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arXiv:2506.17914 · Physical Review Research

Polarons with arbitrary nonlinear electron-phonon interaction

Stefano Ragni, Tomislav Miškić, Thomas Hahn, Nikolay V. Prokof'ev, Osor S. Barišić, Naoto Nagaosa, Cesare Franchini, Andrey S. Mishchenko

Published Nov 12, 2025• 4 citations
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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: (1) quantum interplay of quartic and quadratic interactions that suppresses effects of the quadratic coupling, (2) intermediate-coupling regime with exponential ∝ exp ( α g 2 Ω − 1 / 4 ) scaling of the quasiparticle weight and mass renormalization, and (3) strong-coupling asymptotic behavior.

DOI
10.1103/6127-phps

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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: (1) quantum interplay of quartic and quadratic interactions that suppresses effects of the quadratic coupling, (2) intermediate-coupling regime with exponential ∝ exp ( α g 2 Ω − 1 / 4 ) scaling of the quasiparticle weight and mass renormalization, and (3) strong-coupling asymptotic behavior.
We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or X2{X}^{2}X2 polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling g2{g}_{2}g2​, from the system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ to arbitrary strong repulsion at ${g}_{2}\ensuremath{\gg}1$. The key properties of X2{X}^{2}X2 polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.
We report a breakthrough development allowing one to solve a broad class of polaron problems with highly nonlinear electron-phonon coupling that were considered unsolvable without uncontrolled simplifications in the past using a Monte Carlo technique formulated in the coordinate representation for both the particle and the atomic displacements. The only condition is the sign positivity of the hopping amplitude. The technique dramatically simplifies (with the corresponding efficiency gains) for models with dispersionless phonons. Our study sheds important light on the nature and universality of the most striking qualitative and quantitative effects demonstrated by the ``standard'' (Peierls/Su-Schrieffer-Heeger) model based on the linearized displacement-modulated hopping.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
In most cases, as the strength of electron-phonon coupling increases, the effective mass of polarons typically increases. However, in this paper, we uncover a fascinating phenomenon: the presence of light polarons even within the strong coupling regime, where electron-phonon coupling includes both Holstein and bond types (electron-phonon coupling occurring on the hopping bonds). To investigate this, we employ a diagrammatic Monte Carlo method based on the path-integral formulation of the particle sector and the Fock-state representation of the phonon sector. Our study centers on analyzing the impact of bond electron-phonon coupling on the Holstein polaron's essential properties, including its effective mass, ground-state energy, and the average number of phonons. We examine two distinct scenarios: one where the phonon frequency of both Holstein and bond types is the same, and another where the phonon frequency of the Holstein type is twice as large as that of the bond type. In both of these cases, when the Holstein coupling falls within the light mass regime, we observe minimal changes in the effective mass as a function of bond coupling, compared to the case of the bare bond polaron. However, the behavior of the effective mass undergoes a significant change in the heavy mass regime. Most intriguingly, we discover a nonmonotonic dependence of the effective mass on the bond coupling gB{g}_{B}gB​ when the Holstein coupling gH{g}_{H}gH​ falls within the heavy mass regime. This finding holds promise, particularly in the context of bipolarons, where achieving a light effective mass and a large binding energy to get a compact size is crucial.