Given the interaction part of a general many-body hamiltonian,

$$\hat{H}=\sum_{\alpha, \beta,\gamma,\delta,\sigma,\sigma^\prime}O_{\alpha,\gamma,\sigma}^{\beta,\delta,\sigma^\prime}\hat{c}_{\alpha,\sigma}^\dagger \hat{c}^\dagger_{\beta,\sigma^\prime} \hat{c}_{\delta,\sigma^\prime}\hat{c}_{\gamma,\sigma},$$ I define $$\hat{C}_1=[\hat{H},\hat{c}_{\mu,\sigma}],$$ $$\hat{C}_n=[\hat{H},\hat{C}_{n-1}] \ \text{for every} \ n \gt 1.$$

I was wondering if there's any closed formula for $C_n,$ or at least a nicer recursion relation. $C_1$ and $C_2$ are easy to compute. I've also computed $C_3,$ which is already quite a lenghty and messy calculation! But I haven't seen any pattern, and I haven't been able to write down a general expression for $C_n.$ Is this something known? Is it possible to do this?

  • 1
    $\begingroup$ This may be related to the Baker-Campbell-Hausdorff identity: en.wikipedia.org/wiki/… $\endgroup$ – probably_someone May 7 '18 at 11:20
  • $\begingroup$ I spent some time toying with that, but didn't get anywhere: I only recovered new non-linear recursive relations which weren't any easier. $\endgroup$ – Qwertuy May 7 '18 at 12:59

$\textbf{Hint:}$ This is a twisted way to put perturbation theory in Heisenberg picture.

  1. Define a set of generating operators as $$\hat{G}_{\mu \sigma}\left(z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\underbrace{[\hat{H},[\hat{H},...[\hat{H},\hat{c}^\dagger_{\mu,\sigma}]]...]]}_{n^{\text{th}}-\text{order nested commutator}}~\stackrel{\textbf{BCH}}{=}~e^{z\hat{H}}\hat{c}^{\dagger}_{\mu,\sigma}e^{-z\hat{H}}$$ and $$\hat{\bar{G}}_{\mu \sigma}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\underbrace{[\hat{H},[\hat{H},...[\hat{H},\hat{c}{}_{\mu,\sigma}]]...]]}_{n^{\text{th}}-\text{order nested commutator}}~\stackrel{\textbf{BCH}}{=}~e^{z\hat{H}}\hat{c}_{\mu,\sigma}e^{-z\hat{H}}.$$

  2. Notice $$\frac{\mathrm{d}}{\mathrm{d}z}\hat{G}_{\mu \sigma}(z)~=~[\hat{H},\hat{G}_{\mu \sigma}(z)]$$ and $$\frac{\mathrm{d}}{\mathrm{d}z}\hat{\bar{G}}_{\mu \sigma}(z)=[\hat{H},\hat{\bar{G}}_{\mu \sigma}(z)].$$

  3. Use commutation/anti-commutation relations.

  4. Integrate the resulting equations between $\left[0,z\right]$ and iterate to find Taylor expansion coefficients of $\hat{G}_{\mu \sigma}(z)$ and $\hat{\bar{G}}_{\mu \sigma}(z)$.

  5. Check if you can find any pattern.

  • $\begingroup$ That seems like a nice idea; mine with the BCH was similar. But I don't get anything out of it, I just go in circles recovering the definitions. Was this merely a quick suggestion, or do you actually know that you can really solve the problem with this approach? $\endgroup$ – Qwertuy May 8 '18 at 9:32
  • $\begingroup$ @Qwertuy This suggestion is sort of quick (intuition comes from equation of motion method in Greens function methods, tree expansions in QFT more generally perturbative diagramatic methods for solving nonlinear equations). Although I don't have the explicit answer. $\endgroup$ – Sunyam May 8 '18 at 11:45

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