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arXiv:1801.01132 · Physical review. B./Physical review. B

Exploring the anisotropic Kondo model in and out of equilibrium with alkaline-earth atoms

Márton Kanász-Nagy, Yuto Ashida, Tao Shi, Cătălin Paşcu Moca, Tatsuhiko N. Ikeda, Simon Fölling, J. I. Cirac, Gergely Zaránd, Eugene Demler

Published Apr 26, 2018• 62 citations
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Abstract

We propose a scheme to realize the Kondo model with tunable anisotropy using alkaline-earth atoms in an optical lattice. The new feature of our setup is Floquet engineering of interactions using time-dependent Zeeman shifts, that can be realized either using state-dependent optical Stark shifts or magnetic fields. The properties of the resulting Kondo model strongly depend on the anisotropy of the ferromagnetic interactions. In particular, easy-plane couplings give rise to Kondo singlet formation even though microscopic interactions are all ferromagnetic. We discuss both equilibrium and dynamical properties of the system that can be measured with ultracold atoms, including the impurity spin susceptibility, the impurity spin relaxation rate, as well as the equilibrium and dynamical spin correlations between the impurity and the ferromagnetic bath atoms. We analyze the nonequilibrium time evolution of the system using a variational non-Gaussian approach, which allows us to explore coherent dynamics over both short and long timescales, as set by the bandwidth and the Kondo singlet formation, respectively. In the quench-type experiments, when the Kondo interaction is suddenly switched on, we find that real-time dynamics shows crossovers reminiscent of poor man's renormalization group flow used to describe equilibrium systems. For bare easy-plane ferromagnetic couplings, this allows us to follow the formation of the Kondo screening cloud as the dynamics crosses over from ferromagnetic to antiferromagnetic behavior. On the other side of the phase diagram, our scheme makes it possible to measure quantum corrections to the well-known Korringa law describing the temperature dependence of the impurity spin relaxation rate. Theoretical results discussed in our paper can be measured using currently available experimental techniques.

