Polaron Crossover and Bipolaronic Metal-Insulator Transition in the Holstein model at half-filling

TL;DR

The paper investigates the Holstein model at half filling using DMFT with exact diagonalization to disentangle polaron formation from bipolaronic metal-insulator transitions, comparing spinless and spinful fermions across adiabatic and anti-adiabatic regimes. It combines numerically exact DMFT results with analytic Born-Oppenheimer and Lang-Firsov approaches, showing that polaron formation does not automatically entail insulating behavior and that bipolaron-induced MIT hinges on carrier pairing, with the anti-adiabatic limit mapping to an attractive Hubbard model. The work provides phase diagrams, spectral diagnostics (phonon PDF, phonon DOS, electronic DOS), and a coherent interpretation in terms of a pseudospin Kondo framework, clarifying when lattice polarization and carrier pairing drive metal-insulator transitions. The findings have implications for understanding polaronic effects in correlated electron systems and materials with strong electron-phonon coupling, highlighting regimes where simple polaron pictures fail and where many-body entanglement governs transport and spectral properties.

Abstract

The evolution of the properties of a finite density electronic system as the electron-phonon coupling is increased are investigated in the Holstein model using the Dynamical Mean-Field Theory (DMFT). We compare the spinless fermion case, in which only isolated polarons can be formed, with the spinful model in which the polarons can bind and form bipolarons. In the latter case, the bipolaronic binding occurs through a metal-insulator transition. In the adiabatic regime in which the phonon energy is small with respect to the electron hopping we compare numerically exact DMFT results with an analytical scheme inspired by the Born-Oppenheimer procedure. Within the latter approach,a truncation of the phononic Hilbert space leads to a mapping of the original model onto an Anderson spin-fermion model. In the anti-adiabatic regime (where the phonon energy exceeds the electronic scales) the standard treatment based on Lang-Firsov canonical transformation allows to map the original model on to an attractive Hubbard model in the spinful case. The separate analysis of the two regimes supports the numerical evidence that polaron formation is not necessarily associated to a metal-insulator transition, which is instead due to pairing between the carriers. At the polaron crossover the Born-Oppenheimer approximation is shown to break down due to the entanglement of the electron-phonon state.

Figures (15)

• Figure 1: (color online) DMFT data in the adiabatic regime $\gamma=0.1$ spinless (panels on the left) and $\gamma=0.2$ spinful (panels on the right). In each panel the various Curves refer to different value of $\lambda$ spanning from $0.1$ to $1.8$ in the spinless case and from $0.05$ to $1.1$ in the spinful case and are shifted according $\lambda$ value. The first line show the phonon PDF, the central line the phonon DOS and bottom line the electronic DOS.
• Figure 2: (color online) DMFT data in the antiadiabatic regime $\gamma=2.0$ spinless (panels on the left) and $\gamma=4.0$ spinful (panels on the right). In each panel the various curves refer to different value of $\lambda$ spanning from $0.4$ to $6.5$ in the spinless case and from $0.2$ to $3.0$ in the spinful case and are shifted according $\lambda$ value. The first line show the phonon PDF, the central line the phonon DOS and bottom line the electronic DOS.
• Figure 3: (color online) Phase diagrams of the spinless (left) and spinful (right) Holstein model at half filling. Solid lines: numerical results from DMFT, dashed lines: approximations. Left panel: the bold line is the polaron crossover from bimodality of $P(X)$ and the dotted line is the anti-adiabatic estimate $\alpha^2>1$ for the polaron crossover. Right panel:bold curve is the bipolaronic MIT from vanishing of $Z$, thin solid line the polaron crossover, bold dotted line is the anti-adiabatic prediction for bipolaronic MIT (Eq. (\ref{['eq:MIT_LF']})), light dotted line is the anti-adiabatic estimate $\alpha^2>1/4$ for the polaron crossover.
• Figure 4: a) The equation for $G_0$ (thin line) and $G$ (bold line). Dashed line is the single site impurity propagator ($1/\omega$) bold arrow is the hybridization constant $V_{k}$ double line the bath propagator and x-type insertion the scattering with static displacements field. b) The diagrams expansion of the adiabatic potential.
• Figure 5: (color online) Parameters of the TSPM in the spinless case for $\gamma=0.1$. The spinful case is simply obtained by taking $\lambda_{spinful}=\lambda_{spinless}/2$. The adiabatic polaronic transition $\lambda_c$ is marked by a solid arrow while the adiabatic MIT is marked by a dashed arrow.