Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2206.00898 · Physical review. B./Physical review. B

Variational polaron equations applied to the anisotropic Fröhlich model

Vasilii Vasilchenko, Andriy Zhugayevych, Xavier Gonze

Published Jun 7, 2022• 12 citations
arXivHTMLPDF

Abstract

Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon, and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic nondegenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\mathrm{\ensuremath{\Gamma}}$ point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.

DOI
10.1103/physrevb.105.214301

Related papers

Polarons in the cubic generalized Fröhlich model: Spontaneous symmetry breaking

Vasilii Vasilchenko, Xavier Gonze

Within the variational polaron equation framework, the Fr\"ohlich model for cubic systems with threefold degenerate electronic bands is numerically solved in the strong-coupling regime for a wide range of its input parameters. By comparing the results to the previously reported ones obtained with the Gaussian ansatz approach, the inadequacy of the latter is uncovered, especially when degenerate bands are present in a system. Moreover, the symmetry groups of polaronic solutions in the cubic generalized Fr\"ohlich model without spin-orbit coupling are investigated: We provide and discuss a phase diagram of symmetry groups of ground-state polarons, showing spontaneous symmetry breaking. While the cubic symmetry of the three-band degenerate model Hamiltonian corresponds to the full octahedral group ${O}_{h}$, lowest-energy polarons possess either ${D}_{4h}$ or ${D}_{3d}$ point groups. This phase diagram bears some similarities but differs nevertheless from the one that is obtained by the straight analysis of the band effective masses. The obtained results will provide a firm ground for further exploration of the generalized Fr\"ohlich model and will likely be applicable beyond the model's inherent approximations.

arXiv:2405.01653•Physical review. B./Physical review. B•May 2, 2024•5 citations

All-coupling polaron optical response: Analytic approaches beyond the adiabatic approximation

Serghei N. Klimin, Jacques Tempere, J. T. Devreese

In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Fr\"ohlich polaron is treated, with the goal being to bridge the gap in the validity range that exists between two complementary methods: on the one hand, the memory-function formalism and, on the other hand, the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce diagrammatic quantum Monte Carlo results. To resolve this, we modify the memory-function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman approach in order to take into account a nonquadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation by including in the treatment nonadiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain at $T=0$ by combining the two extended methods agrees well, both qualitatively and quantitatively, with the diagrammatic quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.

arXiv:1606.00605•Physical review. B./Physical review. B•Sep 22, 2016•8 citations

Extending the Feynman variational principle and analytical methods to lattice polarons

Serghei N. Klimin, Jacques Tempere, Matthew Houtput +6 more

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.

arXiv:2603.09609•Physical review. B./Physical review. B•May 27, 2026•0 citations

Strong-Coupling Solutions of the Fröhlich Hamiltonian

Ashok Chatterjee, Soma Mukhopadhyay

This chapter describes some of the most important theories of the strong-coupling polaron. The nature of polarons in the strong-coupling limit is in some sense qualitatively different from the weak or intermediate-coupling polaron—the localization of a polaron being a case in point. Though a qualitative picture of a strong-coupling polaron is easy to visualize as opposed to the weak-coupling polaron, the mathematical formulation of a strongly coupled polaron is rather a difficult problem from the quantum mechanical point of view. In fact, the exact polaron energy in the weak-coupling limit can be obtained from the Frohlich Hamiltonian analytically, while the same is not possible in the strong-coupling limit. In the strong-coupling case, the electron velocity is much larger than the phonon frequency and therefore electron can follow the ion motion adiabatically. The adiabatic expansion of the polaron energy in the strong-coupling regime was reported by Bogoliubov and Tiablikov.

