# Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form

$$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix}$$

where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are symmetric. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above condition is equivalent to the symmetry of $JH$ with

$$J=\begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix},$$

so it would be natural to consider the Gaussian orthogonal ensemble for $JH$. Generically, the eigenvalues come in quadruples: $\lambda, -\lambda, \bar\lambda, -\bar\lambda$

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You also want $B$ and $C$ to be symmetric. – Mark Meckes Jan 31 '13 at 15:20
Thanks. Edited. – Austen Jan 31 '13 at 19:56
Do you want a proof that the eigenvalues converge to the circular law? or a more refined analysis of the local correlation functions? or are you happy with the pictures given above? – David Renfrew Jul 18 '13 at 1:00

## 2 Answers

In the course of a physics project in my group, I have had an opportunity to learn more about the eigenvalue statistics of Hamiltonian matrices. (Our physics problem actually involved skew-Hamiltonian matrices, so I made a small detour, joined by Jonathan Edge & Jan Dahlhaus.)

The ensemble is the one you suggested: $2n\times 2n$ real matrices $H$ with Hamiltonian symmetry and normally distributed elements. It is convenient to rescale the eigenvalues $\varepsilon_k$ of $H$ by a factor $\sqrt{2n}$, and separate the real and imaginary parts:

$(2n)^{-1/2}\varepsilon_k=x_k+iy_k$.

The eigenvalue density in the complex plane $x+iy$ consists of three parts: a two-dimensional density $\rho_{c}(x,y)$ of the complex eigenvalues, a one-dimensional density $\rho_{r}(x)$ of the real eigenvalues and another one-dimensional density $\rho_{i}(y)$ of the imaginary eigenvalues.

Based on numerical experiments, I can offer three conjectures:

1) For large $n$, the rescaled complex eigenvalues $x_k+iy_k$ uniformly cover a disc of unit radius,

$\lim_{n\rightarrow\infty}n^{-1}\rho_{c}(x,y)=2/\pi$ for $x^2+y^2<1$.

2) For large $n$, the rescaled real eigenvalues $x_k$ uniformly cover the interval $-1<x<1$, with density

$\lim_{n\rightarrow\infty}n^{-1/2}\rho_{r}(x)=1/\sqrt{\pi}$.

Therefore the expectation value of the number $n_{r}$ of real eigenvalues satisfies $\lim_{n\rightarrow\infty}n^{-1/2}E[n_{r}]=2/\sqrt{\pi}$.

3) Also the rescaled imaginary eigenvalues $iy_{k}$ have a uniform density in the large-$n$ limit, in the interval $-1<y<1$, but this density is less than the density of the real eigenvalues. The expectation value of the number $n_{i}$ of imaginary eigenvalues satisfies $\lim_{n\rightarrow\infty}n^{-1/2}E[n_{i}]={\rm constant}\approx 0.72$.

Conjectures 1 and 2 were proven by Edelman and collaborators in the absence of Hamiltonian symmetry, so when all $(2n)^{2}$ real matrix elements of $H$ are chosen from independent normal distributions. [This is known as the (real) Ginibre ensemble.] Our numerics suggests that, for large matrices, the Hamiltonian symmetry only affects the (rescaled) eigenvalue distribution within a distance of order $n^{-1/2}$ from the imaginary axis.

By way of illustration, I include a plot of the eigenvalues $\varepsilon$ of $200$ real matrices of size $100\times 100$ (so $n=50$), with normally distributed matrix elements, both with the Hamiltonian symmetry (left) and without (right). These eigenvalues are shown without rescaling, so they cover a disc of radius $\sqrt{2n}=10$.

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It is curious that there seems to be an area with less eigenvalues around the imaginary axis (for Hamiltonian) or the real axis (for Ginibre). Is it an artifact of the numerical methods? I can imagine them having trouble in identifying properly purely imaginary/real eigenvalues. – Federico Poloni May 13 '13 at 8:47
thank you for asking: this is certainly not an artifact; in the Ginibre ensemble the eigenvalue density is known to vanish linearly on approaching the real axis; basically the depletion zone signals the accumulation of eigenvalues on the real axis, as if the eigenvalues are "attracted" towards the real axis, where they condense. It seems that the repulsion from the imaginary axis is stronger than linear, but this is something we have not yet checked. – Carlo Beenakker May 13 '13 at 11:49
I must be missing something then. Doesn't conjecture (1) (proved in a paper) say that they should be uniformly distributed in the limit? – Federico Poloni May 13 '13 at 13:31
@Federico: you're completely correct, I did not properly define the limit in which the density becomes uniform; I've corrected that now; the uniformity applies to the rescaled density; the nonuniformities in the rescaled density are of order $n^{1/2}$, so the factor $n^{-1}$ removes them in the large−$n$ limit. – Carlo Beenakker May 13 '13 at 14:16
Do I understand correctly that both real and imaginary axes have a $\sqrt{n}$ density of eigenvalues, but there is only repulsion around the imaginary axis? – Austen May 13 '13 at 14:56

The exponent $M=e^H\$ of a Hamiltonian matrix $H$ is a symplectic matrix. So you might equivalently ask for the distribution of the eigenvalues $\xi=e^{\lambda}\$ of $M$. There is an extensive literature on this in random matrix theory, I give some pointers below.

The random symplectic matrix $M$ appears as the transfer matrix for the wave equation of a disordered medium, with many applications in optics and electronics. The most natural ensemble for the transfer matrix is inherited from the circular ensemble of the scattering matrix $S$ for the same wave equation. There is a one-to-one algebraic relation between the symplectic transfer matrix $M$ and the unitary scattering matrix $S$. The random matrix ensemble for $S$ is the familiar circular ensemble.

Here are some references to papers on random transfer matrix ensembles. (Notice that the word Hamiltonian has a different meaning in these papers.)

1. S. Bachmann, M. Butz, W. De Roeck, Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law (2012).

2. M. Caselle, U. Magnea, Random matrix theory and symmetric spaces (2004).

3. J. An, Z. Wang, K. Yan, A generalization of random matrix ensembles (2005).

4. P. Devillard, Statistics of transfer matrices for disordered quantum thin metallic slabs (1991).
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Thanks Carlo! I'm not sure that the answer lies in the literature on the transfer matrix, however, as that mostly involves the (real) transmission eigenvalues, rather than the (generically complex) eigenvalues of the matrix H itself. – Austen Feb 1 '13 at 15:36