User:EverettYou/Spectral Function

General Formalism
Spectral function is defined from the imaginary (skew-Hermitian) part of retarded Green's function
 * $$A(\omega)=-2\,\mathrm{Im} G(\omega+i0_+)\equiv i(G(\omega+i0_+)-G^\dagger(\omega+i0_+))$$.

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,
 * $$G(\omega+i0_+)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\frac{A(\omega')}{\omega+i0_+-\omega'}$$,
 * $$G(i\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\frac{A(\omega')}{i\omega-\omega'}$$.

Parity
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
 * $$G(\omega )=\int _0^{\infty }\frac{d\omega '}{2\pi }\frac{2\omega 'A\left(\omega '\right)}{\omega ^2-\omega '^{2}}$$.

If (the real part of) G(-ω)=-G(ω) is odd, then A(-ω)=A(ω) is even and
 * $$G(\omega )=\int _0^{\infty }\frac{d\omega '}{2\pi }\frac{2\omega A\left(\omega '\right)}{\omega ^2-\omega '^{2}}$$.

Diffusive Dynamics
For diffusive dynamics, the Green's function is given by
 * $$G(\omega)=(\omega\eta-H)^{-1}$$,

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
 * $$A(\omega)=-2\,\mathrm{Im}((\omega+i0_+)\eta-H)^{-1}$$.

Diagonal Hamiltonian
Consider the Hamiltonian in its diagonal representation,
 * $$H=\mathrm{diag}_n\epsilon_n$$,

where n labels the energy level $$\epsilon_n$$.

The Green's function is
 * $$G(\omega)=\mathrm{diag}_n\frac{1}{\omega\eta_n-\epsilon_n}$$.

The spectral function is
 * $$A(\omega)=\mathrm{diag}_n\,2\pi\eta_n\delta(\omega-\eta_n\epsilon_n)$$.

SU(2) Hamiltonian
The SU(2) Hilbert space is a dim-2 space equipped with unitary metric $$\eta=\sigma_0$$, 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 $$\sigma_0$$ and $$\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$$,
 * $$H = \epsilon_0\sigma_0+\vec{\epsilon}\cdot\vec{\sigma}

=\epsilon_0\sigma_0+\epsilon_1\sigma_1+\epsilon_2\sigma_2+\epsilon_3\sigma_3$$.

The Green's function is given by
 * $$G(\omega)=((\omega-\epsilon_0)\sigma_0-\vec{\epsilon}\cdot\vec{\sigma})^{-1}=\frac{(\omega-\epsilon_0)\sigma_0+\vec{\epsilon}\cdot\vec{\sigma}}{(\omega-\epsilon_0)^2-\epsilon^2}$$.

The corresponding spectral function reads,
 * $$A(\omega) = \left(\sigma_0+\frac{\vec{\epsilon}\cdot\vec{\sigma}}{\epsilon}\right) \pi\delta(\omega-\epsilon_+) + \left(\sigma_0-\frac{\vec{\epsilon}\cdot\vec{\sigma}}{\epsilon}\right) \pi\delta(\omega-\epsilon_-)$$,

where $$\epsilon_\pm = \epsilon_0\pm\epsilon$$ and $$\epsilon^2=\epsilon_1^2+\epsilon_2^2+\epsilon_3^2$$.

SU(1,1) Hamiltonian
The SU(1,1) Hilbert space is a dim-2 space equipped with metric $$\eta=\sigma_3$$, any Hermitian operator acting on which is a SU(1,1) Hamiltonian. Still take the Hamiltonian in terms of Pauli matrices
 * $$H = \epsilon_0\sigma_0+\vec{\epsilon}\cdot\vec{\sigma}

=\epsilon_0\sigma_0+\epsilon_1\sigma_1+\epsilon_2\sigma_2+\epsilon_3\sigma_3$$.

Note that the metric is not definite. The Green's function is given by
 * $$G(\omega)=(\omega\sigma_3-H)^{-1}=\sigma_3(\omega-H\sigma_3)^{-1}$$.

By introducing $$h_0=\epsilon_3$$, $$h_1=i\epsilon_2$$, $$h_2=-i\epsilon_1$$, $$h_3=\epsilon_0$$, one finds
 * $$H\sigma_3=h_0\sigma_0+\vec{h}\cdot\vec{\sigma}$$,

such that the result in the previous section can be used, yielding
 * $$G(\omega)=\frac{(\omega-\epsilon_3)\sigma_3+\vec{h}\cdot\sigma_3\vec{\sigma}}{(\omega-\epsilon_3)^2-h^2}$$,

and the spectral function
 * $$A(\omega) = \left(\sigma_3+\frac{\vec{h}\cdot\sigma_3\vec{\sigma}}{h}\right) \pi\delta(\omega-h_+) + \left(\sigma_3-\frac{\vec{h}\cdot\sigma_3\vec{\sigma}}{h}\right) \pi\delta(\omega-h_-)$$,

where $$h_\pm = \epsilon_3\pm h$$, $$h^2=\epsilon_0^2-\epsilon_1^2-\epsilon_2^2$$ and
 * $$\vec{h}\cdot\sigma_3\vec{\sigma}=\epsilon_0\sigma_0-\epsilon_1\sigma_1-\epsilon_2\sigma_2$$.

For the SU(1,1) Hamiltonian, its parameters should satisfy the condition $$\epsilon_1^2+\epsilon_2^2\le\epsilon_0^2 $$, otherwise h will be imaginary, and the spectrum will not be stable.

Taking Imaginary Part
Technically the Im is taken by factorizing the denominator and using the identity
 * $$-2\,\mathrm{Im}\frac{1}{x+i0_+}=2\pi\delta(x)$$,

derived from which, the following formula will be useful,
 * $$-2\,\mathrm{Im}\frac{1}{(x+i0_++a)^2-b^2}=\frac{\pi}{b}(\delta(x+a-b)-\delta(x+a+b))$$,
 * $$-2\,\mathrm{Im}\frac{(x+a)}{(x+i0_++a)^2-b^2}=\pi(\delta(x+a-b)+\delta(x+a+b))$$.

Numerical Handling of δ Functions

 * $$\delta(\omega-E)=\left\{\begin{array}{ll}

1/d\omega & \text{if }\omega = E;\\ 0 & \text{if }\omega \neq E.\end{array}\right.$$