Bond-Peierls polaron: Moderate mass enhancement and current-carrying ground state

We study polarons in the one-dimensional Bond-Peierls electron-phonon model in which phonons on bonds of a lattice modulate the hopping of electrons between lattice sites and contrast the results to those known for the breathing-mode Peierls problem. By inspecting the atomic limit, we show that polaronic dressing and mass enhancement of Bond-Peierls polarons depend on the momentum dependence of the phonons. For dispersionless phonons, Bond-Peierls polarons are perfectly localized in the atomic limit, unlike their breathing-mode counterparts because the carrier creates a string of phonon excitations that can only be annihilated via processes that retrace the carrier to its original site. However, inclusion of phonon dispersion leads to a nondivergent polaron mass even in the atomic limit and depending on the form of the phonon dispersion may lead to a transition to a non-zero-momentum ground state akin to that found in the breathing-mode Peierls model.

Figure 1

Polaron dispersion $E_{\mathrm{P}}\left(k\right)-E_{\mathrm{P}}\left(0\right)$, (a) and (b), and quasiparticle weight ${\mathrm{Z}}_{\mathrm{P}}\left(k\right)$, (c) and (d),  in the Bond-Peierls model with dispersionless phonons as a function of $\lambda \in \{0.4,0.8,1.2,1.6,2,2.4,2.8\}$ (from top to bottom) for two different adiabaticity parameters $\gamma =1/16$ and 3/4. The free-particle ($\lambda =0$) dispersion is shown as a dashed black line. The nonmonotonic evolution of $Z_{\mathrm{p}}\left(k\right)$ with $\lambda$ and $k$ in (d) can be understood in terms of an avoided crossing of the polaron band with a one-phonon state at energy $\mathrm{\Omega }$ relative to the band minimum ($k=0$), which occurs for small $\lambda$. Here, what is plotted is $Z$ for the lowest state at a given $k$. For small $\lambda$ and at $k$ beyond the crossing point, this branch is almost entirely of phonon character. As $\lambda$ increases the avoided crossing wave vector moves to the edge of the zone, and for $\lambda \gtrsim 0.5$ this physics becomes irrelevant.

Figure 2

GS properties and polaron effective mass $m_{\mathrm{P}}^{*}$ in the Bond Peierls (BP) model with dispersionless phonons versus that in the breathing-mode Peierls (BMP) model as a function of $\lambda$ for $\gamma =1/16$ and 3/4. $m_0=1/2t$ is the free-electron mass. $\lambda ={\alpha }^2/\mathrm{\Omega }t$ for the Bond-Peierls model, and $\lambda =2{\alpha }^2/\mathrm{\Omega }t$ for the breathing-mode Peierls model.

Figure 3

Spectral functions for $\mathrm{\Omega }=0.5$ and broadening factor $\eta =0.04$ of the (a) Bond- and (b) breathing-mode Peierls models with dispersionless phonons at $t=1$ (top row) and $t=0$ (bottom row) for a variety of different coupling strengths $\alpha .$

Figure 4

Cartoon depicting examples of virtual processes in which the carrier (blue circle) creates phonon excitations (red squares), forming a polaron. (a) A carrier in the Bond-Peierls model with dispersionless phonons moves from site $i\to i+1\to i+2\to ...$ and circles back multiple times (blue arrow) to form a state with 10 total phonons, terminating at site $i+4.$ We observe that there is no way for the carrier to move from site $i$ to some site $j\ne i$ whilst also annihilating all phonons on the lattice via ${\widehat{V}}_{\mathrm{e}-\mathrm{ph}}^{\mathrm{BP}}$, explaining the flat bands in Fig. 3 in the $t=0$ limit. (b) A second-order phonon-mediated process in the breathing-mode Peierls model already permits second nearest-neighbor hopping (note the negative sign, responsible for the sharp transition [10]). We see that unlike in the Bond-Peierls model, the breathing-mode Peierls coupling ${\widehat{V}}_{\mathrm{e}\text{−}\mathrm{ph}}^{\mathrm{BMP}}$ directly mediates carrier mobility.