We present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.We present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.

arXiv:1810.12269Physical review. B./Physical review. BMay 24, 201921 citations

Keldysh effective action theory for universal physics in spin- 12 Kondo dots

Sergey Smirnov, Milena Grifoni

We present a theory for the Kondo spin- $\frac{1}{2}$ effect in strongly correlated quantum dots. The theory is applicable at any temperature and voltage. It is based on a quadratic Keldysh effective action parametrized by a universal function. We provide a general analytical form for the tunneling density of states through this universal function for which we propose a simple microscopic model. We apply our theory to the highly asymmetric Anderson model with $U=\ensuremath{\infty}$ and describe its strong-coupling limit, weak-coupling limit, and crossover region within a single analytical expression. We compare our results with a numerical renormalization group in equilibrium and with a real-time renormalization group out of equilibrium and show that the universal shapes of the linear and differential conductance obtained in our theory and in these theories are very close to each other in a wide range of temperatures and voltages. In particular, as in the real-time renormalization group, we predict that at the Kondo voltage the differential conductance is equal to $2/3$ of its maximum.We present a theory for the Kondo spin-$\frac{1}{2}$ effect in strongly correlated quantum dots. The theory is applicable at any temperature and voltage. It is based on a quadratic Keldysh effective action parametrized by a universal function. We provide a general analytical form for the tunneling density of states through this universal function for which we propose a simple microscopic model. We apply our theory to the highly asymmetric Anderson model with $U=\ensuremath{\infty}$ and describe its strong-coupling limit, weak-coupling limit, and crossover region within a single analytical expression. We compare our results with a numerical renormalization group in equilibrium and with a real-time renormalization group out of equilibrium and show that the universal shapes of the linear and differential conductance obtained in our theory and in these theories are very close to each other in a wide range of temperatures and voltages. In particular, as in the real-time renormalization group, we predict that at the Kondo voltage the differential conductance is equal to $2/3$ of its maximum.

arXiv:1203.4360Physical Review BMar 12, 201326 citations

Quantum-mechanical tunnelling and the renormalization group

A. S. Kapoyannis, N. Tetradis

arXiv:hep-th/0010180Physics Letters ANov 1, 200036 citations

The Real Space Dynamic Renormalization Group

Gene F. Mazenko, Oriol T. Valls

Topics in current physicsJan 1, 19823 citations

Fermionic superoperators for zero-temperature nonlinear transport: Real-time perturbation theory and renormalization group for Anderson quantum dots

