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arXiv:2610.10030

Fourth-order perturbation theory for the Frohlich polaron: Analytic structure of the weak-coupling series and the crossover to strong coupling

S. Das Sarma

Published Oct 7, 2026• 0 citations

Abstract

We calculate the Rayleigh-Schrodinger perturbation series for the ground-state energy and the effective mass of the three-dimensional Frohlich polaron through fourth order in the coupling constant alpha. The third- and fourth-order coefficients of the effective mass are new. Because the Frohlich interaction is a form-bounded perturbation of an isolated nondegenerate ground state at fixed total momentum, the series has a nonzero radius of convergence, and we analyze it with the ratio and Pade methods appropriate to convergent series. The coefficients are strikingly regular and are described by a simple power-law singularity at alpha near 8.5 with an exponent near 1.4. The couplings at which successive truncations of the inverse mass vanish, the first of which is the textbook breakdown value alpha = 6, are slowly converging estimates of this radius. Pade approximants built from the weak-coupling coefficients alone agree with diagrammatic Monte Carlo results for the mass to better than one percent for alpha up to 5. Exact results on the analyticity of the ground state and on the strong-coupling asymptotics imply that the coefficients cannot keep a constant sign, although all known coefficients do, so that the apparent singularity must be a complex-conjugate pair close to the positive real axis. It marks the crossover to strong coupling: the weak-coupling series locates this crossover accurately but cannot be continued through it.

DOI: 10.48550/arxiv.2610.10030

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We calculate the Rayleigh-Schrodinger perturbation series for the ground-state energy and the effective mass of the three-dimensional Frohlich polaron through fourth order in the coupling constant alpha. The third- and fourth-order coefficients of the effective mass are new. Because the Frohlich interaction is a form-bounded perturbation of an isolated nondegenerate ground state at fixed total momentum, the series has a nonzero radius of convergence, and we analyze it with the ratio and Pade methods appropriate to convergent series. The coefficients are strikingly regular and are described by a simple power-law singularity at alpha near 8.5 with an exponent near 1.4. The couplings at which successive truncations of the inverse mass vanish, the first of which is the textbook breakdown value alpha = 6, are slowly converging estimates of this radius. Pade approximants built from the weak-coupling coefficients alone agree with diagrammatic Monte Carlo results for the mass to better than one percent for alpha up to 5. Exact results on the analyticity of the ground state and on the strong-coupling asymptotics imply that the coefficients cannot keep a constant sign, although all known coefficients do, so that the apparent singularity must be a complex-conjugate pair close to the positive real axis. It marks the crossover to strong coupling: the weak-coupling series locates this crossover accurately but cannot be continued through it.

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The Fr\"ohlich polaron confined to move in one space dimension and interacting with a three-dimensional lattice is studied. We calculate exact and approximate results for the ground-state energy and the effective mass. The following exact results were obtained. In (a) the weak-coupling limit for the energy E=-\ensuremath{\alpha}'-[(3/2 \ensuremath{\surd}2 )-1]\ensuremath{\alpha}'

^{2}

, the mass

{\mathit{m}}^{\mathrm{*}}

/

{\mathit{m}}_{\mathit{b}}

=1/2\ensuremath{\alpha}'+[(5/2 \ensuremath{\surd}2 )-1]/4, and the average number of virtual phonons N=1/2\ensuremath{\alpha}'+[(3/ \ensuremath{\surd}2 )-2]\ensuremath{\alpha}'

^{2}

, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}'

^{2}

,

{\mathit{m}}^{\mathrm{*}}

/

{\mathit{m}}_{\mathit{b}}

=2.1254\ensuremath{\alpha}'

^{4}

, and N=0.666 176\ensuremath{\alpha}'

^{2}

. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.

The energy dispersion for arbitrary momentum of a polaron in n dimensions is calculated by the Rayleigh-Schr\"odinger perturbation theory and by a modified Brillouin-Wigner perturbation theory. For the energy and effective mass of the polaron ground state, both methods give results that agree with the results of V. V. Paranjape and P. V. Panjat [Phys. Rev. B 35, 2942 (1987)] to linear order in the polaron coupling constant \ensuremath{\alpha}. The self-energy shifts are progressively weakened with increasing dimension. In particular, this weakening occurs for the structure in the energy dispersion near the momentum value at which the polaron starts decaying via the emission of phonons.

We present results for the solution of the large polaron Fr\"ohlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.

After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.

The one-dimensional optical polaron is treated on the basis of perturbation theory in the weak-coupling limit. A special matrix diagrammatic technique is developed. It is shown how to evaluate all terms of the perturbation theory for the ground-state energy of a polaron to any order by means of this technique. The ground-state energy is calculated up to eighth order of perturbation theory. The effective mass of an electron is obtained up to sixth order of perturbation theory. The radius of convergence of the series obtained is estimated.