### Eigenvalue equation for interacting Green's functions

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Studying the articles "Topological Hamiltonian as an exact tool for topological invariants" (<https://arxiv.org/abs/1207.7341>) and "Simplified Topological Invariants for Interacting Insulators" (<https://arxiv.org/abs/1201.6431>) I stumbled upon the following eigenvalue equation

\begin{align}

G^{-1}(k,i\omega) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)>

\end{align}

where

\begin{align}

G^{-1}(k,i\omega) &= i\omega - H_0(k) - \Sigma(k,i\omega)

\end{align}

is the full Green's function of an interacting problem evaluated at the Matsubara frequencies \(\omega_n = (2n+1)\pi/\beta\) (Probably a similar formular also holds for real frequencies, then just replace \(i\omega \rightarrow \omega\)). \(H_0(k)\) is the Hamiltonian matrix of the noninteracting problem and \(\Sigma(k,i\omega)\) the \(\omega\) dependent self-energy that appears in Dyson's equation \(G = G_0 + G_0 \Sigma G\). On a mathematical level, \(G^{-1}(k,i\omega) \in\ \)GL(N,C) is an invertible complex \(N\times N\) matrix, and \(\{|\alpha(k,i\omega)>\}\) a set of orthogonal eigenvectors of that matrix with eigenvalues \(\mu_\alpha(k,i\omega)\).

The eigenvalue equation can also be written as

\begin{align}

\left(i\omega - H_0(k) - \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)>\\

\left(H_0(k) + \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= -(\mu_\alpha(k,i\omega)+i\omega) |\alpha(k,i\omega)>

\end{align}

where the last line looks pretty similar to the time independent Schroedinger equation of a noninteracting system

\begin{align}

H_0(k) |\alpha(k)> &= \epsilon(k)|\alpha(k)>

\end{align}

with the big difference that the first equation depends on \(\omega\) and the second does not. I want to understand that equation and have the following questions

- What is the meaning of an \(\omega\)-dependent eigenstate?

- What is the meaning of an \(\omega\)-dependent (energy?) eigenvalue?

- Where can I find more about this topic? (Books, articles, ...)

\begin{align}

G^{-1}(k,i\omega) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)>

\end{align}

where

\begin{align}

G^{-1}(k,i\omega) &= i\omega - H_0(k) - \Sigma(k,i\omega)

\end{align}

is the full Green's function of an interacting problem evaluated at the Matsubara frequencies \(\omega_n = (2n+1)\pi/\beta\) (Probably a similar formular also holds for real frequencies, then just replace \(i\omega \rightarrow \omega\)). \(H_0(k)\) is the Hamiltonian matrix of the noninteracting problem and \(\Sigma(k,i\omega)\) the \(\omega\) dependent self-energy that appears in Dyson's equation \(G = G_0 + G_0 \Sigma G\). On a mathematical level, \(G^{-1}(k,i\omega) \in\ \)GL(N,C) is an invertible complex \(N\times N\) matrix, and \(\{|\alpha(k,i\omega)>\}\) a set of orthogonal eigenvectors of that matrix with eigenvalues \(\mu_\alpha(k,i\omega)\).

The eigenvalue equation can also be written as

\begin{align}

\left(i\omega - H_0(k) - \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)>\\

\left(H_0(k) + \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= -(\mu_\alpha(k,i\omega)+i\omega) |\alpha(k,i\omega)>

\end{align}

where the last line looks pretty similar to the time independent Schroedinger equation of a noninteracting system

\begin{align}

H_0(k) |\alpha(k)> &= \epsilon(k)|\alpha(k)>

\end{align}

with the big difference that the first equation depends on \(\omega\) and the second does not. I want to understand that equation and have the following questions

- What is the meaning of an \(\omega\)-dependent eigenstate?

- What is the meaning of an \(\omega\)-dependent (energy?) eigenvalue?

- Where can I find more about this topic? (Books, articles, ...)

### 2 Answers

0

I do not understand the meaning of an expression like

\( \Big( i\omega - H_0(k) - \Sigma(k,i\omega) \Big) | \alpha(k, i\omega) \rangle \)

AFAIU, \( H_0(k) \) and \( \Sigma(k,i\omega) \) are not operators, but scalars, unless you define precisely what you mean.

“On a mathematical level, \( G^{-1}(k,i\omega) \in \) GL(N,C) is an invertible complex \( N \times N \) matrix...” is quite vague for me. That's perhaps the reason you arrive at incomprehensible objects.

\( \Big( i\omega - H_0(k) - \Sigma(k,i\omega) \Big) | \alpha(k, i\omega) \rangle \)

AFAIU, \( H_0(k) \) and \( \Sigma(k,i\omega) \) are not operators, but scalars, unless you define precisely what you mean.

“On a mathematical level, \( G^{-1}(k,i\omega) \in \) GL(N,C) is an invertible complex \( N \times N \) matrix...” is quite vague for me. That's perhaps the reason you arrive at incomprehensible objects.

