The complex symplectic group is the set \begin{equation}\label{eq:20151211a} Sp(2 n , \mathbb{C} ) = \left\{ S \in M_{2 n} ( \mathbb{C} ) \Big| S^T J S = J\right\} \end{equation} Hereby is \( M_{2 n} ( \mathbb{C} )\) the set of complex matrices of size \(2 n \times 2 n\) and \(J \) is the \( 2 n \times 2 n\) matrix \begin{equation}\label{eq:20151211b} J = \left( \begin{array}{cc} 0& 1_n \\ -1_n & 0 \end{array} \right) \end{equation} with \( 1_n \) the identity matrix of size \( n \times n\). Notice that \( Sp(2 n , \mathbb{C} ) \) is defined with the transpose \( S^T \), not with the Hermitian conjugate \( S^\dagger \). One can verify that \( Sp(2 n , \mathbb{C} ) \) is a group with the usual matrix multiplication.

__Definition of dual of a matrix__

If \( A \) is a complex matrix of size \( 2 n \times 2 n \), then its dual \( A^D \) is defined as \begin{equation}\label{eq:20151211c} A^D = J^T A^T J \end{equation} Notice again that this definition uses the transpose, not Hermitian conjugation. The following formulas are easily proved \begin{align*} (AB)^D &= B^D A^D\\ A^{DD} & = A \end{align*} It is also easy to see that \begin{equation*} Sp(2 n , \mathbb{C} ) = \left\{ S \in M_{2 n} ( \mathbb{C} ) \Big| S^D S= 1 \right\} \end{equation*} A self-dual matrix is a complex matrix \( A \) of size \( 2 n \times 2 n \) which satisfies \begin{equation}\label{eq:20151211d} A^D = A \end{equation}

__Definition of GSE__

The GSE is the set of Hermitian, self-dual complex matrices \( H \) of size \( 2 n \times 2 n \) with probability density function \begin{equation}\label{eq:20151211e} P(H) dH = \exp\left( -\frac{1}{2} \mathrm{tr} (H^2 ) \right) dH \end{equation} where \( dH \) is the product over all independent components \( dH_{ij} \). The GSE is invariant under the action \begin{equation}\label{eq:20151211f} H \mapsto S^D H S \end{equation} with \( S \in USp(2 n ) = Sp(2 n , \mathbb{C} ) \cap U(2 n )\) . Hereby is \( U(2 n ) \) the group of unitary matrices of size \(2 n \times 2 n \). It is easy to see that:

- If \( H \) is self-dual, then \( S^D H S \) is also self-dual.
- Because \( S^DS = 1 = S^\dagger S\), it follows that \( S^D = S^\dagger\). If \( H \) is Hermitian, it thus follows that \( S^D H S = S^\dagger H S\) is also Hermitian.

__Block form of \( H \)__

If I write \( H \) as a block matrix with \( n \times n\) blocks, then \( H \) is Hermitian and self-dual if and only if \( H \) has the form \begin{equation}\label{eq:20151211g} H = \left( \begin{array}{cc} A& B \\ B^\dagger & A^T \end{array} \right) \end{equation} with \( A \) a Hermitian matrix of size \( n \times n \) and \( B \) a complex anti-symmetric matrix of size \( n \times n\). The probability density function \eqref{eq:20151211e} is Gaussian, with covariance \begin{equation}\label{eq:20151211h} \mathbb{E} \left[ H_{ij} H_{kl} \right] = \frac{1}{2} \left( \delta_{il} \delta_{jk} + J_{ik} J_{jl} \right) \end{equation} It is easy to prove \eqref{eq:20151211h} if one analyzes the blocks in \eqref{eq:20151211g} case by case.

__Algorithm to generate random matrices from the GSE__

One can draw a random matrix from GSE as follows:

- Draw a random matrix \( A \) from the Gaussian Unitary Ensemble, with normalization \begin{equation*} \mathbb{E} \left[ A_{ij} A_{kl} \right] = \delta_{il} \delta_{jk} \end{equation*}
- Calculate \( H = \frac{1}{2} \left( A + A^D \right) \).

## No comments:

## Post a Comment