C.2 Structure of the Hamiltonian Matrix
427
H m,n =< < m | ˆ
H | n >= (< < m | ˆ
K
† e
i
¯
h π ˆ
J y ) ˆ
H (e
−
i
¯
h π ˆ
J y ˆ
K| n >)
= [< < m |e
i
¯
h π ˆ
J y ˆ
K
† ˆ
H ˆ
Ke
−
i
¯
h π ˆ
J y | n >]
∗
= [< < m | ˆ
H | n >]
∗ , (C.34)
where we have used the fact that ˆ
K commutes with e
−
i
¯
h π ˆ
J y , ˆ
J y commutes with
ˆ
H , and ˆ
H is time reversal invariant, ˆ
K † ˆ
H ˆ
K = ˆ
H . For time reversal invariant
systems with half-integer spin and rotational invariance, the Hamiltonian is real
and symmetric.
(b) Nonrotationally Invariant Systems
For time reversal invariant systems with half-integer spin and with a Hamiltonian
that does not commute with J, it is no longer possible to find a basis of eigenstates
of ˆ
K that are real. Thus, the Hamiltonian matrix is no longer real symmetric. Let
us first consider some properties of the time reversal operator for this case. Let
ˆ
V = e
i
¯
h π ˆ
S y , where ˆ
V is a unitary operator. The time reversal operator is ˆ
K = ˆ
V ˆ
K o .
For half-integer spin,
ˆ
K
2
= ˆ
V ˆ
K o ˆ
V ˆ
K o = ˆ
V ˆ
V
∗
= −1.
(C.35)
Thus, ˆ
V is an antisymmetric unitary operator, ˆ
V T = − ˆ
V . Such an operator cannot
be diagonalized by a unitary transformation (Hua 1963). However, a basis can be
found in which the matrix representation of ˆ
V is given by matrix ¯
Z, where
¯
Z =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 0 . . .
−1 0 0 0 . . .
0 0 0 1 . . .
0 0 −1 0 . . .
. . .
. . .
. . .
. . .
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(C.36)
and ¯
K = ¯
Z ˆ
K o .
We can now find a condition on an arbitrary unitary matrix, ¯
U , that allows it
to commute with the time reversal operator. We require ¯
K ¯
U − ¯
U ¯
K = ¯
Z ˆ
K o ¯
U −
¯
U ¯
Z ˆ
K o = ˆ
K o ¯
Z ¯
U − ˆ
K o ¯
U ∗ ¯
Z = 0. Since ¯
U is unitary, ¯
U † = ¯
U −1 , and we can write
¯
U
T ¯
Z ¯
U = ¯
Z.
(C.37)
Equation (C.37) is the definition of a symplectic unitary matrix. Such matrices
commute with the time reversal operator for systems with half-integer spin. We will
generally denote symplectic unitary matrices as ¯
S, where ¯
S T ¯
Z ¯
S = ¯
Z and ¯
S † = ¯
S −1 .
We next determine what conditions time reversal invariance places on the
Hamiltonian matrix. Let us consider a 2N × 2N-dimensional Hamiltonian matrix
with complex matrix elements. It can be written as an N × N quaternion (see
Appendix E. For example, the (ij )th matrix element of the N × N quaternion
Hamiltonian matrix is given by
427
H m,n =< < m | ˆ
H | n >= (< < m | ˆ
K
† e
i
¯
h π ˆ
J y ) ˆ
H (e
−
i
¯
h π ˆ
J y ˆ
K| n >)
= [< < m |e
i
¯
h π ˆ
J y ˆ
K
† ˆ
H ˆ
Ke
−
i
¯
h π ˆ
J y | n >]
∗
= [< < m | ˆ
H | n >]
∗ , (C.34)
where we have used the fact that ˆ
K commutes with e
−
i
¯
h π ˆ
J y , ˆ
J y commutes with
ˆ
H , and ˆ
H is time reversal invariant, ˆ
K † ˆ
H ˆ
K = ˆ
H . For time reversal invariant
systems with half-integer spin and rotational invariance, the Hamiltonian is real
and symmetric.
(b) Nonrotationally Invariant Systems
For time reversal invariant systems with half-integer spin and with a Hamiltonian
that does not commute with J, it is no longer possible to find a basis of eigenstates
of ˆ
K that are real. Thus, the Hamiltonian matrix is no longer real symmetric. Let
us first consider some properties of the time reversal operator for this case. Let
ˆ
V = e
i
¯
h π ˆ
S y , where ˆ
V is a unitary operator. The time reversal operator is ˆ
K = ˆ
V ˆ
K o .
For half-integer spin,
ˆ
K
2
= ˆ
V ˆ
K o ˆ
V ˆ
K o = ˆ
V ˆ
V
∗
= −1.
(C.35)
Thus, ˆ
V is an antisymmetric unitary operator, ˆ
V T = − ˆ
V . Such an operator cannot
be diagonalized by a unitary transformation (Hua 1963). However, a basis can be
found in which the matrix representation of ˆ
V is given by matrix ¯
Z, where
¯
Z =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 0 . . .
−1 0 0 0 . . .
0 0 0 1 . . .
0 0 −1 0 . . .
. . .
. . .
. . .
. . .
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(C.36)
and ¯
K = ¯
Z ˆ
K o .
We can now find a condition on an arbitrary unitary matrix, ¯
U , that allows it
to commute with the time reversal operator. We require ¯
K ¯
U − ¯
U ¯
K = ¯
Z ˆ
K o ¯
U −
¯
U ¯
Z ˆ
K o = ˆ
K o ¯
Z ¯
U − ˆ
K o ¯
U ∗ ¯
Z = 0. Since ¯
U is unitary, ¯
U † = ¯
U −1 , and we can write
¯
U
T ¯
Z ¯
U = ¯
Z.
(C.37)
Equation (C.37) is the definition of a symplectic unitary matrix. Such matrices
commute with the time reversal operator for systems with half-integer spin. We will
generally denote symplectic unitary matrices as ¯
S, where ¯
S T ¯
Z ¯
S = ¯
Z and ¯
S † = ¯
S −1 .
We next determine what conditions time reversal invariance places on the
Hamiltonian matrix. Let us consider a 2N × 2N-dimensional Hamiltonian matrix
with complex matrix elements. It can be written as an N × N quaternion (see
Appendix E. For example, the (ij )th matrix element of the N × N quaternion
Hamiltonian matrix is given by