CRC Press eBooks•Oct 9, 2018•0 citations

Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

Nikolai Leopold, David Mitrouskas, Simone Rademacher +2 more

We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order $α^2$.

arXiv:2005.02098•Pure and Applied Analysis•Dec 31, 2021•15 citations

GROUND-STATE ENERGY OF FRÖHLICH ELECTRON-PHONON SYSTEM

Chih-Yuan Lu

Recently Feynman's path-integral formalism of the Fröhlich optical polaron problem is generalized, by which it is easy and natural to get the second-order perturbation result in the weak-coupling case and the Pekar's result in the strongcoupling case, even in the crudest ground-state approximation. By using numerical method to the equations which are given by the ground-state approximation, the ground-state energy of the electron-optical phonon system can obtained for the overall range of coupling strength. In addition, a self-consistent set of equations is derived for a central optimized potential without using the ground-state approximation. This effective local, central potential is the solution of a linear integral equation which was similar to the extensive work of optimized-potential model (OPM) in atomic physics. By using OPM formalism, no particular choice of interaction form is taken, therefore, except the approximation by Jensen's inequality, the ground-state energy of the electron-optical phonon interacting system can be obtained without any approximation.

Le Journal de Physique Colloques•Dec 1, 1981•0 citations
Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon, and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr"ohlich model with anisotropic nondegenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr"ohlich electron-phonon matrix elements at \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr"ohlich model.
Within the variational polaron equation framework, the Fr\"ohlich model for cubic systems with threefold degenerate electronic bands is numerically solved in the strong-coupling regime for a wide range of its input parameters. By comparing the results to the previously reported ones obtained with the Gaussian ansatz approach, the inadequacy of the latter is uncovered, especially when degenerate bands are present in a system. Moreover, the symmetry groups of polaronic solutions in the cubic generalized Fr\"ohlich model without spin-orbit coupling are investigated: We provide and discuss a phase diagram of symmetry groups of ground-state polarons, showing spontaneous symmetry breaking. While the cubic symmetry of the three-band degenerate model Hamiltonian corresponds to the full octahedral group Oh{O}_{h}Oh​, lowest-energy polarons possess either D4h{D}_{4h}D4h​ or D3d{D}_{3d}D3d​ point groups. This phase diagram bears some similarities but differs nevertheless from the one that is obtained by the straight analysis of the band effective masses. The obtained results will provide a firm ground for further exploration of the generalized Fr\"ohlich model and will likely be applicable beyond the model's inherent approximations.
In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Fr\"ohlich polaron is treated, with the goal being to bridge the gap in the validity range that exists between two complementary methods: on the one hand, the memory-function formalism and, on the other hand, the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce diagrammatic quantum Monte Carlo results. To resolve this, we modify the memory-function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman approach in order to take into account a nonquadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation by including in the treatment nonadiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain at T=0T=0T=0 by combining the two extended methods agrees well, both qualitatively and quantitatively, with the diagrammatic quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.
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.
This chapter describes some of the most important theories of the strong-coupling polaron. The nature of polarons in the strong-coupling limit is in some sense qualitatively different from the weak or intermediate-coupling polaron—the localization of a polaron being a case in point. Though a qualitative picture of a strong-coupling polaron is easy to visualize as opposed to the weak-coupling polaron, the mathematical formulation of a strongly coupled polaron is rather a difficult problem from the quantum mechanical point of view. In fact, the exact polaron energy in the weak-coupling limit can be obtained from the Frohlich Hamiltonian analytically, while the same is not possible in the strong-coupling limit. In the strong-coupling case, the electron velocity is much larger than the phonon frequency and therefore electron can follow the ion motion adiabatically. The adiabatic expansion of the polaron energy in the strong-coupling regime was reported by Bogoliubov and Tiablikov.
We consider the Fröhlich Hamiltonian with large coupling constant ααα. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order α2α^2α2. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order α2α^2α2.
Recently Feynman's path-integral formalism of the Fröhlich optical polaron problem is generalized, by which it is easy and natural to get the second-order perturbation result in the weak-coupling case and the Pekar's result in the strongcoupling case, even in the crudest ground-state approximation. By using numerical method to the equations which are given by the ground-state approximation, the ground-state energy of the electron-optical phonon system can obtained for the overall range of coupling strength. In addition, a self-consistent set of equations is derived for a central optimized potential without using the ground-state approximation. This effective local, central potential is the solution of a linear integral equation which was similar to the extensive work of optimized-potential model (OPM) in atomic physics. By using OPM formalism, no particular choice of interaction form is taken, therefore, except the approximation by Jensen's inequality, the ground-state energy of the electron-optical phonon interacting system can be obtained without any approximation.