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arXiv:2603.07290 · Journal of Physics Condensed Matter

Coherent-state ansatz for the Holstein polaron in one and two dimensions

Connor M. Walsh, Igor Boettcher, F. Marsiglio

Published Aug 24, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1361-648x/ae9dd1

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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.
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.
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.
Employing a largely unbiased variational exact diagonalization technique, we analyze the consequences of longer-ranged electron hopping and electron-phonon interaction on polaron formation in one dimension. Having at our disposal the accurate ground-state energy and wave function, we calculate and discuss various physical quantities, such as the renormalized band structure, effective mass, wave-function renormalization factor, phonon dressing, and Drude weight, characterizing the properties of the polaronic quasiparticle. We demonstrate that the electron-phonon coupling affects the relative strength of the nearest neighbor (NN) and next nearest neighbor (NNN) hopping processes in a dynamic way. Most notably we observe that the minimum of the polaron band, occurring at a finite momentum for large negative ratio between NN and NNN transfer, jumps to zero momentum as the electron-phonon coupling exceeds a critical one, thereby causing a rather sharp polaron transition in the one-dimensional extended Holstein model. The signatures of this transition are seen in the effective mass and polaron mobility, and therefore should be easily detectable by transport measurements.
We utilize an exact variational numerical procedure to calculate the ground state properties of a polaron in the presence of a Rashba-like spin-orbit interaction. Our results corroborate previous work performed with the momentum average approximation and with weak-coupling perturbation theory. We find that spin-orbit coupling increases the effective mass in the regime with weak electron-phonon coupling, and decreases the effective mass in the regimes of intermediate and strong electron-phonon coupling. Analytical strong-coupling perturbation theory results confirm our numerical results in the small-polaron regime. A large amount of spin-orbit coupling can lead to a significant lowering of the polaron effective mass.
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.
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.