Fate of orbitons coupled to phonons

The key feature of an orbital wave or orbiton is a significant dispersion, which arises from exchange interactions between orbitals on distinct sites. We study the effect of a coupling between orbitons and phonons in one dimension using continuous unitary transformations. Already for intermediate values of the coupling, the orbiton band width is strongly reduced and the spectral density is dominated by an orbiton-phonon continuum. However, we find sharp features within the continuum and an orbiton-phonon antibound state above. Both show a significant dispersion and should be observable experimentally.

Figure 1

Dispersion $\omega \left(k\right)$ of the dressed orbitons for ${\omega }_{\mathrm{ph}}=W$ using the truncation $({}^{2}V,e=9)$. Inset: Effective band width $W_{\mathrm{eff}}$, as obtained by CUT and by SOPT. For small values of $g$, one has $\delta W_{\mathrm{eff}}\propto -g^2$ for $W<{\omega }_{\mathrm{ph}}$ and $\delta W_{\mathrm{eff}}\propto -g^{4∕3}$ for $W={\omega }_{\mathrm{ph}}$ in 1D.

Figure 2

Effect of the different truncation schemes on the orbiton dispersion $\omega \left(k\right)$ for ${\omega }_{\mathrm{ph}}=W$. Upper panel: The difference $\delta \omega \left(k\right)=\omega ({}^2\Gamma ,e=9)-\omega ({}^{2}V,e=9)$ at constant extension $e=9$. Lower panel: $\delta \omega \left(k\right)=\omega ({}^{2}V,e=7)-\omega ({}^{2}V,e=9)$ at constant structure $S={}^{2}V$.

Figure 3

(Color online) Spectral weights of the orbiton quasiparticle band and of the orbiton-plus-$n$-phonon continua $(n=1–4)$ for ${\omega }_{\mathrm{ph}}∕W=1$ using the truncation $({}^{2}V,e=9)$. Inset: The sum of the weights of the five considered contributions obeys the sum rule very well. Lines are guides to the eyes only.

Figure 4

The $k$-resolved spectral weight of the orbiton quasiparticle band (top) and of the orbiton-plus-one-phonon sideband (bottom) for ${\omega }_{\mathrm{ph}}=W$ using the truncation $({}^{2}V,e=9)$.

Figure 5

Effect of the truncation on the $k$-resolved spectral weight of the orbiton quasiparticle band for ${\omega }_{\mathrm{ph}}∕W=1$. Upper panel: The difference $\delta I=I({}^2\Gamma ,e=9)-I({}^{2}V,e=9)$ at constant extension $e=9$. Lower panel: $\delta I=I({}^{2}V,e=7)-I({}^{2}V,e=9)$ at constant structure $S={}^{2}V$.

Figure 6

Effect of the truncation on the $k$-resolved spectral weight of the orbiton-plus-one-phonon sideband for ${\omega }_{\mathrm{ph}}∕W=1$. Upper panel: The difference $\delta I=I({}^2\Gamma ,e=9)-I({}^{2}V,e=9)$ at constant extension $e=9$. Lower panel: $\delta I=I({}^{2}V,e=7)-I({}^{2}V,e=9)$ at constant structure $S={}^{2}V$.

Figure 7

(Color online) Spectral density of one orbiton (red/gray) and the orbiton-plus-one-phonon continuum (black) for ${\omega }_{\mathrm{ph}}∕W=1$ using the truncation $({}^{2}V,e=9)$. An orbiton-phonon antibound state (green/light gray) is shown where it has gained substantial weight. (a) $g∕W=1∕8$. (b) $g∕W=1∕4$. (c) $g∕W=3∕8$. (d) $g∕W=1∕2$. The $\delta$ functions are broadened by $\Gamma =0.01W$. The dashed lines depict the orbiton-plus-one-phonon continuum in SOPT for $k=-\pi$.