[v/2]

derived from the x2-distribution. From this definition, it is found that the joint density distribution of the eigenvalues is given by11

P (xi, X2,..., Xn ) = Cn exp i -4)Tl lxJ - Xi I

\ k=i ' j>i

■ lp

Although some results have already been obtained,15,16 analytical derivations of exact expressions describing eigenvalue statistical properties for arbitrary values of i are still lacking. Results, however, can be inferred in the two extreme limits i ^ 0 and i ^to and, also, in the asymptotic regime of large matrices. For instance, since

lim fv(y) = 25(y),

H0 becomes diagonal with eigenvalues behaving independently. Therefore, in this limit, the i -ensemble becomes a Poissonian ensemble of uncorrelated levels. Regarding the infinite limit, as the two first moments of the distribution fv (y) are

7 r[(v + i)/2] „»i

(y) =--> ,,

r(v/2) V 2

( y2) =

it follows that for large i, that is large v, the average (y> diverges while the variance vanishes. Under this circunstance, the diagonal elements of Hp 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 Hp distribute themselves on the real axis with the same density the Gaussian ensemble.11 This means they occupy, as proved below, a compact support defined by Wigner semi-circle law12

p(x) = -iV2nP - x2. np

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

p(f) = j dou^ exp(-1 - «I) p(y)5

which, after a simple calculation, leads to

P(t) =-2-1p exp(_t2). (10)

r[(p + 1)/2] F

After the rescaling, s = t/ (t), of the variable such that (s) = 1, the distribution P(s) with

r(1 + p/2)

(t) = ' +p'' (11) W T[(1 + p )/2] ' ;

that generalizes the so-called Wigner surmise is obtained. In agreement with the above discussion, P(s) has the limits

P (s) = 2exp( --) (12)

n \ n J

when 3 ^ 0 and when p ^to

P(s) = 5 (s - 1). (13)

The distribution P(s) fit reasonably well the spacing between eigenvalues of large matrices.

III. THE PSEUDO-HERMITIAN ^-ENSEMBLE

Considering the general case of a non-Hermitian tridiagonal matrix, H, with diagonal a = (an, ..., ai), upper sub-diagonal b = (bn _ i,..., bi), and lower sub-diagonal b = (cn _ i,..., ci), 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 n whose elements are given by

,. , , A bn — 1 bn_1bn_2 bn_1bn_2...b1 \

diag(n) = 11,-,-,...,-I (14)

\ cn_ 1 cn_ icn_ 2 cn_ 1 Cn —2...c 1 /

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 bc\ are positive then all eigenvalues of H are real.

Proof. Define the invertible diagonal matrix n 2 whose elements are obtained by taking the square roots of the elements of n, that is

diag(n2) = 1, /1 J^2, bn_ibn_(15) \ V cn_1 V cn_icn_2 y cn_icn_2...ci J

then, the matrix

K = n2Hn_1 = n_2 (nHn_^ n2 = n_2Htn1 = Kt (16)

is Hermitian with diagonal a and sub-diagonal (^/bn—1cn—1, ...^b1c1). □

Let us now introduce the random non-Hermitian matrix Hp obtained from Hp by allowing the elements of the two sub-diagonals to be different though sorted from the same f(n_i)p(y) distribution. In this case, the probability of matrices with zero elements is negligible, and, from the above theorems, Hip is a pseudo-Hermitian matrix and its eigenvalues are real. Moreover, the eigenvalues

of Hiß are the same of the matrix

(N (0, 1) Kn-i)ß

I)ß N (0, 1) K(n—2)ß

1 A. _ 1

Kß = n2Hißi] 2 =

K2ß N (0, 1)

where kv are distributed according to the K-distribution

Kß N (0, 1)

gv (z) =

r2(v/2)

K0 (2z2

where K0(x) is the modified Bessel function. This distribution has the limit

lim gv (z) = 2S(z)

such that H0 coincides with H0 when i ^ 0. In the other extreme, the moments

r[(2v + 1)/4] r(v/2)

<z2> =

r[(v + 1)/2] r(v/2)

v>>1 v -> —

implies that Hß also becomes rigid when ß ^to. Therefore, the non-Hermitian and the Hermitian beta ensembles coincide in the two extreme limits.

For large matrix sizes, the eigenvalues of the matrices Kß, and a fortiori those of Hß, are asymptotically distributed on the real axis as those of Hß, 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 P0 = 1, the recurrence relation

Pn (x) = (an — x)Pn-1(x) — bn—1 cn — 1 Pn—2(x)-

In particular, applying to Hi and , the independence of their elements can be used to define an average characteristic polynomial (Pn> constructed by taking the average

< Pn > = (<an > —x )< Pn —1> — <bn — 1cn—1>< Pn—2>

of Eq. (22). Considering first the Hermitian case, Hß, <an> = 0 and

<bn — 1cn — 1> =

which replaced in Eq. (23) gives

X(n — 1)ß

2\ (n — 1)

<Pn > = —x <Pn—1> — ß <Pn—2>

Comparing this relation with the recurrence relation of the Hermite polynomials, the identification

follows.11 Therefore, eigenvalues of 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

<bn— 1cn —1> = <K(n —1)ß > =

r ß+1) r (^ ß)

Pn > = -

FIG. 1. For f = 0.5 and f = 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.

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 f-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 f = 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 2x2 case, replacing in Eq. (9) the distribution ff(y) by gf (y), the spacing distribution between the eigenvalues

t2fe-t r1 dv (t2v\ f , (t2v\

P(t) = r2,f/2)4>-vnI Tr=v»p(-)v K0\-)- (28)

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

P(s) - -s2f log(s). (29)

We remark that logarithmic dependent repulsion of pseudo-Hermitian matrices has already been reported.18 On the other hand, for large values of s, making in integral the change of variable v = 2u/t2 we find that P(s) decays exponentially as

P(s) - e-1 (t)2s2 (30)

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

FIG. 2. The spacing distributions calculated with Eqs. (10) (dashed lines) and (28) (full lines), are compared for the values 0.5 (black), 1 (red), 2 (blue), and 5 (green) of fi.

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 fi 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 fi = 5 in Fig. 1.

In conclusion, we have shown that by an appropriate removal of the Hermitian condition, the fi-ensemble of tridiagonal matrices becomes a model of pseudo-Hermitian matrices with real eigenvalues. This extension of the fi-ensemble parallels in RMT the extension of quantum mechanics to incorporate complex Hamiltonian with real eigenvalues.

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