Coherent-state ansatz for the Holstein polaron in one and two dimensions

Abstract

The Holstein model often serves as an archetype for electron-phonon interactions and polaron formation in solids. However, precise descriptions of the Holstein polaron are difficult when the phonon frequency is small and the electron-phonon coupling is strong, due to the presence of many phonons in the ground state. We present a semi-analytical approximation that consists of a variational ansatz with clouds of phonons surrounding the electron in the form of coherent states. This becomes particularly simple and exact in the Lang-Firsov limit. We determine the domain of validity away from this limit, and further explore the improvement achieved with a removal of the requirement that the phonon clouds form coherent states. Both approximations work extremely well at strong coupling, and both work surprisingly well also at weak coupling. The coherent-state ansatz provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy and effective mass.

Figures (8)

• Figure 1: Plot of the numerically exact ground-state wavefunction for a 1D chain with $\omega_{\rm E}=0.2t$ and $\lambda=1.5$, showing the amplitudes of states belonging to the five families depicted on the right. Dashed lines connect points within a given family and serve as a guide to the eye. States belonging to these families contribute the majority of the total probability density, with $\sum_{\nu} |c_{\nu}|^2=0.997$ for the five families depicted here. On the right, arrows represent the location of the electron, while the number of phonons on a given lattice site is indicated in the attached box, with $n=0,1,\dots$ labelling individual states within each family. Sites not explicitly labelled contain no phonons. As explained in the main text, the actual states used are Bloch superpositions, over all sites on the lattice, of the states depicted here. Except for those in the blue family (top), each state has an equivalent counterpart, obtained by reflecting across the electron site, which is also included in the appropriate family.
• Figure 2: Ground-state polaron energy $E_0$ as a function of coupling strength $\lambda$, in 1D (left) and 2D (right), for $\omega_{\rm E}/t=0.1,\ 0.2,\ 0.5,\ 1.0,$ and $2.0$. Results from the CSA (dashed lines) and RHS (solid lines) are almost identical everywhere, and deviate most from the exact BT results (circles) for low phonon frequencies and in the intermediate coupling regime. In 2D, results from both approximations remain very close to the exact results, in contrast to the 1D case. Note the apparent kink in the 2D energy curve for small frequencies. While the exact curves are analytic everywhere, the crossover between weak- and strong-coupling regimes is extremely abrupt, a fact which is captured well by both the CSA and the RHS.
• Figure 3: Effective mass $m^*$ as a function of coupling strength $\lambda$, in 1D (left) and 2D (right), for $\omega_{\rm E}/t=0.1,\ 0.2,\ 0.5,\ 1.0,$ and $2.0$. Unlike the 1D case, the crossover from a weak- to a strong-coupling polaron in 2D is extremely abrupt, especially for small $\omega_{\rm E}$. Despite the abrupt crossover, the RHS mass prediction (solid lines) and exact BT results (circles) vary smoothly with $\lambda$. This can be seen more clearly in the inset, where the crossover region for $\omega_{\rm E}=0.2t$ is isolated. On the other hand, the CSA results (dashed lines) exhibit a discontinuous jump between weak- and strong-coupling regimes at a critical coupling strength $\lambda_{\rm c}$. The value of $\lambda_{\rm c}$ decreases with decreasing $\omega_{\rm E}$ in 1D, but tends to a finite value in 2D.
• Figure 4: Plot of the largest components of the ground-state wavefunction for $\omega_{\rm E}=0.2t$ in 2D, at strong coupling (here $\lambda=1.25$). The colour of the lines and symbols indicates which of the families from \ref{['fig_1D_states']} is represented. For the BT wavefunction (open circles), we only show amplitudes for the states belonging to the five families included in the RHS (filled squares). These families contribute $\sum_\nu |c_{\nu}|^2=0.9989$ of the total BT probability, with the largest amplitude among the states not included here being $0.0019$. Results from the CSA are shown with a dashed line for clarity, but the amplitude is defined only at integer values of $n$. The blue and orange curves agree well for all three methods. The inset shows the lowest-lying curves in more detail. While these do not match precisely, the overall shape of the coherent states, as well as the relative importance of the families, are consistent across all methods. This demonstrates the power of the restricted Hilbert space approximation.
• Figure 5: Plot of all components of the RHS and CSA ground-state wavefunctions for $\omega_{\rm E}=0.2t$ in 2D. The three panels represent, from left to right, the weak-coupling regime ($\lambda=0.5$), the crossover point ($\lambda=0.92$), and the strong-coupling regime ($\lambda=1.25$). The colour of each curve indicates the family from \ref{['fig_1D_states']} to which it corresponds. For $n=0$ there is no meaningful distinction between the families, so the entire contribution is assigned to the blue family. In the left and right panels, results from the RHS (square symbols) and CSA (dashed lines) are nearly indistinguishable. Near the crossover, however, the RHS smoothly interpolates between the two types of polaron, while the CSA selects one of the two regimes.