In these equations, \(H_0(k)\), \(\Sigma(k,i\omega)\) as well as \(G^{-1}(k,i\omega)\) are all matrices acting on the vectors \(|\alpha(k,i\omega)>\).

written
4 months ago by
LidoDiCamaiore

It's better to start from a very simple problem; like a non-interacting problem or a two-level mixing (or hybridization of two levels). Then try to represent or interpret the Green's function as a matrix. You might then get an idea.

For me, it is totally unclear what is meant by that matrix representation.

For me, it is totally unclear what is meant by that matrix representation.

written
4 months ago by
AlQuemist

I agree with the Boss; the statement

\(\Big( i\omega - H_0(k) - \Sigma(k,i\omega) \Big) | \alpha(k, i\omega) \rangle\)

is quite ambiguous. If it is indeed, as you say, a multiplication of matrices with vectors, then this implies that the operators and states have been projected to a specific basis, and in doing this one sees that there will be integrals involved (due to inserting a complete set of states). These integrals are missing in the above equation.

\(\Big( i\omega - H_0(k) - \Sigma(k,i\omega) \Big) | \alpha(k, i\omega) \rangle\)

is quite ambiguous. If it is indeed, as you say, a multiplication of matrices with vectors, then this implies that the operators and states have been projected to a specific basis, and in doing this one sees that there will be integrals involved (due to inserting a complete set of states). These integrals are missing in the above equation.

written
4 months ago by
Master Dragon

In my case I have \(H=H_{MF}+H_{fl}\). The quadratic part can be written in matrix form as

\begin{align}

H_{MF} &= \sum_{k}

\begin{pmatrix} c_k^\dagger \\ f_k^\dagger \end{pmatrix}

\underbrace{\begin{pmatrix} \xi_k^c\cdot 1 & b^* V_0\cdot\Phi(\vec{k}) \\ b V_0\cdot\Phi(\vec{k}) & \xi_k^f\cdot 1

\end{pmatrix}}_{=H_0(k)}

\begin{pmatrix} c_k \\ f_k \end{pmatrix}

\end{align}

The self energy \(\Sigma\), on the other hand, is obtained by perturbation theory and there are plenty of sums and integrals involved. But in the mentioned eigenvalue equation, I don't care how to calculate it in practice, I just act as if I knew it. It does not really matter.

I have the suspicion, however, that the equation does not have much physical meaning in the first place, but is used as a mathematical trick in the mentioned papers. Did you have a look at them? They are indeed interesting papers.

\begin{align}

H_{MF} &= \sum_{k}

\begin{pmatrix} c_k^\dagger \\ f_k^\dagger \end{pmatrix}

\underbrace{\begin{pmatrix} \xi_k^c\cdot 1 & b^* V_0\cdot\Phi(\vec{k}) \\ b V_0\cdot\Phi(\vec{k}) & \xi_k^f\cdot 1

\end{pmatrix}}_{=H_0(k)}

\begin{pmatrix} c_k \\ f_k \end{pmatrix}

\end{align}

The self energy \(\Sigma\), on the other hand, is obtained by perturbation theory and there are plenty of sums and integrals involved. But in the mentioned eigenvalue equation, I don't care how to calculate it in practice, I just act as if I knew it. It does not really matter.

I have the suspicion, however, that the equation does not have much physical meaning in the first place, but is used as a mathematical trick in the mentioned papers. Did you have a look at them? They are indeed interesting papers.

written
4 months ago by
LidoDiCamaiore

You seem to be mixing apples and coconuts ;)

The ingredients of the Hamiltonian are

The self-energy is a ‘simple’ scalar-valued

At any case, you cannot represent the

The ingredients of the Hamiltonian are

**operators**on the Fock space.The self-energy is a ‘simple’ scalar-valued

**function**.At any case, you cannot represent the

*quartic*(or higher-order parts) of a Hamiltonian with matrices. The reason is fundamental: Matrices represent*linear*transformations (operators) while such interactions are*nonlinear*operators.
written
4 months ago by
AlQuemist

First: you can gather the functions, that the self-energy is, in a matrix if you have different particle types.

Second: I do not represent the Hamiltonian by matrices, but the Green's function. And that comes straight from Dyson's equation

Second: I do not represent the Hamiltonian by matrices, but the Green's function. And that comes straight from Dyson's equation

written
4 months ago by
LidoDiCamaiore

Just make clear what you want to

*achieve*with such a “matrix-method”. Perhaps, then we can find a proper way to do that.
written
4 months ago by
AlQuemist

Aaaand... I think the problem is solved by now, you can read everything in my thesis later ;)

written
4 months ago by
LidoDiCamaiore

Well then; good luck!

But that remains still unclear.

But that remains still unclear.

written
4 months ago by
AlQuemist

0

Actually, notice that taking this statement literally,

“On a mathematical level, \( G^{-1}(k,i\omega) \in \) GL(N,C) is an invertible complex \( N \times N \) matrix...”

implies that

“On a mathematical level, \( G^{-1}(k,i\omega) \in \) GL(N,C) is an invertible complex \( N \times N \) matrix...”

implies that

*any*function, \( f(\mathbf{x}, y) \in GL(N,C) \)! This is absurd.
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