Determination of the phase shifts for interacting electrons connected to reservoirs

TL;DR

This paper introduces a general method to determine phase shifts in interacting quantum-dot systems using NRG eigenvalues, applicable without specific Hamiltonian assumptions, and demonstrates its use in calculating conductance in a triple quantum dot system.

Contribution

The authors develop a versatile approach to extract phase shifts from NRG fixed-point eigenvalues, applicable to a broad class of quantum impurity models without assuming electron-hole symmetry.

Findings

Identified Kondo plateaus in conductance at odd electron numbers.

Observed wide conductance minima at even electron numbers.

Showed parallel conductance can be deduced from two phase shifts.

Abstract

We describe a formulation to deduce the phase shifts, which determine the ground-state properties of interacting quantum-dot systems with the inversion symmetry, from the fixed-point eigenvalues of the numerical renormalization group (NRG). Our approach does not assume the specific form of the Hamiltonian nor the electron-hole symmetry, and it is applicable to a wide class of quantum impurities connected to noninteracting leads. We apply the method to a triple dot which is described by a three-site Hubbard chain connected to two noninteracting leads, and calculate the dc conductance away from half-filling. The conductance shows the typical Kondo plateaus of Unitary limit in some regions of the gate voltages, at which the total number of electrons N_el in the three dots is odd, i.e., N_el =1, 3 and 5. In contrast, the conductance shows a wide minimum in the gate voltages corresponding to even number of electrons, N_el = 2 and 4. We also discuss the parallel conductance of the triple dot connected transversely to four leads, and show that it can be deduced from the two phase shifts defined in the two-lead case.