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.
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.
In the study of open quantum systems, the polaron transformation has recently attracted a renewed interest as it offers the possibility to explore the strong system-bath coupling regime. Despite this interest, a clear and unambiguous analysis of the regimes of validity of the polaron transformation is still lacking. Here we provide such a benchmark, comparing second order perturbation theory results in the original untransformed frame, the polaron frame, and the variational extension with numerically exact path integral calculations of the equilibrium reduced density matrix. Equilibrium quantities allow a direct comparison of the three methods without invoking any further approximations as is usually required in deriving master equations. It is found that the second order results in the original frame are accurate for weak system-bath coupling; the results deteriorate when the bath cut-off frequency decreases. The full polaron results are accurate for the entire range of coupling for a fast bath but only in the strong coupling regime for a slow bath. The variational method is capable of interpolating between these two methods and is valid over a much broader range of parameters.
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.
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.
Abstract We propose a simple variational solution for calculating one-particle spectral functions in lattice models of spinless metals with strong electron-phonon coupling. It is based on a generalization of the Momentum Average variational approximation for single polarons, combined with the assumption that the other fermions in the system are locked into an inert Fermi sea. We expect the method to be accurate for fermion addition spectral functions in metals with a small Fermi energy (nearly empty band), and for fermion removal spectral functions in metals with a large Fermi energy (nearly full band), provided that the characteristic phonon frequency is not too small. Both these regions are far from the region where the Migdal theorem holds, thus our results offer new insights into polaronic behavior in a largely unexplored part of the parameter space. Here, we show results for the Holstein coupling in one-dimension and present ways to gauge their accuracy, but ultimately this will need to be verified against numerical calculations. This variational method can be extended straightforwardly to higher dimensions and other forms of electron-phonon coupling.
The Feynman path-integral variational approach to the polaron problem, along with the associated Feynman-Hellwarth-Iddings-Platzman (FHIP) linear-response mobility theory, provides a computationally amenable method to predict the frequency-resolved temperature-dependent charge-carrier mobility, and other experimental observables in polar semiconductors. We show that the FHIP mobility theory predicts non-Drude transport behavior, and shows remarkably good agreement with the recent diagrammatic Monte Carlo mobility simulations of Mishchenko et al. [Phys. Rev. Lett. 123, 076601 (2019)] for the abstract Fr\"ohlich Hamiltonian. We extend this method to multiple phonon modes in the Fr\"ohlich model action. This enables a slightly better variational solution, as inferred from the resulting energy. We carry forward this extra complexity into the mobility theory, which shows a richer structure in the frequency and temperature-dependent mobility, due to the different phonon modes activating at different energies. The method provides a computationally efficient and fully quantitative method of predicting polaron mobility and response in real materials.