30 research outputs found
Multiloop functional renormalization group for general models
We present multiloop flow equations in the functional renormalization group
(fRG) framework for the four-point vertex and self-energy, formulated for a
general fermionic many-body problem. This generalizes the previously introduced
vertex flow [F. B. Kugler and J. von Delft, Phys. Rev. Lett. 120, 057403
(2018)] and provides the necessary corrections to the self-energy flow in order
to complete the derivative of all diagrams involved in the truncated fRG flow.
Due to its iterative one-loop structure, the multiloop flow is well-suited for
numerical algorithms, enabling improvement of many fRG computations. We
demonstrate its equivalence to a solution of the (first-order) parquet
equations in conjunction with the Schwinger-Dyson equation for the self-energy
Fermi-edge singularity and the functional renormalization group
We study the Fermi-edge singularity, describing the response of a degenerate
electron system to optical excitation, in the framework of the functional
renormalization group (fRG). Results for the (interband) particle-hole
susceptibility from various implementations of fRG (one- and two-
particle-irreducible, multi-channel Hubbard-Stratonovich, flowing
susceptibility) are compared to the summation of all leading logarithmic (log)
diagrams, achieved by a (first-order) solution of the parquet equations. For
the (zero-dimensional) special case of the X-ray-edge singularity, we show that
the leading log formula can be analytically reproduced in a consistent way from
a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic
structure, we show that this derivation relies on fortuitous partial
cancellations special to the form of and accuracy applied to the X-ray-edge
singularity and does not generalize
Multiloop functional renormalization group that sums up all parquet diagrams
We present a multiloop flow equation for the four-point vertex in the
functional renormalization group (fRG) framework. The multiloop flow consists
of successive one-loop calculations and sums up all parquet diagrams to
arbitrary order. This provides substantial improvement of fRG computations for
the four-point vertex and, consequently, the self-energy. Using the X-ray-edge
singularity as an example, we show that solving the multiloop fRG flow is
equivalent to solving the (first-order) parquet equations and illustrate this
with numerical results
Counting Feynman diagrams via many-body relations
We present an iterative algorithm to count Feynman diagrams via many-body
relations. The algorithm allows us to count the number of diagrams of the exact
solution for the general fermionic many-body problem at each order in the
interaction. Further, we apply it to different parquet-type approximations and
consider spin-resolved diagrams in the Hubbard model. Low-order results and
asymptotics are explicitly discussed for various vertex functions and different
two-particle channels. The algorithm can easily be implemented and generalized
to many-body relations of different forms and levels of approximation
RG transport theory for open quantum systems: Charge fluctuations in multilevel quantum dots in and out of equilibrium
We present the real-time renormalization group (RTRG) method as a method to
describe the stationary state current through generic multi-level quantum dots
with a complex setup in nonequilibrium. The employed approach consists of a
very rudiment approximation for the 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 numerically 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
theoretically in quantum dot systems
Computing local multipoint correlators using the numerical renormalization group
Local three- and four-point correlators yield important insight into strongly
correlated systems and have many applications. However, the nonperturbative,
accurate computation of multipoint correlators is challenging, particularly in
the real-frequency domain for systems at low temperatures. In the accompanying
paper, we introduce generalized spectral representations for multipoint
correlators. Here, we develop a numerical renormalization group (NRG) approach,
capable of efficiently evaluating these spectral representations, to compute
local three- and four-point correlators of quantum impurity models. The key
objects in our scheme are partial spectral functions, encoding the system's
dynamical information. Their computation via NRG allows us to simultaneously
resolve various multiparticle excitations down to the lowest energies. By
subsequently convolving the partial spectral functions with appropriate
kernels, we obtain multipoint correlators in the imaginary-frequency Matsubara,
the real-frequency zero-temperature, and the real-frequency Keldysh formalisms.
