Dual coupling effective band model for polarons

Nondiagonal couplings to a bosonic bath completely change polaronic dynamics, from the usual diagonally coupled paradigm of smoothly varying properties. We study, using analytic and numerical methods, a model having both diagonal Holstein and nondiagonal Su-Schrieffer-Heeger (SSH) couplings. The critical coupling found previously in the pure SSH model, at which the $k=0$ effective mass diverges, now becomes a transition line in the coupling constant plane—the form of the line depends on the adiabaticity parameter. Detailed results are given for the quasiparticle and ground-state properties, over a wide range of couplings and adiabaticity ratios. The new paradigm involves a destabilization, at the transition line, of the simple Holstein polaron to one with a finite ground-state momentum, but with everywhere a continuously evolving band shape. No “self-trapping transition” exists in any of these models. The physics may be understood entirely in terms of competition between different hopping terms in a simple renormalized effective band theory. The possibility of further transitions is suggested by the results.

Figure 1

Results for the 1D Holstein model, using DMC (symbols) and MA (lines). (a) Polaron energy $E_k$ and (b) its $qp$ weight $Z_k$, for ${\lambda }_H=0.25$; (c) polaron energy $E_k$ and (d) its $qp$ weight $Z_k$, for ${\lambda }_H=1.0$; (e) polaron ground-state energy $E_{gs}$ and (f) its inverse effective mass $m/m^{*}(k=0)$ vs ${\lambda }_H$ at $k=0$. In all cases, $t_o=1, {\mathrm{\Lambda }}_o\equiv {\mathrm{\Omega }}_o/\left(4t_o\right)=0.125$, and ${\lambda }_{\text{SSH}}=0$. DMC data are from Ref. [46].

Figure 2

Results for the 1D SSH model with ${\mathrm{\Lambda }}_o\equiv {\mathrm{\Omega }}_o/\left(4t_o\right)=0.125$ from BDMC, MA, and first- and second-order RS and WB perturbation theory. (a) Shifted polaron energy $E_k-E_0$ and (b) its $qp$ weight $Z_k$, for ${\lambda }_{\text{SSH}}=0.25,0.5,1,1.094,1.21,1.96$, with BDMC (symbols) and MA (lines). (c) Polaron ground-state energy $E_{\text{gs}}$, (d) the ground-state momentum $k_{\text{gs}}$, (e) the ground-state $qp$ weight $Z_{\text{gs}}$, and (f) the inverse effective mass $m/m^{*}(k=k_{\text{gs}})$, all as functions of ${\lambda }_{\text{SSH}}$. (c)–(f) The same color code described in the legend of (c). For BDMC results, error bars are shown only when they are larger than the size of the symbols. In all cases, $t_o=1$ and ${\lambda }_H=0$.

Figure 3

Results for the 1D SSH model with ${\mathrm{\Lambda }}_o=0.75$. We show the same quantities as in Fig. 2; in (a) and (b) we show results for ${\lambda }_{\text{SSH}}$ ranging from 0.25 to 4.0.

Figure 4

Results for the 1D SSH model, with ${\mathrm{\Lambda }}_o=25$. We show the same quantities as in Fig. 2; in (a) and (b) we show results for ${\lambda }_{\text{SSH}}$ ranging from 0.5 to 3.5.

Figure 5

Transition line ${\lambda }_{\text{SSH}}^{*}$ vs ${\mathrm{\Omega }}_o$ in the pure SSH model, separating the weak-coupling regime, with its nondegenerate ground state at $k_{\text{gs}}=0$, from the strong-coupling regime with its doubly degenerate ground state at $\pm k_{\text{gs}}\ne 0$. The inset shows the same data over a wider range of phonon energies. Results from BDMC, MA, and perturbational RS and WB of both first and second order are shown. Other parameters are ${\lambda }_H=0,t_o=1$.

Figure 6

Effective hopping constants $t_1$ (squares) and $-4t_2$ (circles) in the SSH model, obtained from a fit of $E_k$ as described by Eq. (17), for (a) ${\mathrm{\Omega }}_o=0.5t_o$, (b) ${\mathrm{\Omega }}_o=3t_o$, (c) ${\mathrm{\Omega }}_o=4t_o$, and (d) ${\mathrm{\Omega }}_o=100t_o$. Triangles, shown only in (a), represents ${\sum }_{n=1}^5{n}^2{t}_n$. See text for more details.

Figure 7

Polaron properties for the combined H/SSH model. We show the same plots as those shown in Fig. 2 for the pure SSH model, but now with a finite diagonal coupling ${\lambda }_H=0.25$. We choose ${\mathrm{\Lambda }}_o=0.75,t_o=1$. In (a) and (b), the values of ${\lambda }_{\text{SSH}}$ are shown in the legend.

Figure 8

The same plots as in Fig. 7, but now for ${\lambda }_H=0.5$. As before, ${\mathrm{\Lambda }}_o=0.75,t_o=1$. In (a) and (b), the values of ${\lambda }_{\text{SSH}}$ are shown in the legend.

Figure 9

Transition coupling ${\lambda }_{\text{SSH}}^{*}$ vs ${\mathrm{\Omega }}_o$. MA results are shown for ${\lambda }_H=0$ (circles), ${\lambda }_H=0.5$ (squares), and ${\lambda }_H=1$ (triangles). Big circles show BDMC data for ${\lambda }_H=0$. The inset compares these values with those predicted by second-order RS perturbation theory.

Figure 10

Transition coupling ${\lambda }_H^{*}$ vs ${\mathrm{\Omega }}_o$ for the H/SSH model. MA results are shown for ${\lambda }_{\text{SSH}}=0.7$ (full line) and ${\lambda }_{\text{SSH}}=0.8$ (dashed line).

Figure 11

Diagrams contributing to $G^{i,\text{first}}(\tau ,k)$. Relevant indices are only shown for the the first few diagrams. Each phonon propagator $i$ has a momentum $q_i$, and each electron propagator $j$ has a momentum $k_j=k-{\sum }_{i^{\prime }\in {\left\{j\right\}}_{i^{\prime }}}{q}_{i^{\prime }}$, where ${\left\{j\right\}}_{i^{\prime }}$ is the set of phonons propagators covering the electron propagator $i^{\prime }$.

Figure 12

Drawing first- and second-order self-energy diagrams with a more complicated (bold) propagator composed of the bare propagator and first-order Green's function diagrams. The bold propagator is shown as a dashed thick line. The diagrams are assumed to be in frequency space such that the length of a propagator does not matter.

Figure 13

Forbidden and allowed self-energy bold diagrams up to third order.

Figure 14

Algorithm to check if a self-energy diagram is allowed in BDMC. Each phonon is assigned a unique number and each propagator has a list of phonons covering it. Diagrams with each list being unique are allowed, and diagrams with repeated lists are forbidden.