Quantum-entanglement aspects of polaron systems

We describe quantum entanglement inherent to the polaron ground states of coupled electron-phonon (or, more generally, particle-phonon) systems based on a model comprising both local (Holstein-type) and nonlocal (Peierls-type) couplings. We study this model using a variational method supplemented by the exact numerical diagonalization on a system of finite size. By way of subsequent numerical diagonalization of the reduced density matrix, we determine the particle-phonon entanglement as given by the von Neumann and linear entropies. Our results are strongly indicative of the intimate relationship between the particle localization/delocalization and the particle-phonon entanglement. In particular, we find a compelling evidence for the existence of a nonanalyticity in the entanglement entropies with respect to the Peierls-coupling strength. The occurrence of such nonanalyticity—not accompanied by an actual quantum phase transition—reinforces analogous conclusion drawn in several recent studies of entanglement in the realm of quantum-dissipative systems. In addition, we demonstrate that the entanglement entropies saturate inside the self-trapped region where the small-polaron states are nearly maximally mixed.

Figure 1

Comparison of the ground-state energies obtained using the variational method of Cataudella et al. (Ref. 55) (circles) and the method of the present work (solid curve) in the $\varphi =0$ case: (a) $t/\omega =1.0$ and (b) $t/\omega =2.0$.

Figure 2

Phonon distribution sitewise in the polaron ground state for $t/\omega =1.0$ and $\varphi =1.3$ ($N=6$ and $M=10$). The states $\mathbf{m}$ with the same total number of phonons $m={\sum }_i{m}_i$ are indicated by the bins at the bottom portion of the plot.

Figure 3

(a) Variational ground-state energy, (b) the von Neumann entropy, and (c) linear entropy as functions of dimensionless local $\left(g\right)$ and nonlocal $\left(\varphi \right)$ coupling strengths, for $t/\omega =1.0$ and $N=32$.

Figure 4

The von Neumann entropy $S$, obtained variationally, as a function of dimensionless local $\left(g\right)$ and nonlocal $\left(\varphi \right)$ coupling strengths. Displayed are results for (a) $t/\omega =0.25$ and (b) $t/\omega =2.0$, with $N=32$.

Figure 5

(a) Ground-state energy, (b) the von Neumann entropy, and (c) linear entropy for $t/\omega =1.0$ and $g=0$, as calculated using the variational (var.) and exact diagonalization (e.d.) approaches. $N$ is the number of sites while $M$ stands for the maximal number of phonons used for exact diagonalization. The arrows indicate the entropies for the maximally mixed reduced density matrix.