Polarons with arbitrary nonlinear electron-phonon interaction

TL;DR

This paper addresses the challenge of solving polaron models with arbitrary nonlinear electron-phonon interactions by introducing Numeric X-propagators (NXP), an exact, approximation-free framework that combines X-propagator tabulation with Diagrammatic Monte Carlo for general $V_{\rm ep}(x)$ and $t_{ij}(x_i,x_j)$. It explicitly applies the method to a double-well local EPI, revealing three regimes: a quantum-interplay regime with possible suppression of EPI, an intermediate-coupling regime with universal exponential scaling of the quasiparticle weight and mass ($Z \propto \exp(\xi_Z g_2\Omega^{-1/4})$, $m^*/m \propto \exp(-\xi_m g_2\Omega^{-1/4})$), and a strong-coupling asymptotic single-site limit. The results highlight the limitations of approximation schemes like MA and ED in the adiabatic regime, and demonstrate the method's accuracy across a broad parameter range, including small phonon frequencies; the work also outlines extensions to dispersive phonons and first-principles EPI profiles, enabling reliable predictions for quantum paraelectrics, ferroelectrics, and halide perovskites. Overall, NXP provides a robust, scalable tool for exact treatment of nonlinear EPI in realistic materials, with significant implications for transport and optical properties in strongly anharmonic systems.

Abstract

We develop an exact computational method based on numerical X-propagators for solving polaron models with arbitrary nonlinear couplings of local vibration modes to the electron density and magnitude of the hopping amplitude. Our approach covers various polaron models, some of which were impossible to treat by any existing approximation-free techniques. Moreover, it remains efficient in the most relevant but computationally challenging regime of phonon frequencies much smaller than the electron bandwidth. As a case study, we consider the double-well type nonlinear model with quadratic ($g_2<0$) and quartic ($g_4>0$) interactions describing a broad class of technologically important materials, such as quantum paraelectric compounds and halide perovskites. We observe, depending on the model parameters, three qualitatively different regimes: (i) quantum interplay of quartic and quadratic interactions which suppresses effects of the quadratic coupling, (ii) intermediate-coupling regime with exponential $\propto \exp(αg_2 Ω^{-1/4})$ scaling of the quasiparticle weight and mass renormalization, and (iii) strong-coupling asymptotic behavior.

Figures (12)

• Figure 1: Potential energy profiles described by Eq. \ref{['loc3']} when only $g_2$ and $g_4$ are nonzero in $V_{\rm ep}(x)$. $g_4 = 0.1$ and $\Omega = 0.5$ are kept fixed and four values of $g_2$ are shown, representative of different physical regimes that will be probed in the remainder of the text.
• Figure 2: (modified from supplement in stefano23). Green's function (\ref{['GFGF']}) diagrams in one-dimensional space (index $i$) and imaginary time ($\tau$) plane. Horizontal solid red lines with arrows are bare Green's functions (\ref{['upgf']}), while vertical lines with arrows are hopping events $t$. $\tilde{U}$ (double-line dashed arcs with arrows) and $U$ (single-line dashed arcs with arrows) are X-propagators with and without the electron on a given site, respectively. $U_0$ (dotted lines with arrows) is the limiting value of the X-propagator in the ground state technique (\ref{['u0']}).
• Figure 3: Energy renormalization for $t=1$, $g_4=0.1$ as a function of $g_2$ for different values of $\Omega$. Points are our numeric data. Error bars, if not shown, are smaller than the symbol size. Momentum Average (MA) method data AdolphsPRB for $\Omega=t/2$ are shown by the green solid line, and Exact Diagonalization (ED) method data for $\Omega=t$, $\Omega=t/2$, and $\Omega=t/4$ are shown by the gray dotted, black dashed, and orange dash-dotted lines, respectively.
• Figure 4: Quasiparticle weight $Z$. For notations, see the Fig. \ref{['fig:e']} caption. Upper panel data are shown over a broad range of $g_2/t$ variation. Lower panel is focusing on the small $g_2$ region.
• Figure 5: Effective mass renormalization $m^*/m$. For notations, see the caption of Fig. \ref{['fig:e']}. Upper panel data are shown over a broad range of $g_2/t$ variation. Lower panel is focusing on the small $g_2$ region.