Bound excited states of Fröhlich polarons in one dimension

TL;DR

This work analyzes the one-dimensional Fröhlich polaron, showing that an arbitrarily large number of bound excited states emerge in the strong-coupling limit and deriving analytic expressions for their frequencies. Using a Lee-Low-Pines transformation to obtain a phonon-only Hamiltonian and a semi-stochastic Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method, the authors compute ground- and excited-state energies in an effectively infinite phonon space, achieving sign-problem amelioration via walker annihilation. They establish weak- and strong-coupling expansions for the ground state, map the excited-state spectrum as a function of the coupling $\alpha$, and quantify spectral weights and phonon content to identify observable bound states near the threshold $\alpha \approx 1.73$. The results provide a computational framework for accessing bound polaron states in models with infinite phonon spaces and motivate extensions to higher dimensions and experimental platforms.

Abstract

The one-dimensional Fröhlich model describing the motion of a single electron interacting with optical phonons is a paradigmatic model of quantum many-body physics. We predict the existence of an arbitrarily large number of bound excited states in the strong coupling limit and calculate their excitation energies. Numerical simulations of a discretized model demonstrate the complete amelioration of the projector Monte Carlo sign problem by walker annihilation in an infinite Hilbert space. They reveal the threshold for the occurrence of the first bound excited states at a value of $α\approx 1.73$ for the dimensionless coupling constant. This puts the threshold into the regime of intermediate interaction strength. We find a significant spectral weight and increased phonon number of the bound excited state at threshold.

Figures (9)

• Figure 1: The ground state energy $E_0$ of the one-dimensional Fröhlich polaron, calculated using FCIQMC for various coupling strengths, compared with theoretical expansions in the weak [Eq. \ref{['eq:weak']}] and strong coupling [Eq. \ref{['eq:strong']}] regimes. The FCIQMC calculations used $L=6l_0$ and $k_c=12\pi/l_0$.
• Figure 2: Energies of excited states relative to the ground state for the one-dimensional Fröhlich model with zero total momentum. The grey shaded region shows the phonon continuum starting $1\hbar\omega_\mathrm{LO}$ above the ground state. The dashed horizontal line indicates the strong-coupling prediction for the first bound excited state from Eq. \ref{['eq:oddexcited']}. Symbols and error bars show finite system numerical results calculated using FCIQMC for two different box sizes ($L = 6l_0$, green, and $L = 10l_0$, purple) and classified by parity (circles for even and crosses for odd parity). The states inside the phonon continuum are discretized and strongly affected by the finite box size. A discretization-insensitive even state with energy below the phonon continuum for $\alpha \gtrapprox 1.73$ provides numerical evidence for a bound excited state of the phonon dressing cloud. The inset shows a detail near the emergence of the bound state and includes additional data for box size $L = 8l_0$.
• Figure 3: Spectral weight $Z_j^{(0)}$ of the three lowest energy states vs. the coupling strength $\alpha$ for $k_c=4\pi/l_0$ and two different box sizes. For $\alpha> 1.73$ marked by the vertical dashed line the state $|\mathrm{B}\rangle$ becomes a bound excited state. The dotted line is the first order perturbation theory result $1-\alpha/2$ for the ground state.
• Figure 4: (a) The ground state energy for $\alpha = 2$ with $L = 6l_0$ as a function of the cutoff $k_c$. (b) The difference between the ground state energy and the extrapolated energy at infinite $k_c$, on a log-log plot. The trend suggests an algebraic convergence of the energy, with an error in the energy proportional to approximately $k_c^{-1}$.
• Figure 5: (a) The ground state energy for $\alpha = 2$ with $k_c=10\pi/l_0$, for various box sizes $L$. (b) The difference between the ground state energy and the extrapolated energy at infinite $L$, on a semi-log plot. In this case, the data imply an exponential convergence of the energy.