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User:EverettYou/Spectral Function

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General Formalism

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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,

,
.

Parity

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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

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If (the real part of) G(-ω)=-G(ω) is odd, then A(-ω)=A(ω) is even and

.

Diffusive Dynamics

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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

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Diagonal Hamiltonian

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Consider the Hamiltonian in its diagonal representation,

,

where n labels the energy level .

The Green's function is

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The spectral function is

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SU(2) Hamiltonian

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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 ,

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The Green's function is given by

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The corresponding spectral function reads,

,

where and .

SU(1,1) Hamiltonian

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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

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Note that the metric is not definite. The Green's function is given by

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By introducing , , , , one finds

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such that the result in the previous section can be used, yielding

,

and the spectral function

,

where , and

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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

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Appendix

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Taking Imaginary Part

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Technically the Im is taken by factorizing the denominator and using the identity

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derived from which, the following formula will be useful,

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Numerical Handling of δ Functions

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See Also

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Reference

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