arXiv:0805.4525 · Physical Review E

Perturbation theory without diagrams: The polaron case

R. Rosenfelder

Published Jan 13, 2009• 3 citations

Abstract

Higher-order perturbative calculations in quantum (field) theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines two new coefficients are obtained corresponding to a four- and five-loop calculation. Several analytical and numerical procedures have been implemented which were crucial for obtaining reliable results.

DOI: 10.1103/physreve.79.016705

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arXiv:0805.4525 · Physical Review E

Perturbation theory without diagrams: The polaron case

R. Rosenfelder

Published Jan 13, 2009• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.79.016705

Related papers

Direct path-integral treatment of the polaron problem

Arkady L. Kholodenko, Karl F. Freed

Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$ -order terms in the polaron coupling constant are obtained from $n\mathrm{th}$ -order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$ -order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.

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n\mathrm{th}

-order terms in the polaron coupling constant are obtained from

n\mathrm{th}

-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a

2n\mathrm{th}

-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.

On the basis of an exact perturbational expression for the interacting one-particle Green function

G

corresponding to bosons / fermions in terms of the bare interaction potential

v

and permanents / determinants of the non-interacting one-particle Green function

G_0

, we deduce four recursive perturbation expansions for the self-energy

Σ

. With

W

denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of

Σ

in terms of (i) self-energy diagrams and

(v, G_0)

, (ii)

G

-skeleton self-energy diagrams and

(v, G)

, (iii)

W

-skeleton self-energy diagrams and

(W, G_0)

, and (iv)

G

- and

W

-skeleton self-energy diagrams and

(W,G)

. For the calculation of

W

, we rely on a similar exact perturbational expression for the interacting two-particle Green function

G_2

as for

G

. From this expression, we deduce four recursive perturbation expansions for the polarization function

P

, necessary for the calculation of

W

, that are similar to those for

Σ

specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-

\tfrac{1}{2}

fermions on a lattice in arbitrary

d

spatial dimensions. In two further appendices, we present methods and short programs for determining the

ν

th-order diagrams corresponding to the perturbation expansions of

G

in terms of

(v,G_0)

, and

Σ

in terms of

(v,G_0)

and

(v,G)

on the basis of the cycle decompositions of the elements of the symmetric group

S_{2ν}