Fermionic renormalization group methods for transport through inhomogeneous Luttinger liquids

Abstract

We compare two fermionic renormalization group methods which have been used to investigate the electronic transport properties of one-dimensional metals with two-particle interaction (Luttinger liquids) and local inhomogeneities. The first one is a poor man's method setup to resum ``leading-log'' divergences of the effective transmission at the Fermi momentum. Generically the resulting equations can be solved analytically. The second approach is based on the functional renormalization group method and leads to a set of differential equations which can only for certain setups and in limiting cases be solved analytically, while in general it must be integrated numerically. Both methods are claimed to be applicable for inhomogeneities of arbitrary strength and to capture effects of the two-particle interaction, such as interaction dependent exponents, up to leading order. We critically review this for the simplest case of a single impurity. While on first glance the poor man's approach seems to describe the crossover from the ``perfect'' to the ``open chain fixed point'' we collect evidence that difficulties may arise close to the ``perfect chain fixed point''. Due to a subtle relation between the scaling dimensions of the two fixed points this becomes apparent only in a detailed analysis. In the functional renormalization group method the coupling of the different scattering channels is kept which leads to a better description of the underlying physics.

25 pages, accepted for publication in NJP, remarks added on the poor man's RG treatment of the Y-junction and the Breit-Wigner line shapes

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Cited by in corpus (20)

• Derivation of exact flow equations from the self-consistent parquet relations
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• Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model
• Numerical Renormalization Group at marginal spectral density: application to quantum tunneling in Luttinger liquids
• Steady-state spectra, current and stability diagram of a quantum dot: a non-equilibrium Variational Cluster Approach
• Chiral Y-junction of Luttinger liquid wires at weak coupling: lines of stable fixed points
• Functional renormalization group in Floquet space
• Dual fermionic variables and renormalization group approach to junctions of strongly interacting quantum wires
• Universality at work -- the local sine-Gordon model, lattice fermions, and quantum circuits
• On the nature of the Schmid transition in a resistively shunted Josephson junction
• A systematic stability analysis of the renormalisation group flow for the normal-superconductor-normal junction of Luttinger liquid wires
• Effective low-energy Hamiltonians for interacting nanostructures
• Transport across a system with three p-wave superconducting wires: effects of Majorana modes and interactions
• Two-wire Junction of Inequivalent Tomonaga-Luttinger Liquids
• A Luttinger Liquid coupled to Ohmic-class environments
• Electron-electron interaction effects on transport through mesoscopic superconducting hybrid junctions
• Conductance of correlated many-fermion systems from charge fluctuations
• Conductance of inhomogeneous Luttinger liquids with a finite bandwidth
• Breaking a "poor man" RG approach in the Luttinger liquid with one impurity
• A functional renormalization group approach to electronic structure calculations for systems without translational symmetry