General Formalism[edit]
Spectral function is defined from the imaginary (skew-Hermitian) part of retarded Green's function
.
The spectral function contains full information of the Green's function. Both the retarded function and the Matsubara function can be restored from the spectral function,
,
.
As related by the Kramers-Kronig relation, the real part of G and the spectral function A are of opposite parity.
If (the real part of) G(-ω)=G(ω) is even, then A(-ω)=-A(ω) is odd and
.
If (the real part of) G(-ω)=-G(ω) is odd, then A(-ω)=A(ω) is even and
.
Diffusive Dynamics[edit]
For diffusive dynamics, the Green's function is given by
,
where H is the Hamiltonian governs the diffusion rate, and the metric η is the matter number operator. η is always positive definite for fermion system, but not necessarily for boson system.
The spectral function is therefore
.
Diagonal Hamiltonian[edit]
Consider the Hamiltonian in its diagonal representation,
,
where n labels the energy level
.
The Green's function is
.
The spectral function is
.
SU(2) Hamiltonian[edit]
The SU(2) Hilbert space is a dim-2 space equipped with unitary metric
, any Hermitian operator acting on which is a SU(2) Hamiltonian. The Hamiltonian can be represented by the 2×2 matrix, which can be in general decomposed into Pauli matrices
and
,
.
The Green's function is given by
.
The corresponding spectral function reads,
,
where
and
.
SU(1,1) Hamiltonian[edit]
The SU(1,1) Hilbert space is a dim-2 space equipped with metric
, any Hermitian operator acting on which is a SU(1,1) Hamiltonian. Still take the Hamiltonian in terms of Pauli matrices
.
Note that the metric is not definite. The Green's function is given by
.
By introducing
,
,
,
, one finds
,
such that the result in the previous section can be used, yielding
,
and the spectral function
,
where
,
and
.
For the SU(1,1) Hamiltonian, its parameters should satisfy the condition
, otherwise h will be imaginary, and the spectrum will not be stable.
Wave Dynamics[edit]
Appendix[edit]
Taking Imaginary Part[edit]
Technically the Im is taken by factorizing the denominator and using the identity
,
derived from which, the following formula will be useful,
,
.
Numerical Handling of δ Functions[edit]
![{\displaystyle \delta (\omega -E)=\left\{{\begin{array}{ll}1/d\omega &{\text{if }}\omega =E;\\0&{\text{if }}\omega \neq E.\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a5e15b715878a281312e8c71c1b8973d9ba168)
See Also[edit]
Reference[edit]
- Gerald D. Mahan (2000). Many-Particle Physics (3rd Edition). Kluwer Academic/Plenum Pulishers. ISBN 0-306-46338-5.