Exact Treatment of Exciton-Polaron Formation by Diagrammatic Monte Carlo Simulations

We develop an approximation-free diagrammatic Monte Carlo technique to study fermionic particles interacting with each other simultaneously through both an attractive Coulomb potential and bosonic excitations of the underlying medium. Exemplarily we apply the method to the long-standing exciton-polaron problem and present numerically exact results for the wave function, ground-state energy, binding energy and effective mass of this quasiparticle. Focusing on the electron-hole pair bound-state formation, we discuss various limiting cases of a generic exciton-polaron model. The frequently used instantaneous approximation to the retarded interaction due to the exchange of phonons is found to be of very limited applicability.

Figure 1

A typical diagram for $G_{\mathbf{p},\mathbf{k}}^{M=0}(\tau )$. Solid (dashed) lines represent E(H) propagators, solid circles (squares) designate Coulomb (QP-phonon) interactions, and dotted lines are the phonon propagators. Imaginary time runs from left to right.

Figure 2

The way how momentum conservation is ensured (b)–(c) when a nondiagonal phonon is added to configuration (a). Diagrams with a nondiagonal phonon line and no Coulomb vertices are prohibited by momentum conservation in (3).

Figure 3

Binding energy ${\epsilon }_{\mathrm{EX}-\mathrm{P}}$ (in units of the gap $E_g$) and effective mass $m_{\mathrm{EX}-\mathrm{P}}$ (in units of $m_0=E_{c,v}/6$) as functions of $\mathrm{E}/\mathrm{H}$-phonon coupling $\lambda$ for $\Omega =0.25E_g$. The inset shows the EX-P energy $E_{\mathrm{g}.\mathrm{s}.}$ (circles) compared to $2E_{\mathrm{HP}}$ (squares) at $U/E_g=1$. Statistical error bars are smaller than symbol size. Dot-dashed lines indicate the HP strong- and weak-coupling results.

Figure 4

Internal structure of an EX-P in the mass-symmetric model with $U=1.5E_g$ and $\lambda =0.195<{\lambda }_{*}$ (1st column), $\lambda =0.223\approx {\lambda }_{*}$ (2nd column), $\lambda =0.29>{\lambda }_{*}$ (3rd column). First row shows the integrated $Z$-factors, second (third) row displays the reduced distributions of $W(\mathbf{P},\mathbf{K})$, Eq. (5) $w(K_z)\equiv \int d^{3}Pd{K}_{x}d{K}_{y}W(\mathbf{P},\mathbf{K})$ ($w(P_z)\equiv \int d^{3}Kd{P}_{x}d{P}_{y}W(\mathbf{P},\mathbf{K})$).

Figure 5

Exact DMC EX-P energy and effective mass (symbols with dotted lines, statistical errors are smaller than the symbol size) compared to results for the EAHM (solid lines). Data obtained for $U=E_g$ and $E_c=E_v=3E_g$.

Figure 6

Phase diagram of the mass-asymmetric EX-P model with $E_c=3E_g$, $E_v=\eta E_c$, and $\Omega =E_g$. Solid circles indicate the exact numerical solution in the static impurity limit $\eta =0$ [19, 20], while the solid line is calculated in the adiabatic approximation [18]. Dashed lines are guides to the eye.