Analytic treatment of a polaron in a nonparabolic conduction band

Abstract

We develop and compare several analytical approximations for the polaron problem in finite-width, non-parabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. We revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results over the whole range of momenta within the Brillouin zone. All methods are applied to the Holstein model and are benchmarked against numerically exact calculations, including Diagrammatic Monte Carlo (both our calculations and preceding works), exact diagonalization, and density-matrix renormalization-group results. The analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.

Figures (9)

• Figure 1: Ground state energy of a Holstein polaron in 1D for $\omega_{0}=0.5t$, calculated by different methods: Diagrammatic Monte Carlo Berciu2006Goodvin2006 (full dots), Diagrammatic Monte Carlo of the present work (triangles), the modified Feynman variational method (solid curve), the reduced Feynman variational method (dotted curve), the Momentum Average approximation MA$^{\left( 0\right) }$ (dashed curve) Berciu2006Goodvin2006, method of canonical transformations (dot-dot-dashed curve), and the RS perturbation theory Berciu2006Goodvin2006 (short-dashed line).
• Figure 2: Ground state energy of a Holstein polaron in 1D for $\omega_{0}=0.1t$, calculated by different methods: Diagrammatic Monte Carlo Berciu2006Goodvin2006 (full dots), Diagrammatic Monte Carlo of the present work (triangles), the modified Feynman variational method (solid curve), the reduced Feynman variational method (dotted curve), the momentum-average approximation MA$^{\left( 0\right) }$ and MA$^{\left( 2\right) }$ (dashed and dot-dashed curves, respectively) Berciu2006Goodvin2006, method of canonical transformations (dot-dot-dashed curve), and the RS perturbation theory Berciu2006Goodvin2006 (short-dashed line).
• Figure 3: Ground state energy of a polaron with the Rashba spin-orbit coupling with the phonon frequency $\omega_{0}=t$ and the SO potential $V_{s}=0$ (a) and $V_{s}=t$ (b) calculated using different methods: the reduced Feynman approach (solid curves), the exact diagonalization method Li2011 (dashed curves), the method of canonical transformations (dotted curves), the RS perturbation theory (dot-dashed curves), the next-to-leading order (dot--dot-dashed curves) and leading-order (short-dashed curves) Lang-Firsov approximations.
• Figure 4: Ground state energy of a polaron with the Rashba SO coupling with the phonon frequency $\omega_{0}=0.1t$ and the SO potential $V_{s}=0$ (a) and $V_{s}=t$ (b). The notations are the same as in Fig. \ref{['GSSO-1']}.
• Figure 5: Energy difference $E_{pol}-E_{0}$ as a function of the SO coupling potential $V_{s}$ for different values of the coupling strength $\lambda$. Curves: results of the calculation within the reduced Feynman approximation. Symbols: results of the momentum-average approximation MA$^{\left( 2\right) }$Covaci2009.