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The Journal of Physical Chemistry Letters

Variational Polaron Theory for Ground States of Strongly Coupled Light–Matter and Electron–Phonon Systems

Nguyen Thanh Phuc

Published Sep 21, 2026• 0 citations
Publisher

Abstract

Strong light-matter and electron-phonon coupling generate ground states dressed by virtual bosonic excitations, making bare-state truncations and perturbative treatments unreliable in the ultrastrong-coupling regime. We introduce a nonperturbative variational ground-state framework based on a state-dependent polaron transformation, combined with a product-state ansatz and a second-order perturbative correction for residual matter-boson entanglement. We show that the optimized transformed frame becomes asymptotically decoupled at infinite coupling, because the leading linear coupling is canceled while off-diagonal matter transitions are suppressed by displaced-oscillator overlaps. The approach is asymptotically correct in both weak- and strong-coupling limits and remains accurate in the intermediate regime, where fixed polaron transformations are least reliable. Dicke-model benchmarks reproduce ground-state energies, fidelities, and the superradiant transition, with second-order energy errors below 0.2%. Holstein-model benchmarks yield errors below 0.5% and clarify how translational symmetry affects wave function quality. This dressed-basis framework enables nonperturbative modeling of strongly coupled light-matter and electron-phonon systems.

DOI
10.1021/acs.jpclett.6c02011

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Strong light-matter and electron-phonon coupling generate ground states dressed by virtual bosonic excitations, making bare-state truncations and perturbative treatments unreliable in the ultrastrong-coupling regime. We introduce a nonperturbative variational ground-state framework based on a state-dependent polaron transformation, combined with a product-state ansatz and a second-order perturbative correction for residual matter-boson entanglement. We show that the optimized transformed frame becomes asymptotically decoupled at infinite coupling, because the leading linear coupling is canceled while off-diagonal matter transitions are suppressed by displaced-oscillator overlaps. The approach is asymptotically correct in both weak- and strong-coupling limits and remains accurate in the intermediate regime, where fixed polaron transformations are least reliable. Dicke-model benchmarks reproduce ground-state energies, fidelities, and the superradiant transition, with second-order energy errors below 0.2%. Holstein-model benchmarks yield errors below 0.5% and clarify how translational symmetry affects wave function quality. This dressed-basis framework enables nonperturbative modeling of strongly coupled light-matter and electron-phonon systems.
Strong light-matter and electron-phonon coupling generate ground states dressed by virtual bosonic excitations, making bare-state truncations and perturbative treatments unreliable in the ultrastrong-coupling regime. We introduce a nonperturbative variational ground-state framework based on a state-dependent polaron transformation, combined with a product-state ansatz and a second-order perturbative correction for residual matter-boson entanglement. We show that the optimized transformed frame becomes asymptotically decoupled at infinite coupling, because the leading linear coupling is canceled while off-diagonal matter transitions are suppressed by displaced-oscillator overlaps. The approach is asymptotically correct in both weak- and strong-coupling limits and remains accurate in the intermediate regime, where fixed polaron transformations are least reliable. Dicke-model benchmarks reproduce ground-state energies, fidelities, and the superradiant transition, with second-order energy errors below 0.2%. Holstein-model benchmarks yield errors below 0.5% and clarify how translational symmetry affects wave-function quality. This dressed-basis framework enables nonperturbative modeling of strongly coupled light-matter and electron-phonon systems.
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 for the ground-state energy at strong coupling, and both work surprisingly well also at weak-coupling. For the effective mass, however, both approximations have difficulty, particularly in the crossover region. Moreover, the coherent-state ansatz (CSA) retains an unphysical jump in the crossover region. Nonetheless, the CSA provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy.
Recent development of theoretical method for dissipative quantum transport has achieved notable progresses in the weak or strong electron-phonon coupling regime. However, a generalized theory for dissipative quantum transport in a wide parameter space had not been established. In this work, a variational polaron theory for dissipative quantum transport in a wide range of electron-phonon coupling is developed. The optimal polaron transformation is determined by the optimization of the Feynman-Bogoliubov upper bound of free energy. The free energy minimization ends up with an optimal mean-field Hamiltonian and a minimal interaction Hamiltonian. Hence, second-order perturbation can be applied to the transformed system, resulting in an accurate and efficient method for the treatment of dissipative quantum transport with different electron-phonon coupling strength. Numerical benchmark calculation on a single site model coupled to one phonon mode is presented.
Polaron ground states and finite-temperature dynamics remain challenging to simulate because existing methods struggle to combine nonperturbative accuracy, systematic improvability, and scalability from models to materials-specific Hamiltonians. We introduce a translationally invariant variational coupled-cluster (CC) theory for polarons, termed delocalized CC (dCC), with closed-form energies at cost as low as O(N3)\mathcal{O}(N^3)O(N3) and no phonon-number cutoff. dCC accurately describes the ground states of the one- and two-dimensional Holstein and Su--Schrieffer--Heeger (optical and bond) models and the Fr{ö}hlich model, in close agreement with density matrix renormalization group (DMRG) and diagrammatic Monte Carlo benchmarks. A projected tangent-space response formalism built on the same ansatz yields electron-addition spectral functions and optical conductivities at zero and finite temperature. The resulting spectra agree well with DMRG, Lanczos, and neural-network quantum-state benchmarks while retaining a physically interpretable excitation hierarchy, and extend to two-dimensional lattices at finite temperature beyond the practical reach of these methods. The same framework applies directly to \textit{ab initio} electron--phonon matrix elements, yielding LiF electron- and hole-polaron binding energies that match state-of-the-art many-body calculations. These results establish dCC as a unified variational framework for polaron ground states and dynamics, from model systems to real materials.
In this work, we develop two many-body quantum electrodynamic methods to calculate the ground-state energies of strongly coupled light-matter molecular systems. Specifically, we extend the second-order many-body Green's function theory (GF2) for electronic systems to incorporate electron-boson couplings. We employ two ansätze to treat the bosonic part of the system, namely the coherent-state (CS) and Lang-Firsov (LF) transformed vacuum state. These are combined with the GF2 method to construct two new approaches, which we refer to as CS-GF2 and LF-GF2. We benchmark CS- and LF-GF2 by studying various molecular systems inside an optical cavity. We investigate H2\mathrm{H}_2H2​ and LiH\mathrm{LiH}LiH potential energy surfaces, keto-enol tautomerization energy barrier, van der Waals interactions between two H2\mathrm{H_2}H2​ molecules and the torsional potential energy surface of the ethylene molecule, C2H4\mathrm{C_2H_4}C2​H4​. Both methods provide similar energy profiles, with relatively small differences between CS-GF2 and LF-GF2. Comparisons with the reference calculations also reveal remaining limitations at stretched bond lengths and near the twisted ethylene configuration. We additionally benchmark CS-GF2 against exact diagonalization for a two-site Hubbard-Holstein model and observe that it closely reproduces the coupling-induced energy change in the weakly interacting regime while improving upon CS-MP2 in the strongly interacting regime. Overall, the results demonstrate that CS-GF2 and LF-GF2 provide self-consistent descriptions of correlated molecular electron-boson systems, offering a many-body Green's function alternative for studying ground-state cavity-modified systems.
We develop and compare several analytical approximations for the ground state and zero-temperature polaron dispersion in finite-width, nonparabolic 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. In addition, 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. All methods are applied to the Holstein model on a tight-binding lattice and are benchmarked against numerically exact calculations, including diagrammatic Monte Carlo (both our calculations and preceding works) and exact diagonalization results. Furthermore, 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.