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arXiv:1609.09750 · Physical review. B./Physical review. B

Realistic quantum critical point in one-dimensional two-impurity models

Benedikt Lechtenberg, Fabian Eickhoff, Frithjof B. Anders

Published Jul 7, 2017• 10 citations
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Abstract

We show that the two-impurity Anderson model exhibits an additional quantum critical point at infinitely many specific distances between both impurities for an inversion symmetric one-dimensional dispersion. Unlike the quantum critical point previously established, it is robust against particle-hole or parity symmetry breaking. The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, protected. A finite single-particle tunneling $t$ or an applied uniform gate voltage will drive the system across the quantum critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functions at low temperature, which might be useful for future spintronics devices. A local parity conservation will prevent the spin-spin correlation function from decaying to its equilibrium value after spin manipulations.

DOI
10.1103/physrevb.96.041109

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We show that the two-impurity Anderson model exhibits an additional quantum critical point at infinitely many specific distances between both impurities for an inversion symmetric one-dimensional dispersion. Unlike the quantum critical point previously established, it is robust against particle-hole or parity symmetry breaking. The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, protected. A finite single-particle tunneling ttt or an applied uniform gate voltage will drive the system across the quantum critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functions at low temperature, which might be useful for future spintronics devices. A local parity conservation will prevent the spin-spin correlation function from decaying to its equilibrium value after spin manipulations.
We show that the two impurity Anderson model exhibit an additional quantum critical point at infinitely many specific distances between both impurities for an inversion symmetric 1D dispersion. Unlike the quantum critical point previously established by Jones and Varma, it is robust against particle-hole or parity symmetry breaking. The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, topological protected. A finite single particle tunneling ttt or an applied uniform gate voltages will drive the system across the quantum critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functions at low temperature which might be useful for future spintronics devices. A local parity conservation will prevent the spin-spin correlation function to decay to its equilibrium value after spin-manipulations.
We show that the RKKY interaction in the two-impurity Anderson model comprise two contributions: a ferromagnetic part stemming from the symmetrized hybridization functions and an antiferromagnetic part. We demonstrate that this antiferromagnetic contribution can also be generated by an effective local tunneling term between the two impurities. This tunneling can be analytically calculated for particle-hole symmetric impurities. Replacing the full hybridization functions by the symmetric part and this tunneling term leads to the identical low-temperature fixed point spectrum in the numerical renormalization group. Compensating this tunneling term allows us to restore the Varma-Jones quantum critical point between a strong-coupling phase and a local singlet phase even in the absence of particle-hole symmetry in the hybridization functions. We analytically investigate the spatial frequencies of the effective tunneling term based on the combination of the band dispersion and the shape of the Fermi surface. Numerical renormalization group calculations provide a comparison of the distance-dependent tunneling term and the local spin-spin correlation function. Deviations between the spatial dependency of the full spin-spin correlation function and the textbook RKKY interaction are reported.
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 apply a combination of numerical renormalization group (NRG) and renormalized perturbation theory (RPT) to a model of two quantum dots (impurities) described by two Anderson impurity models hybridized to their respective baths. The dots are coupled via a direct Coulomb interaction U12{U}_{12}U12​ and a spin exchange interaction JJJ. The model has two types of quantum critical points, one at J=JcJ={J}_{c}J=Jc​ to a local singlet state and one at U12=U12c{U}_{12}={U}_{12}^{c}U12​=U12c​ to a locally charge ordered state. The renormalized parameters which determine the low energy behavior are calculated from the NRG. The results confirm the values predicted from the RPT on the approach to the critical points, which can be expressed in terms of a single energy scale T∗{T}^{*}T∗ in all cases. This includes cases without particle-hole symmetry, and cases with asymmetry between the dots, where there is also a transition at J=JcJ={J}_{c}J=Jc​. The results give a comprehensive quantitative picture of the behavior of the model in the low energy Fermi liquid regimes, and some of the conclusions regarding the emergence of a single energy scale may apply to a more general class of quantum critical points, such as those observed in some heavy fermion systems.
We study by Wilson numerical renormalization group the spectral properties of a two-orbital Anderson impurity model in the presence of an exchange splitting that follows either regular or inverted Hund's rules. The phase diagram contains a non-Fermi-liquid fixed point separating a screened phase, where conventional Kondo effect occurs, from an unscreened one, where the exchange splitting takes care of quenching the impurity degrees of freedom. On the Kondo screened side close to this fixed point the impurity density of states shows a narrow Kondo peak on top of a broader resonance. This narrow peak transforms in the unscreened phase into a narrow pseudogap inside the broad resonance. Right at the fixed point only the latter survives. The fixed point is therefore identified by a jump of the density of states at the chemical potential. We also consider the effect of several particle-hole symmetry-breaking terms. We show that particle-hole perturbations that simply shift the orbital energies do not wash out the fixed point, unlike those perturbations that hybridize the two orbitals. Consequently the density-of-state jump at the chemical potential remains finite even away from particle-hole symmetry. In other words, the pseudogap stays pinned at the chemical potential, although it is partially filled in. We also discuss the relevance of these results for lattice models that map onto this Anderson impurity model in the limit of large lattice coordination. Upon approaching the Mott metal-insulator transition, these lattice models necessarily enter a region with a local criticality that reflects the impurity non-Fermi-liquid fixed point. However, unlike the impurity, the lattice can get rid of the single-impurity fixed-point instability by spontaneously developing bulk coherent symmetry-broken phases, which we identify for different lattice models.
The system of several (N)(N)(N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel NNN-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the NNN-impurity S=1∕2S=1∕2S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin S=N∕2S=N∕2S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the NNN-impurity Anderson model to the exact Bethe ansatz results of a S=N∕2S=N∕2S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective S=N∕2S=N∕2S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities NNN. We demonstrate the robustness of the spin N∕2N∕2N∕2 ground state as well as of the associated S=N∕2S=N∕2S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.