Nonequilibrium functional renormalization group with frequency-dependent vertex function: A study of the single-impurity Anderson model

Abstract

We investigate nonequilibrium properties of the single impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening Γ/2 can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi liquid behaviour that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions U 3Γ. In equilibrium we find excellent agreement with NRG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with TD-DMRG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.

Analytic continuation of functional renormalization group equations

“…The infrared regulator was chosen to depend on spatial derivatives only while the fluctuations in the time directions have been regulated by compactification. A large amount of literature exists also on applications of different renormalization group techniques to transport through a zero-dimensional quantum system such as a quantum dot which is coupled to appropriate reservoirs [27][28][29][30][31][32][33][34][35][36][37][38].…”

Nonequilibrium transport theory of the singlet-triplet transition: Perturbative approach

“…In spite of the simplicity of this "iterative perturbation theory", the results obtained with it show remarkable agreement with the ones obtained with more advanced techniques for several equilibrium systems. 26 Unfortunately, a well established and exact non-equilibrium impurity solver being not available yet, one can only judge the validity of this perturbative approach by possibly comparing with results obtained by some reliable equilibrium methods, and by investigating the internal consistency of its. As discussed in Section II E, the range of applicability turns out to be limited to small to moderate values of U and J.…”

“…Although theoretical physicists devoted a lot of effort to design methods that are able to tackle these difficulties, so far none of the proposed methods proved to be entirely successful in describing strongly interacting non-equilibrium systems: Monte Carlo methods are presently unable to reach the required precision, 20,21 Bethe Ansatz methods can be used for a few models only, and are still in an experimental stage, [22][23][24] and perturbative renormalization group methods can only reach a particular region of the parameter space. [25][26][27][28] Maybe numerical renormalization group methods are currently the most reliable techniques to study these non-equilibrium systems, 24,29,30 however, they scale very badly with the number of states involved, and to compute the transport through just two levels in the presence of interaction seems to be numerically too demanding.…”

Nonequilibrium transport through a Josephson quantum dot

“…57. It can be thought of as an additional artificial (metallic) reservoir that is coupled to the dot via a hybridization constant Λ, which assumes the role of the flow parameter flowing from ∞ to 0.…”

“…The functional RG approach to mesoscopic transport 54 was earlier generalized to study nonequilibrium setups with metallic leads in the steady state [55][56][57][58][59] as well as ground state properties of quantum dots with superconducting leads such as the Josephson current. 36,45,59,60 We here combine these two extensions and thus further advance the method.…”