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arXiv:2102.13035 · Physical review. B./Physical review. B

Many-body perturbation theory for the superconducting quantum dot: Fundamental role of the magnetic field

V. Janiš, Jiawei Yan

Published Jun 28, 2021• 5 citations
arXivHTMLPDF

Abstract

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated 0 phase. It controls the critical behavior of the $0\text{\ensuremath{-}}\ensuremath{\pi}$ quantum transition and lifts the degeneracy in the $\ensuremath{\pi}$ phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in 0 and $\ensuremath{\pi}$ phases. While the magnetic susceptibility vanishes in the 0 phase it becomes divergent in the $\ensuremath{\pi}$ phase at zero temperature.

DOI
10.1103/physrevb.103.235163

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We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated 0 phase. It controls the critical behavior of the 0\ensuremath-\ensuremathπ0\text{\ensuremath{-}}\ensuremath{\pi}0\ensuremath-\ensuremathπ quantum transition and lifts the degeneracy in the \ensuremathπ\ensuremath{\pi}\ensuremathπ phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in 0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ phases. While the magnetic susceptibility vanishes in the 0 phase it becomes divergent in the \ensuremathπ\ensuremath{\pi}\ensuremathπ phase at zero temperature.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
The system of several (N)(N)(N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel NNN-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the NNN-impurity S=1∕2S=1∕2S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin S=N∕2S=N∕2S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the NNN-impurity Anderson model to the exact Bethe ansatz results of a S=N∕2S=N∕2S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective S=N∕2S=N∕2S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities NNN. We demonstrate the robustness of the spin N∕2N∕2N∕2 ground state as well as of the associated S=N∕2S=N∕2S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.
We consider strongly correlated regimes which emerge at low temperature in coupled quantum dot (or magnetic impurity) systems. In strongly correlated systems a single particle description fails to explain the observed behaviour, so we resort to many body methods. We describe our system using a 2-impurity Anderson model and develop a numerical renormalisation group procedure which provides non-perturbative insight into the low energy behaviour, through calculation of dynamic quantities. We combine this approach with renormalised perturbation theory, thus acquiring a picture of how the Hamiltonian and interactions change at low energies. These approaches are first used to study the emergence of a Kondo effect with an SU(4) symmetry in capacitively-coupled double quantum dot systems. We classify the 'types' of SU(4) symmetry which can emerge and show how an experimentalist might achieve such emergence through tuning their system. We provide a way of distinguishing between the SU(2) and SU(4) Kondo regimes by considering the conductance. We also study a quantum critical point which occurs in the Heisenberg coupled quantum dot/impurity model. There is an anomalous entropy contributed by the impurities in this regime which is indicative of an uncoupled Majorana Fermion. We calculate dynamic quantities in regimes with different symmetries and establish correspondence with the 2-channel Kondo model. We formulate possible pictures of the underlying mechanisms of the critical point and construct a Majorana fermion model for the case with particle-hole symmetry, which explains the non-Fermi liquid energy levels and degeneracies obtained. We conjecture that a Majorana zero mode is present, and that this is responsible for the anomalous entropy.
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where UUU is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed{E}_{d}Ed​, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT\mathrm{NRG}+\mathrm{RPT}NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed{E}_{d}Ed​ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in UUU agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of UUU. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.
A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices based on the parquet equations. We explain in detail how to reduce the full set of parquet equations so as not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation and discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. Finally, we demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.