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Many-body correlations and Majorana fermions in quantum impurity problems

Joelson F. Silva

Published Mar 3, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.14393/ufu.te.2021.182

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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.
The prospect of using semiconductor quantum dots as an experimental tool to distinguish Majorana zero modes (MZMs) from other zero-energy excitations such as Kondo resonances has brought up the fundamental question of whether topological superconductivity and the Kondo effect can coexist in these systems. Here, we study the Kondo effect in a quantum dot coupled to a metallic contact and to a pair of MZMs. We consider a situation in which the MZMs are spin polarized in opposite directions. By using numerical renormalization group calculations and scaling analysis of the renormalization group equations, we show that the Kondo effect takes place at low temperatures, regardless of the coupling to the MZMs. Interestingly, we find that the Kondo singlet essentially decouples from the MZMs such that the residual impurity entropy can show local non-Fermi-liquid properties characteristic of the single Majorana excitations. This offers the possibility of tuning between Fermi-liquid and non-Fermi-liquid regimes simply by changing the quantum-dot-MZM couplings.
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
We study a quantum impurity coupled to the edge states of a two-dimensional helical topological superconductor, that is, to a pair of counterpropagating Majorana fermion edge channels with opposite spin polarizations. For an impurity described by the Anderson impurity model, we show that the problem maps onto a variant of the interacting resonant two-level model which, in turn, maps onto the ferromagnetic Kondo model. Both magnetic and nonmagnetic impurities are considered. For magnetic impurities, an analysis relying on bosonization and the numerical renormalization group shows that the system flows to a fixed point characterized by a residual ln2\mathrm{ln}2ln2 entropy and anisotropic static and dynamical impurity magnetic susceptibilities. For nonmagnetic impurities, the system flows instead to a fixed point with no residual entropy and we find diamagnetic impurity response at low temperatures. We comment on the Schrieffer-Wolff transformation for problems with nonstandard conduction band continua and on the differences which arise when we describe the impurities by either Anderson or Kondo impurity models.
The competition between the Kondo screening and indirect magnetic exchange in systems with two magnetic impurities coupled to a conventional sss-wave superconductor gives rise to a nontrivial ground-state phase diagram. Here, we utilize the density matrix renormalization group (DMRG) method and exploit the non-Abelian spin-SU(2) symmetry to study the phase diagram for two quantum spin-12\frac{1}{2}21​ impurities locally exchange coupled to large one-dimensional chains. The nonlocal interimpurity exchange is treated as an emergent Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We find qualitatively different phase diagrams for impurity spins coupled to sites with odd or even distances ddd on the chain and a partial Kondo-screened spin-doublet phase that extends over the whole range of local exchange couplings JJJ in the limit of weak superconducting pairing strength $\mathrm{\ensuremath{\Delta}}$. Our numerical DMRG studies are complemented by exact diagonalization of small (quantum-box) systems and by perturbative-in-JJJ computations of the ddd- and $\mathrm{\ensuremath{\Delta}}$-dependent RKKY interaction. It is thereby demonstrated that the specific system geometry is essential for our understanding of magnetic impurity interactions in superconducting hosts, and thus for insights into the control of quantum-state properties in nanoparticle systems and topological superconductivity.
The inherent complexity of interacting quantum many-body systems poses an outstanding challenge to both theory and experiment. Especially in the presence of strong electronic correlations, highly interesting and perplexing physical phenomena can occur. \nIn this thesis, we focus on three different examples of strongly correlated electron systems in which different types of defects occur. First, we investigate the Heisenberg-Kitaev model formulated on a triangular lattice. Using a mixture of numerical and analytical techniques we map out the entire phase diagram for the classical and quantum models. We provide an analytical foundation to the intriguing Z2-vortex ground state, in which strong spin-orbit coupling leads to the formation of a lattice of topological point defects. This state was observed previously in classical Monte Carlo simulations. We furthermore propose the iridate Ba3IrTi2O9 to be a prime candidate for the realization of such a state. \nThe second part deals with the physics of a defect in the form of a localized magnetic moment which is embedded into a metallic environment: the Kondo effect. Although this effect has been a cornerstone of condensed matter physics for more than 50 years, its properties in real-space are still not fully understood. What is the Kondo screening cloud---the extended many-body state of entangled conduction electrons? We present numerical results in 1D and 2D for the charge density oscillations created by the impurity. We find that the entire RG flow of the problem is recovered in these oscillations, elucidating the internal structure of the screening cloud. \nFinally, we investigate the competition between the Kondo effect and Majorana physics. Majorana bound states are highly interesting objects which exhibit unusual statistics and could be used as a building block of a topological quantum computer. Recently, signatures of their existence were observed in experiment, and we here examine how Kondo physics (which might play a role in real systems) interact with such Majorana bound states.