From short to long distances with Gell-Mann--Low Renormalization group

Vanja Dunjko, Maxim Olshanii

While Padé approximation is a general method for improving convergence of series expansions, Gell-Mann--Low renormalization group normally relies on the presence of special symmetries. We show that in the single-variable case, the latter becomes an integral Hermite-Padé approximation, needing no special symmetries. It is especially useful for interpolating between expansions for small values of a variable and a scaling law of known exponent for large values. As an example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order term in the short-distance expansion, we find a remarkable agreement with known ab initio numerical results.

V. V. Afonin, Victor Yu. Petrov

Using a previously derived effective theory we explore the conductance in the Luttinger model with one impurity. A new approach to the renormalization group (RG) analysis of this model is developed. It is based on the original Gell-Mann-Low formulation of RG. We sum up infrared logarithmic contributions to conductance in the leading and a few subsequent approximations. We analyze the validity of the widely used 'poor man's' scaling approach and find that it is applicable only in the leading approximation. Our results for corrections to this approximation are different from results obtained in other papers. This should be expected beforehand, as the Gell-Mann-Low function of the model is not regularization scheme invariant. For this reason the observed quantity (e.g. conductance) can not satisfy the Gell-Mann-Low equation beyond the 'leading-log' approximation as it is supposed in the 'poor man's' approach. We formulate the method to calculate the conductance from the renormalized hamiltonian in post-leading approximations and match our results to the case of weak impurity where the answer is known in any order of electron-electron interaction.

Lucas Böttcher, Hans Jürgen Herrmann

Gabor Forgács