DOI
10.1103/physrevb.97.155156

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We propose a scheme to realize the Kondo model with tunable anisotropy using alkaline-earth atoms in an optical lattice. The new feature of our setup is Floquet engineering of interactions using time-dependent Zeeman shifts, that can be realized either using state-dependent optical Stark shifts or magnetic fields. The properties of the resulting Kondo model strongly depend on the anisotropy of the ferromagnetic interactions. In particular, easy-plane couplings give rise to Kondo singlet formation even though microscopic interactions are all ferromagnetic. We discuss both equilibrium and dynamical properties of the system that can be measured with ultracold atoms, including the impurity spin susceptibility, the impurity spin relaxation rate, as well as the equilibrium and dynamical spin correlations between the impurity and the ferromagnetic bath atoms. We analyze the nonequilibrium time evolution of the system using a variational non-Gaussian approach, which allows us to explore coherent dynamics over both short and long timescales, as set by the bandwidth and the Kondo singlet formation, respectively. In the quench-type experiments, when the Kondo interaction is suddenly switched on, we find that real-time dynamics shows crossovers reminiscent of poor man's renormalization group flow used to describe equilibrium systems. For bare easy-plane ferromagnetic couplings, this allows us to follow the formation of the Kondo screening cloud as the dynamics crosses over from ferromagnetic to antiferromagnetic behavior. On the other side of the phase diagram, our scheme makes it possible to measure quantum corrections to the well-known Korringa law describing the temperature dependence of the impurity spin relaxation rate. Theoretical results discussed in our paper can be measured using currently available experimental techniques.
We demonstrate that laser excitations can coherently induce a novel Kondo effect in ultracold atoms in optical lattices. Using a model of alkaline-earth fermions with two orbitals, it is shown that the optically coupled two internal states are dynamically entangled to form the Kondo-singlet state, overcoming the heating effect due to the irradiation. Furthermore, a lack of SU(N) symmetry in the optical coupling provides a peculiar feature in the Kondo effect, which results in spin-selective renormalization of effective masses. We also discuss the effects of interorbital exchange interactions, and reveal that they induce novel crossover or reentrant behavior of the Kondo effect owing to control of the coupling anisotropy. The laser-induced Kondo effect is highly controllable by tuning the laser strength and the frequency, and thus offers a versatile platform to study the Kondo physics using ultracold atoms.
Motivated by recent progress on ultracold alkaline-earth atoms towards the goal of simulating Kondo physics, in this work we exactly solve the few-body problem of one and two trapped fermions in one dimension interacting with a localized impurity under tunable spin-exchange interaction. It is found that depending on the sign of the spin-exchange coupling, i.e., ferromagnetic (FM) or antiferromagnetic (AFM), the attractive and repulsive branches can hold different magnetic structures. For the two-fermion case, we demonstrate the Kondo screening effect for the attractive branch with AFM coupling, and show that such screening is absent for the ground state with FM coupling. Moreover, we find a sequence of FM upper branches in the AFM coupling side. These FM states are orthogonal to all other attractive branches and their wave functions feature a full spin-charge separation. The effect of an additional contact interaction and the extension of our results to many particles are also discussed. This work reveals the intriguing physics uniquely associated with the spin-exchange interaction in the few-body point of view, which can be readily explored in the experiment of ultracold alkaline-earth atoms.
Motivated by recent experiments on molecular quantum dots we investigate the relaxation of pure spin states when coupled to metallic leads. Under suitable conditions these systems are well described by a ferromagnetic Kondo model. Using two recently developed theoretical approaches, the time-dependent numerical renormalization group and an extended flow equation method, we calculate the real-time evolution of a Kondo spin into its partially screened steady state. We obtain exact analytical results which agree well with numerical implementations of both methods. Analytical expressions for the steady state magnetization and the dependence of the long-time relaxation on microscopic parameters are established. We find the long-time relaxation process to be much faster in the regime of anisotropic Kondo couplings. The steady state magnetization is found to deviate significantly from its thermal equilibrium value.
We propose a scheme to investigate the interplay between the Kondo-exchange interaction and the quantum spin Hall effect with ultracold fermionic alkaline-earth-metal atoms trapped in two-dimensional optical lattices using ultracold collision and laser-assisted tunneling. In the strong Kondo-coupling regime, although the loop trajectory of the mobile atom disappears, collective dynamics of an atom pair in two clock states can exhibit an unexpected spin-dependent cyclotron orbit in a plaquette, realizing the quantum spin Hall effect of the Kondo singlet. We demonstrate that the collective cyclotron dynamics of the spin-zero Kondo singlet is governed by an effective Harper-Hofstadter model in addition to second-order diagonal tunneling.
We study the nonequilibrium formation of a spin screening cloud that accompanies the quenching of a local magnetic moment immersed in a Fermi sea at zero temperature. Based on high-precision density matrix renormalization-group results for the interacting single-impurity Anderson model, we discuss the real-time evolution after a quantum quench in the impurity-reservoir hybridization using time-evolving block decimation. We report emergent length and time scales in the spatiotemporal structure of nonlocal correlation functions in the spin and the charge density channel. At equilibrium, our data for the correlation functions and the extracted length scales show good agreement with existing results, as do local time-dependent observables at the impurity. In the time-dependent data, we identify a major signal which defines a ``light cone'' moving at the Fermi velocity and a ferromagnetic component in its wake. Inside the light cone we find that the structure of the nonequilibrium correlation functions emerges on two time scales. Initially, the qualitative structure of the correlation functions develops rapidly at the lattice Fermi velocity. Subsequently the spin correlations converge to the equilibrium results on a much larger time scale. This process sets a dynamic energy scale, which we identify to be proportional to the Kondo temperature. Outside the light cone we observe two different power-law decays of the correlation functions in space, with time- and interaction-strength-independent exponents.