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 $\epsilon _{1}^{2}+\epsilon _{2}^{2}\leq \epsilon _{0}^{2}$, otherwise h will be imaginary, and the spectrum will not be stable.

Appendix

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

$\delta (\omega -E)=\left\{{\begin{array}{ll}1/d\omega &{\text{if }}\omega =E;\\0&{\text{if }}\omega \neq E.\end{array}}\right.$

See Also

• Spectral theory

Reference

• Gerald D. Mahan (2000). Many-Particle Physics (3rd Edition). Kluwer Academic/Plenum Pulishers. ISBN 0-306-46338-5.