arXiv:1604.04704 · Journal of Physics Condensed Matter

Breaking a one-parameter ‘poor man’s’ scaling approach in the Luttinger liquid

V. V. Afonin, Victor Yu. Petrov

Published Jul 23, 2018• 2 citations

Abstract

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.

DOI: 10.1088/1361-648x/aad515

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arXiv:1604.04704 · Journal of Physics Condensed Matter

Breaking a one-parameter ‘poor man’s’ scaling approach in the Luttinger liquid

V. V. Afonin, Victor Yu. Petrov

Published Jul 23, 2018• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1361-648x/aad515

Related papers

Conductance through a potential barrier embedded in a Luttinger liquid: Nonuniversal scaling at strong coupling

D. N. Aristov, P. Wölfle

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arXiv:cond-mat/0509021Physical Review BJan 26, 200648 citations

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction

{g}_{2}

, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale-independent terms is emphasized. The latter depend on the regularization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group $\ensuremath{\beta}$-function are identified. A universal part of the $\ensuremath{\beta}$-function is given by a ladder series and summed to all orders in

{g}_{2}

. First nonuniversal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.

1∕2

fermions. The underlying approximations are devised for weak interactions and arbitrary impurity strengths, and have been checked by comparing with density-matrix renormalization-group data. We present results for the density of states, the density profile, and the linear conductance. Two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.