R. B. Saptsov, M. R. Wegewijs

We study electron quantum transport through a strongly interacting Anderson quantum dot at finite bias voltage and magnetic field at zero temperature using the real-time renormalization group (RT-RG) in the framework of a kinetic (generalized master) equation for the reduced density operator. To this end, we further develop the general, finite-temperature real-time transport formalism by introducing field superoperators that obey fermionic statistics. This direct second quantization in Liouville Fock space strongly simplifies the construction of operators and superoperators that transform irreducibly under the Anderson-model symmetry transformations. The fermionic field superoperators naturally arise from the univalence (fermion-parity) superselection rule of quantum mechanics for the total system of quantum dot plus reservoirs. Expressed in these field superoperators, the causal structure of the perturbation theory for the effective time-evolution superoperator kernel becomes explicit. Using the constraints of the causal structure, we construct a parametrization of the exact effective time-evolution kernel for which we analytically find the eigenvectors and eigenvalues in terms of a minimal set of only 30 independent coefficients. The causal structure also implies the existence of a fermion-parity protected eigenvector of the exact Liouvillian, explaining a recently reported result on adiabatic driving [Contreras-Pulido et al., Phys. Rev. B 85, 075301 (2012)] and generalizing it to arbitrary order in the tunnel coupling $\ensuremath{\Gamma}$. Furthermore, in the wide-band limit, the causal representation exponentially reduces the number of diagrams for the time-evolution kernel. The remaining diagrams can be identified simply by their topology and are manifestly independent of the energy cutoff term by term. By an exact reformulation of this series, we integrate out all infinite-temperature effects, obtaining an expansion targeting only the nontrivial, finite-temperature corrections, and the exactly conserved transport current follows directly from the time-evolution kernel. From this new series, the previously formulated RT-RG equations are obtained naturally. We perform a complete one-plus-two-loop RG analysis at finite voltage and magnetic field, while systematically accounting for the dependence of all renormalized quantities on both the quantum dot and reservoir frequencies. Using the second quantization in Liouville space and symmetry restrictions, we obtain analytical RT-RG equations, which can be solved numerically in an efficient way, and we extensively study the model parameter space, excluding the Kondo regime where the one-plus-two-loop approach is obviously invalid. The incorporated renormalization effects result in an enhancement of the inelastic cotunneling peak, even at a voltage $\ensuremath{\sim}$magnetic field $\ensuremath{\sim}$tunnel coupling $\ensuremath{\Gamma}$. Moreover, we find a tunnel-induced nonlinearity of the stability diagrams (Coulomb diamonds) at finite voltage, both in the single-electron tunneling and inelastic cotunneling regime.We study electron quantum transport through a strongly interacting Anderson quantum dot at finite bias voltage and magnetic field at zero temperature using the real-time renormalization group (RT-RG) in the framework of a kinetic (generalized master) equation for the reduced density operator. To this end, we further develop the general, finite-temperature real-time transport formalism by introducing field superoperators that obey fermionic statistics. This direct second quantization in Liouville Fock space strongly simplifies the construction of operators and superoperators that transform irreducibly under the Anderson-model symmetry transformations. The fermionic field superoperators naturally arise from the univalence (fermion-parity) superselection rule of quantum mechanics for the total system of quantum dot plus reservoirs. Expressed in these field superoperators, the causal structure of the perturbation theory for the effective time-evolution superoperator kernel becomes explicit. Using the constraints of the causal structure, we construct a parametrization of the exact effective time-evolution kernel for which we analytically find the eigenvectors and eigenvalues in terms of a minimal set of only 30 independent coefficients. The causal structure also implies the existence of a fermion-parity protected eigenvector of the exact Liouvillian, explaining a recently reported result on adiabatic driving [Contreras-Pulido et al., Phys. Rev. B 85, 075301 (2012)] and generalizing it to arbitrary order in the tunnel coupling $\ensuremath{\Gamma}$. Furthermore, in the wide-band limit, the causal representation exponentially reduces the number of diagrams for the time-evolution kernel. The remaining diagrams can be identified simply by their topology and are manifestly independent of the energy cutoff term by term. By an exact reformulation of this series, we integrate out all infinite-temperature effects, obtaining an expansion targeting only the nontrivial, finite-temperature corrections, and the exactly conserved transport current follows directly from the time-evolution kernel. From this new series, the previously formulated RT-RG equations are obtained naturally. We perform a complete one-plus-two-loop RG analysis at finite voltage and magnetic field, while systematically accounting for the dependence of all renormalized quantities on both the quantum dot and reservoir frequencies. Using the second quantization in Liouville space and symmetry restrictions, we obtain analytical RT-RG equations, which can be solved numerically in an efficient way, and we extensively study the model parameter space, excluding the Kondo regime where the one-plus-two-loop approach is obviously invalid. The incorporated renormalization effects result in an enhancement of the inelastic cotunneling peak, even at a voltage $\ensuremath{\sim}$magnetic field $\ensuremath{\sim}$tunnel coupling $\ensuremath{\Gamma}$. Moreover, we find a tunnel-induced nonlinearity of the stability diagrams (Coulomb diamonds) at finite voltage, both in the single-electron tunneling and inelastic cotunneling regime.

arXiv:1207.3207Physical Review BDec 18, 201271 citations

We present a theory for the Kondo spin-

\frac{1}{2}

effect in strongly correlated quantum dots. The theory is applicable at any temperature and voltage. It is based on a quadratic Keldysh effective action parametrized by a universal function. We provide a general analytical form for the tunneling density of states through this universal function for which we propose a simple microscopic model. We apply our theory to the highly asymmetric Anderson model with $U=\ensuremath{\infty}$ and describe its strong-coupling limit, weak-coupling limit, and crossover region within a single analytical expression. We compare our results with a numerical renormalization group in equilibrium and with a real-time renormalization group out of equilibrium and show that the universal shapes of the linear and differential conductance obtained in our theory and in these theories are very close to each other in a wide range of temperatures and voltages. In particular, as in the real-time renormalization group, we predict that at the Kondo voltage the differential conductance is equal to

2/3

of its maximum.