Quantum Impurity Problems in the Framework of Natural Orbitals

Rong-Qiang He, Zhong-Yi Lu

One of the most important and accurate approaches for studying interacting many-electron correlated systems is the quantum renormalization group (RG) procedure, upon which we propose a concept in the framework of natural orbitals so that we can generalize the RG into general orbital space. To demonstrate its effectiveness, we apply this new approach on a quantum cluster-impurity model with four impurities in comparison with the exact diagonalization method. We also find a shortcut to dramatically improving this RG procedure. A profound impact on the cluster dynamical mean-field theory and multi-impurity/orbital Kondo problems is further discussed.

Gergely Zaránd

Arne Ludwig

Wen-Jing 文静 Zhang 张, Ru Zheng, Rong-Qiang He +1 more

Abstract In a superconductor embedded with a quantum magnetic impurity, the Kondo effect is involved, leading to the competition between the Kondo singlet phase and the superconductivity phase. By means of the natural orbitals renormalization group (NORG) method, we revisit the problem of a quantum magnetic impurity coupled with a conventional s-wave superconductor. Here we present a detailed study focusing on the impurity spin polarization and susceptibility, the Kondo screening cloud, as well as the number and structures of the active natural orbitals (ANOs). In the superconducting phase, the impurity spin is partially polarized, indicating that the impurity remains partially screened by the quantum fluctuations. Furthermore, the impurity spin susceptibility becomes divergent, resulting from the presence of residual local moment formed at the impurity site. Correspondingly, a non-integral (incomplete) Kondo cloud is formed, although the ground state is a spin doublet in this phase. In comparison, the Kondo cloud is complete in the Kondo singlet phase as expected. We also quantify the critical point, where the quantum phase transition from a Kondo singlet phase to a superconducting phase occurs, which is consistent with that in previous works. On the other hand, it is illustrated that only one ANO emerges in both quantum phases. The structures of the ANO, projected into both the real space and momentum space, are distinct in the Kondo singlet phase from that in the superconducting phase. More specifically, in the Kondo singlet phase, the ANO keeps fully active with half-occupied, and the superconducting gap has negligible influence on its structure. On the contrary, in the superconducting phase, the ANO tends to be inactive and its structure changes significantly as the superconducting gap increases. Additionally, our investigation demonstrates that the NORG method is reliable and convenient to solve the quantum impurity problems in superconductors as well, which will promote further theoretical studies on the Kondo problems in such systems using numerical methods.

Maxime Debertolis

Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.

Sergey Bravyi, David Gosset