F.3 Gaussian Symplectic Ensemble (GSE)
485
The eigenvalue nearest neighbor spacing distribution for GUE is
P
c (s)ds =
2
π
s
2 exp
−
s 2
2
ds.
(F.102)
F.3 Gaussian Symplectic Ensemble (GSE)
Systems with spin
1
2 particles that are not rotationally invariant but are time reversal
invariant are governed by Hamiltonians, ¯
H Q , that are quaternion real matrices.
Systems with strong spin-orbit coupling and no external magnetic field fall in this
category.
F.3.1 Quaternion Real Matrices: GSE
As described in Appendix E, an N×N quaternion is an N ×N matrix whose
matrix elements consist of 2×2 matrices. Therefore, such a matrix has 2N×2N
complex scalar matrix elements. The Hamiltonian that governs the GSE is an N ×N
quaternion real matrix and has the structure
¯
H Q =
⎛
⎜
⎜
⎜
⎜
⎝
¯
h 1,1 ¯
h 1,2 . . . ¯
h 1,N
¯
h
†
1,2
¯
h 2,2 . . . ¯
h 2,N
. . .
. . .
. . .
. . .
¯
h
†
1,N
¯
h
†
2,N . . . ¯
h N,N
⎞
⎟
⎟
⎟
⎟
⎠
,
(F.103)
where
¯
h j,j =
h
(0)
j,j 0
0 h
(0)
j,j
and ¯
h i,j =
h
(0)
i,j + ih
(1)
i,j h
(2)
i,j + ih
(3)
i,j
−h
(2)
i,j + ih
(3)
i,j h
(0)
i,j − ih
(1)
i,j
,
(F.104)
and the quantities h
(α)
i,j (for α = 0, 1, 2, 3) are real numbers. The spectrum of
energy eigenvalues of a quaternion real matrix is twofold degenerate (Kramers’
degeneracy). The matrix ¯
H Q has 2N diagonal matrix elements (only N of which
are independent) and 4×
1
2 N(N − 1) independent off-diagonal elements, for a total
of 2N 2 − N independent matrix elements. The invariant metric for the N ×N
quaternion, ¯
H Q , takes the form
(ds)
2
H Q
= 2
N
j =1
(dh
(0)
j,j )
2
+ 4
1≤i 3
α=0
(dh
(α)
i,j )
2 .
(F.105)