We present exemplary results for the connected four-point correlators of the
Anderson impurity model, and for resonant inelastic x-ray scattering (RIXS)
spectra of related impurity models. Our method can treat temperatures and
frequencies -- imaginary or real -- of all magnitudes, from large to
arbitrarily small ones.Comment: See also the jointly published paper [F. B. Kugler, S.-S. B. Lee, and
J. von Delft, Phys. Rev. X 11, 041006 (2021); arXiv:2101.00707
Flavor fluctuations in 3-level quantum dots: Generic SU(3)-Kondo fixed point in equilibrium and non-Kondo fixed points in nonequilibrium
We study a -level quantum dot in the singly occupied cotunneling regime
coupled via a generic tunneling matrix to several multi-channel leads in
equilibrium or nonequilibrium. We derive an effective model where also each
reservoir has three channels labelled by the quark flavors , and
with an effective d.o.s. polarized w.r.t. an eight-dimensional -spin
corresponding to the eight generators of . In equilibrium we perform a
standard poor man scaling analysis and show that tunneling via virtual
intermediate states induces flavor fluctuations on the dot which become
-symmetric at a characteristic and exponentially small low-energy scale
. Using the numerical renormalization group (NRG) we study in detail the
linear conductance and confirm the -symmetric Kondo fixed point with
universal conductance for various tunneling setups by tuning the
level spacings on the dot. In contrast to the equilibrium case, we find in
nonequilibrium that the fixed point model is not -symmetric but
characterized by rotated -spins for each reservoir with total vanishing sum.
At large voltage we analyse the -spin magnetization and the current in
golden rule as function of a magnetic field for the isospin of the up/down
quark and the level spacing to the strange quark. As a smoking gun to detect
the nonequilibrium fixed point we find that the curve of zero -spin
magnetization has a particular shape on the dot parameters. We propose that our
findings can be generalized to the case of quantum dots with an arbitrary
number of levels.Comment: 24 pages, 8 figure
Analytic continuation of multipoint correlation functions
Conceptually, the Matsubara formalism (MF), using imaginary frequencies, and
the Keldysh formalism (KF), formulated in real frequencies, give equivalent
results for systems in thermal equilibrium. The MF has less complexity and is
thus more convenient than the KF. However, computing dynamical observables in
the MF requires the analytic continuation from imaginary to real frequencies.
The analytic continuation is well-known for two-point correlation functions
(having one frequency argument), but, for multipoint correlators, a
straightforward recipe for deducing all Keldysh components from the MF
correlator had not been formulated yet. Recently, a representation of MF and KF
correlators in terms of formalism-independent partial spectral functions and
formalism-specific kernels was introduced by Kugler, Lee, and von Delft [Phys.
Rev. X 11, 041006 (2021)]. We use this representation to formally elucidate the
connection between both formalisms. We show how a multipoint MF correlator can
be analytically continued to recover all partial spectral functions and yield
all Keldysh components of its KF counterpart. The procedure is illustrated for
various correlators of the Hubbard atom.Comment: 56 pages, 8 figure
Real-frequency quantum field theory applied to the single-impurity Anderson model
A major challenge in the field of correlated electrons is the computation of
dynamical correlation functions. For comparisons with experiment, one is
interested in their real-frequency dependence. This is difficult to compute, as
imaginary-frequency data from the Matsubara formalism require analytic
continuation, a numerically ill-posed problem. Here, we apply quantum field
theory to the single-impurity Anderson model (AM), using the Keldysh instead of
the Matsubara formalism with direct access to the self-energy and dynamical
susceptibilities on the real-frequency axis. We present results from the
functional renormalization group (fRG) at one-loop level and from solving the
self-consistent parquet equations in the parquet approximation. In contrast to
previous Keldysh fRG works, we employ a parametrization of the four-point
vertex which captures its full dependence on three real-frequency arguments. We
compare our results to benchmark data obtained with the numerical
renormalization group and to second-order perturbation theory. We find that
capturing the full frequency dependence of the four-point vertex significantly
improves the fRG results compared to previous implementations, and that solving
the parquet equations yields the best agreement with the NRG benchmark data,
but is only feasible up to moderate interaction strengths. Our methodical
advances pave the way for treating more complicated models in the future.Comment: 25 pages, 20 figure