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arXiv:1710.09473 · Physical review. B./Physical review. B

Spin-1 two-impurity Kondo problem on a lattice

Andrew Allerdt, Rok Žitko, Adrian E. Feiguin

Published Jan 5, 2018• 5 citations
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Abstract

We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on a square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling ${J}_{K}$, longitudinal anisotropy $D$, and transverse anisotropy $E$. In the isotropic case, two impurities on opposite (the same) sublattices have a singlet (triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively fourfold-degenerate ground state, i.e., two decoupled impurities. For large enough ${J}_{K}$ the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear nonperturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persist and the two effects coexist: the impurities are underscreened, and the dangling spin-$1/2$ degrees of freedom are responsible for the interimpurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasiclassical correlations.

DOI
10.1103/physrevb.97.045103

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arXiv:2406.05029•Physical review. B./Physical review. B•Aug 9, 2024•1 citation
We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on a square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling JK{J}_{K}JK​, longitudinal anisotropy DDD, and transverse anisotropy EEE. In the isotropic case, two impurities on opposite (the same) sublattices have a singlet (triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively fourfold-degenerate ground state, i.e., two decoupled impurities. For large enough JK{J}_{K}JK​ the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear nonperturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persist and the two effects coexist: the impurities are underscreened, and the dangling spin-1/21/21/2 degrees of freedom are responsible for the interimpurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasiclassical correlations.
We analyze the two-Anderson-impurity problem, in the strong-Coulomb-repulsion limit, by means of variational wave functions whose equations we solve analytically. We found two pairs of doublet states, one odd and one even with respect to the midplane between the impurities. These doublets make a significatively strong use of the hybridization terms of the Hamiltonian and have a much larger Kondo-like correlation energy than a single impurity (the Kondo energy). Furthermore, these doublets combine to form a supersinglet. This supersinglet makes use of the remaining hybridization probability to improve its energy with respect to the doublet energy, but for low-dimensional systems and at the interimpurity distances where the doublets are fully developed, its energy gain is exponentially small. The interaction behind the doublets is a first-order (in the effective Kondo coupling) one; it also generates a parallel alignment of the impurity spins. This ferromagnetic impurity-impurity response is thus generated without resort to the RKKY interaction. The effective range of this first-order Doublet interaction is also much larger than the one of the RKKY interaction.
We study the ground state properties of a single magnetic impurity in the honeycomb lattice. To understand the non-perturbative properties, we use the density matrix renormalization group method where the honeycomb lattice model is mapped onto an effective one-dimensional model by means of the Lanczos recursion technique. We apply this method to three different models: (i) a magnetic adatom on the honeycomb lattice, (ii) a substitutional magnetic impurity in the honeycomb lattice, and (iii) a defect in the honeycomb lattice. Here, we focus only on the particle-hole symmetric cases and find that the ground state of model (i) is qualitatively different from the others: the magnetic impurity spin behaves like an isolated spin in model (i) while it is screened in models (ii) and (iii) by conduction electrons to form a Kondo singlet like state.
The two-channel Kondo (2CK) effect with its exotic ground state properties has remained difficult to realize in physical systems. At low energies, a quantum impurity with orbital degree of freedom, like a proton bound in an interstitial lattice space, comprises a 3-level system with a unique ground state and (at least) doubly degenerate rotational excitations with excitation energy Delta0Delta_0Delta0​. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings JJJ), while the electron spin is conserved. We show by extensive numerical renormalization group (NRG) calculations that without fine-tuning of parameters this system exhibits a 2CK fixed point, due to Kondo correlations in the excited-state doublet whose degeneracy is stabilized by the host lattice parity, while the channel symmetry (electron spin) is guaranteed by time reversal symmetry. We find a pronounced plateau in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between the high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the 2CK ground state value, S(0)=kBlnsqrt2S(0) = k_B lnsqrt{2}S(0)=kB​lnsqrt2. This indicates a downward renormalization of the doublet below the non-interacting ground state, thus realizing the 2CK fixed point, in agreement with earlier conjectures. We mapped out the phase diagram of the model in the J−Delta0J-Delta_0J−Delta0​ plane. The Kondo temperature TKT_KTK​ shows non-monotonic JJJ-dependence, characteristic for 2CK systems.newline indent Beside the two-channel Kondo effect of the model, we also study the single-channel version, which is realized by applying a strong magnetic field to the conduction band electrons so that their degeneracy is lifted and consequently having only one kind of electrons scattering off the impurity. This single-channel case is easier to analyze since the Hilbert space is not as large as that of the 2CK. We equally find a downward renormalization of the excited state energy by the Kondo correlations in the SU(2) doublet. In a wide range of parameter values this stabilizes the single-channel Kondo fixed point and a phase diagram is also mapped out for the model. In the single-channel version a plateau is found in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the single-channel Kondo ground state value, S(0)=kBln1S(0) = k_B ln 1S(0)=kB​ln1.
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.
Lattice density functional theory (LDFT) is used to investigate spin excitations in the single-impurity Anderson model. In this method, the single-particle density matrix ${\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}$ with respect to the lattice sites replaces the wave function as the basic variable of the many-body problem. A recently developed two-level approximation (TLA) to the interaction-energy functional $W[\ensuremath{\gamma}]$ is extended to systems having spin-polarized density distributions and bond orders. This allows us to investigate the effect of external magnetic fields and, in particular, the important singlet-triplet gap $\ensuremath{\Delta}E$, which determines the Kondo temperature. Applications to finite Anderson rings and square lattices show that the gap $\ensuremath{\Delta}E$ as well as other ground-state and excited-state properties are very accurately reproduced. One concludes that the spin-polarized TLA is reliable in all interaction regimes, from weak to strong correlations, for different hybridization strengths and for all considered impurity valence states. In this way the efficiency of LDFT to account for challenging electron-correlation effects is demonstrated.
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.