39 research outputs found
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
Emergent Properties of the Periodic Anderson Model: a High-Resolution, Real-Frequency Study of Heavy-Fermion Quantum Criticality
We study paramagnetic quantum criticality in the periodic Anderson model
(PAM) using cellular dynamical mean-field theory, with the numerical
renormalization group (NRG) as an impurity solver. The PAM describes an
itinerant band hybridizing with a localized band. At , it exhibits
a hybridization tuned Kondo breakdown quantum critical point (KB-QCP) from a
Kondo to an RKKY phase. At the KB-QCP, the band changes character from
itinerant to mainly localized, while the band remains itinerant. We
elucidate its nature in detail by performing a high-resolution, real-frequency
study of dynamical quantities. NRG allows us to study the quantum critical
non-Fermi-liquid (NFL) regime located between . Surprisingly,
self-consistency is essential to stabilize the NFL and the QCP. The
Fermi-liquid (FL) scale decreases towards and vanishes at the QCP. At
, we find the following properties. The quasiparticle (QP) weight
decreases continuously as the QCP is approached from either side,
vanishing only at the QCP. Therefore, is nonzero in both the Kondo and
the RKKY phase; hence, the FL QP comprise and electrons in both phases.
The Fermi surface (FS) volumes in the two phases differ. Whereas the large-FS
Kondo phase has a usual two-band structure, the small-FS RKKY phase has an
unexpected three-band structure. We provide a detailed analysis of
quasiparticle properties of both the Kondo and the RKKY phase. The FS
reconstruction is accompanied by the appearance of a Luttinger surface (LS) on
which the self-energy diverges. The FS and LS volumes are related to the
density by a generalized Luttinger sum rule. We interpret the small FS volume
and the emergent LS as evidence for -electron fractionalization in the RKKY
phase. Our Hall coefficient and specific heat are in good qualitative agreement
with experiment.Comment: 38 pages, 26 figure
Minimax optimization of entanglement witness operator for the quantification of three-qubit mixed-state entanglement
We develop a numerical approach for quantifying entanglement in mixed quantum
states by convex-roof entanglement measures, based on the optimal entanglement
witness operator and the minimax optimization method. Our approach is
applicable to general entanglement measures and states and is an efficient
alternative to the conventional approach based on the optimal pure-state
decomposition. Compared with the conventional one, it has two important merits:
(i) that the global optimality of the solution is quantitatively verifiable,
and (ii) that the optimization is considerably simplified by exploiting the
common symmetry of the target state and measure. To demonstrate the merits, we
quantify Greenberger-Horne-Zeilinger (GHZ) entanglement in a class of
three-qubit full-rank mixed states composed of the GHZ state, the W state, and
the white noise, the simplest mixtures of states with different genuine
multipartite entanglement, which have not been quantified before this work. We
discuss some general properties of the form of the optimal witness operator and
of the convex structure of mixed states, which are related to the symmetry and
the rank of states
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