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arXiv:2303.14331 · Physical review. B./Physical review. B

Natural orbitals renormalization group approach to a spin- 12 impurity interacting with two helical liquids

Ru Zheng, Rong-Qiang He, Zhong-Yi Lu

Published Mar 24, 2023• 2 citations
arXivHTMLPDF

Abstract

Using the natural orbitals renormalization group, we studied the problem of a localized spin-$\frac{1}{2}$ impurity coupled to two helical liquids via the Kondo interaction in a quantum spin Hall insulator, based on the Kane-Mele model defined in a finite zigzag graphene nanoribbon. We investigated the influence of the Kondo couplings with the helical liquids on both the static and dynamic properties of the ground state. The number and distinct spatial structures of the active natural orbitals (ANOs), which play essential roles in constructing the ground-state wave function, were first analyzed. Our numerical results indicate that two ANOs emerge, equal to the number of helical liquids. Specifically, at the coupling symmetry point, both ANOs are fully active with their spatial structures being, respectively, constituted by the different helical liquids. In comparison, when deviating from the symmetry point, only one ANO remains fully active, which is dominantly constructed by the helical liquid with the larger Kondo coupling. Local screening of the impurity, described by the impurity spin polarization and susceptibility, was further studied. It shows that at the coupling symmetry point, the impurity is maximally polarized and the spin susceptibility reaches the maximum. On the contrary, the impurity tends to be screened without polarization when the Kondo couplings deviate well from the symmetry point. The Kondo screening cloud, manifested by the spin correlation between the impurity and the conduction electrons, was finally explored. It is demonstrated that the Kondo cloud is mainly formed by the helical liquid with the larger Kondo coupling to the impurity. On the other hand, the spin-orbital coupling breaks the symmetry in spatial distribution of the spin correlation, leading to anisotropy in the Kondo cloud.

DOI
10.1103/physrevb.107.115149

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Using the natural orbitals renormalization group, we studied the problem of a localized spin-12\frac{1}{2}21​ impurity coupled to two helical liquids via the Kondo interaction in a quantum spin Hall insulator, based on the Kane-Mele model defined in a finite zigzag graphene nanoribbon. We investigated the influence of the Kondo couplings with the helical liquids on both the static and dynamic properties of the ground state. The number and distinct spatial structures of the active natural orbitals (ANOs), which play essential roles in constructing the ground-state wave function, were first analyzed. Our numerical results indicate that two ANOs emerge, equal to the number of helical liquids. Specifically, at the coupling symmetry point, both ANOs are fully active with their spatial structures being, respectively, constituted by the different helical liquids. In comparison, when deviating from the symmetry point, only one ANO remains fully active, which is dominantly constructed by the helical liquid with the larger Kondo coupling. Local screening of the impurity, described by the impurity spin polarization and susceptibility, was further studied. It shows that at the coupling symmetry point, the impurity is maximally polarized and the spin susceptibility reaches the maximum. On the contrary, the impurity tends to be screened without polarization when the Kondo couplings deviate well from the symmetry point. The Kondo screening cloud, manifested by the spin correlation between the impurity and the conduction electrons, was finally explored. It is demonstrated that the Kondo cloud is mainly formed by the helical liquid with the larger Kondo coupling to the impurity. On the other hand, the spin-orbital coupling breaks the symmetry in spatial distribution of the spin correlation, leading to anisotropy in the Kondo cloud.
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.
The problem of two magnetic impurities in a normal metal exposes the two opposite tendencies in the formation of a singlet ground state, driven respectively by the single-ion Kondo effect with conduction electrons to screen impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction between the two impurities to directly form impurity spin singlet. However, whether the competition between these two tendencies can lead to a quantum critical point has been debated over more than two decades. Here, we study this problem by applying the newly proposed natural orbitals renormalization group method to a lattice version of the two-impurity Kondo model with a direct exchange KKK between the two impurity spins. The method allows for unbiased access to the ground state wave functions and low-lying excitations for sufficiently large system sizes. We demonstrate the existence of a quantum critical point, characterized by the power-law divergence of impurity staggered susceptibility with critical exponent $\ensuremath{\gamma}=0.60(1)$, on the antiferromagnetic side of KKK when the interimpurity distance RRR is even lattice spacing, while a crossover behavior is recovered when RRR is odd lattice spacing. These results have ultimately resolved the long-standing discrepancy between the numerical renormalization group and quantum Monte Carlo studies, confirming a link of this two-impurity Kondo critical point to a hidden particle-hole symmetry predicted by the local Fermi liquid theory.
We investigate the entanglement properties of a Kondo system undergoing a transition to a state with a quenched magnetic impurity, using the density matrix renormalization group method. We focus on a two-channel spin-1 Kondo impurity with single-ion anisotropy, where a quantum phase transition occurs between two topologically distinct local Fermi liquids. In the fully screened Kondo phase, realized at lower anisotropies, the entangled region surrounding the magnetic impurity mimics the Kondo screening cloud, although its length does not follow the conventional behavior. In contrast, beyond the transition, the system enters a non-Landau Fermi liquid phase with a markedly different entanglement structure: As the impurity is quenched and disentangled from the rest of the system due to the breakdown of the Kondo effect, the two conduction channels---coupled only through the impurity---develop a significant degree of entanglement with one another. Our findings demonstrate that a quenched magnetic impurity can passively and efficiently mediate entanglement between spatially separated conduction bands.
We introduce a block Lanczos (BL) recursive technique to construct quasi-one-dimensional models, suitable for density-matrix renormalization group (DMRG) calculations, from single- as well as multiple-impurity Anderson models in any spatial dimensions. This new scheme, named BL-DMRG method, allows us to calculate not only local but also spatially dependent static and dynamical quantities of the ground state for general Anderson impurity models without losing elaborate geometrical information of the lattice. We show that the BL-DMRG method can be easily extended to treat a multiorbital Anderson impurity model where not only inter- and intraorbital Coulomb interactions but also Hund's coupling and pair hopping interactions are included. We also show that the symmetry adapted BL bases can be utilized, when it is appropriate, to reduce the computational cost. As a demonstration, we apply the BL-DMRG method to three different models for graphene with a structural defect and with a single hydrogen or fluorine absorbed, where a single Anderson impurity is coupled to conduction electrons in the honeycomb lattice. These models include (i) a single adatom on the honeycomb lattice, (ii) a substitutional impurity in the honeycomb lattice, and (iii) an effective model for a single carbon vacancy in graphene. Our analysis of the local dynamical magnetic susceptibility and the local density of states at the impurity site reveals that, for the particle-hole symmetric case at half-filling of electron density, the ground state of model (i) behaves as an isolated magnetic impurity with no Kondo screening, while the ground state of the other two models forms a spin-singlet state where the impurity moment is screened by the conduction electrons. We also calculate the real-space dependence of the spin-spin correlation functions between the impurity site and the conduction sites for these three models. Our results clearly show that, reflecting the presence or absence of unscreened magnetic moment at the impurity site, the spin-spin correlation functions decay as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}3}$, differently from the noninteracting limit ($\ensuremath{\propto}$ ${r}^{\ensuremath{-}2}$), for model (i) and as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}4}$, exactly the same as the noninteracting limit, for models (ii) and (iii) in the asymptotic rrr, where rrr is the distance between the impurity site and the conduction site. Finally, based on our results, we shed light on recent experiments on graphene where the formation of local magnetic moments as well as the Kondo-like behavior have been observed.
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.
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