The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wave function. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite-size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.
The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wave function. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite-size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.
We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local moment and strong-coupling ground states, and is related to a change in the Luttinger volume of the conduction bath. Our results offer fresh insight on the nature of the emergent many-particle entanglement within the Kondo cloud, and pave the way for further investigations in more exotic contexts such as the fixed point of the over-screened multi-channel Kondo problem.
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.
In this thesis we study the physical properties of quantum magnetic impurities in con- densed matter systems. More precisely, we are interested in the quantum many-body phenomena emerging in metallic quantum wires with spin-orbit coupling (SOC) in the presence of magnetic impurities. In particular, we investigate Kondo effect in topological superconducting quantum wires sustaining Majorana zero modes (MZMs) bound to their edges. The first part of this work is dedicated to a general digression on the formation of localized magnetic moments in solid state materials and the emergence of the Kondo effect in single-impurity systems. We further study a system composed by a single level quantum dot coupled to a metallic reservoir and to two MZMs with opposite polarizations. The system is described by a version of single impurity Anderson model (SIAM) adapted to account for the MZMs. To access the Kondo physics we used the projection approach to derive an effective Kondo-like Hamiltonian from the SIAM. We show the coexistence of the many-body Kondo singlet and a local Majorana singlet which results from the robustness of the Kondo fixed points even in the presence local Majorana operators. Moreover, we show that the free MZMs of the system are responsible for the impurity residual entropy S res = N/ 2 k B ln(2), where N is the number of free MZMs. This result reveals a fractional residual entropy for an odd number of free MZMs associated with a local non-Fermi liquid behavior. Both results were confirmed by the perturbative Anderson’s Poor Man’s scaling (PMS), and the nonperturbative Wilson’s numerical renormalization group (NRG). The second part of the thesis is dedicated to the low-temperature physics of two-impurity systems. We revisit the well known Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two spatially separated magnetic impurities in a quantum wire with SOC. To compute the RKKY coupling we use the original Ruderman-Kittel approach, based on the traditional second-order perturbation theory. This allows us find also the Dzaloshinskii-Moriya and Ising interactions between the impurities, both originated from the SOC. Our results show that due to momentum-spin-lock, provided by the SOC, the RKKY coupling does not decay with the inverse of the inter-impurities distance, as in the usual case. Instead, the coupling survives even at large distances oscillating with a characteristic wave-vector 2k R, where k R is proportional to the Rashba SOC.
We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.
Dynamical quantum impurities in metals induce electronic correlations in real space that are difficult to simulate due to their multiscale nature, so that only $s\text{\ensuremath{-}}\mathrm{wave}$ scattering in clean metallic hosts has been investigated so far. However, screening clouds should show anisotropy due to lack of full rotational invariance in two- and three-dimensional lattices, while inherent disorder will also induce spatial inhomogeneities. To tackle these challenges, we present an efficient and robust algorithm based on the recursive generation of natural orbitals defined as eigenvectors of the truncated single-particle density matrix. This method provides well-converged many-body wave functions on lattices with up to tens of thousands of sites, bypassing some limitations of other approaches. The algorithm is put to the test by investigating the charge screening cloud around an interacting resonant level, both on clean and disordered lattices, achieving accurate spatial resolution from short to long distances. We thus demonstrate strong anisotropy of spatial correlations around an adatom in the half-filled square lattice. Taking advantage of the efficiency of the algorithm, we further compute the disorder-induced distribution of Kondo temperatures over several thousands of random realizations, at the same time gaining access to the full spatial profile of the screening cloud in each sample. While the charge screening cloud is typically shortened due to the polarization of the impurity by the disorder potential, we surprisingly find that rare disorder configurations preserve the long-range nature of Kondo correlations in the electronic bath.
We disentangle all the individual degrees of freedom in the quantum impurity problem to deconstruct the Kondo singlet, both in real and energy space, by studying the contribution of each individual free electron eigenstate. This is a problem of two spins coupled to a bath, where the bath is formed by the remaining conduction electrons. Being a mixed state, we resort to the ``concurrence'' to quantify entanglement. We identify ``projected natural orbitals'' that allow us to individualize a single-particle electronic wave function that is responsible of more than 90 of the impurity screening. In the weak coupling regime, the impurity is entangled to an electron at the Fermi level, while in the strong coupling regime, the impurity counterintuitively entangles mostly with the high energy electrons and disentangles completely from the low-energy states carving a ``hole'' around the Fermi level. This enables one to use concurrence as a pseudo order parameter to compute the characteristic ``size'' of the Kondo cloud, beyond which electrons are weakly correlated to the impurity and are dominated by the physics of the boundary.