Short notes on the Kondo effect and its potential realization in ultracold atomic gases

The Kondo effect is one of the hallmarks of condensed-matter physics. It describes the peculiar interactions between previously non-interacting Fermions, which are induced by a single spin impurity at a certain temperature. Despite (or maybe because of) its large interest as a benchmark for various theoretical frameworks, it is typically quite hard to find accessible introductions in the literature. Here, I will give a very naive interpretation of the Kondo effect and discuss its possible observation in ultracold atomic gases.

A short history and a naive picture of the Kondo effect

The Kondo model was originally proposed in 1964 to describe the appearance of a resistive minimum at a characteristic temperature $T_K$ in certain alloys (Citation: , ) (). Resistance Minimum in Dilute Magnetic Alloys. Progress of Theoretical Physics, 32(1). 37–49. . In this model, a localized magnetic impurity of spin S = 1/2 is interacting with a metal host of non-interacting Fermions with spin s = 1/2. The interaction is described through an anti-ferromagnetic exchange term $H_I = J\mathbf{s}(0)\cdot \mathbf{S}$ with $J > 0$. Because of the anti-ferrmagnetic interaction, we can identify the groundstate of the system as the singlet state, where impurity and electron of opposite spins.

At high temperatures, this coupling can be well described by the weak scattering of the Fermions on the impurity and the resistivity is decreasing as the temperature is lowered. Jun Kondo discovered in his seminal paper that the perturbative approach fails in the vicinity of the Kondo temperature $T_K$ as higher order perturbative terms diverge in this regime. Ever since, the Kondo model has inspired a large number of different theoretical approaches to explain the properties of this strongly correlated regime (Citation: , ) (). The Kondo Problem to Heavy Fermions. Cambridge University Press. , (Citation: , ) (). The Kondo problem : Fancy mathematical techniques versus simple physical ideas. or (Citation: N.A.) (N.A.). (n.d.). Condensed matter: strongly correlated electrons - IOPscience. Retrieved from . (The review by Nozieres (Citation: , ) (). The Kondo problem : Fancy mathematical techniques versus simple physical ideas. is really remarkable as it was maybe the only paper I found that really explained the phyiscal ideas in a remotely understandable way to me).

A Fermi  sea of s = 1/2 particles is  (orange) is coupled to a localized impurity (blue), whose only degree of freedom is its spin orientation S = 1/2. They interact through an anti-ferromagnetic exchange interaction.  a) For low temperatures, the interaction leads to the formation of a spin singlet between a Fermion and the impurity. The high energy cost of creating an excitation in the region of the impurity blocks it entirely from the dynamics of the Fermi sea. The Fermions can now be treated as a Fermi liquid with the impurity region removed. b) For intermediate temperatures the singlet gets deconfined and virtual excitations in the impurity region are allowed. As they involve the neighboring  Fermions, they can lead to strong interactions between the previously non-interacting particles. These processes are typically only describable by non-perturbative techniques. c) For high temperatures, the interaction with the impurity is  well described by perturbative scattering theory and the system is once again well described by a Fermi liquid.

As sketched in the figure, the following picture has emerged from these works. For low temperatures, $T\ll T_K$, the impurity spin is confined in a many-body singlet ground state of extension $\xi_K = \frac{\hbar v_F}{k_B T_K}$ , called the Kondo screening cloud. As the spin of the impurity is fully screened, the remaining electrons simply scatter on the region and the usual scattering can take place. For intermediate temperatures, the impurity is still bound in the singlet state, but virtual excitation due to Fermions hopping in the impurity region are allowed (see Fig. 1 b). The virtual excitation by one Fermion polarizes the impurity and this polarization is felt by the neighboring Fermions. Therefore, this process gives rise to interactions between the originally non-interacting Fermions. In the strong coupling regime, $T\sim T_K$, these interactions are so strong that they cannot be treated perturbatively. For high temperatures, $T\gg T_K$, the electrons are only weakly scattered by the impurity and the system is again well described by a Fermi liquid (see Fig. 1 c).

Several decades after its first discovery in metals, the Kondo effect has been observed in a multitude of solid-state systems such as quantum dots (Citation: & al., ) , , , , & (). Kondo effect in a single-electron transistor. Nature, 391(6663). 156–159. , (Citation: , ) (). A Tunable Kondo Effect in Quantum Dots. Science, 281(5376). 540–544. , (Citation: & al., ) , , & (). A quantum dot in the limit of strong coupling to reservoirs. Physica B: Condensed Matter, 256-258. 182–185. , carbon nano-tubes (Citation: & al., ) , & (). Kondo physics in carbon nanotubes. Nature, 408(6810). 342–346. , and single molecules (Citation: & al., ) , , , & (). Kondo resonance in a single-molecule transistor. Nature, 417(6890). 725–729. , (Citation: & al., ) , , , , , , , , , & (). Coulomb blockade and the Kondo effect in single-atom transistors. Nature, 417(6890). 722–725. , confirming the presented equilibrium picture. Dynamical and spatial properties, which would be observable on time scales of $t \ll \frac{\hbar}{k_B T_K} \sim$ps, remain inaccessible in these solid-state systems. Such measurement would be however highly desirable, as the Kondo model (and its generalization in form of the Anderson model) is nowadays one of the standard benchmarks for calculations on dynamical and real space properties, where our theoretical understanding is much less complete than in the equilibrium case (Citation: & al., ) , , & (). Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Reviews of Modern Physics, 83(3). 863–883. , (Citation: & al., ) , & (). Buildup of the Kondo effect from real-time effective action for the Anderson impurity model. Physical Review B, 94(4). . As several aspect of these open questions are captured in the properties of the debated Kondo screening cloud (Citation: & al., ) , , & (). How to Directly Measure a Kondo Cloud’s Length. Physical Review Letters, 110(24). , it would be of one of the key objectives for any new experiment with ultracold atomic gases to provide precise experimental studies of the Kondo screening cloud and its dynamics with ultracold quantum gases.

Ultracold atoms as experimental platform for impurity models

The experimental achievement of fabricating gases of ultracold atoms (either Bosons or Fermions) opened the way to study a multitude of fundamental physical phenomena that were otherwise very difficult to realize (Citation: & al., ) , & (). Many-body physics with ultracold gases. Reviews of Modern Physics, 80(3). 885–964. . They have already proven to be a fantastic tool for the study of several simple impurity problems resulting in numerous intriguing quantum effects. In the simplest case, the impurities are static. They form a static disorder, which first leads to the well-known diffusion, but can eventually result in Anderson localization via the quantum interference between different diffusive paths (Citation: , ) (). Absence of Diffusion in Certain Random Lattices. Physical Review, 109(5). 1492–1505. . Even sixty years after its initial prediction fundamental questions remain, especially in the higher dimensional case, and ultracold atoms are now a well established tool to investigate this long-standing problem (Citation: & al., ) , , , , , , , , & (). Direct observation of Anderson localization of matter waves in a controlled disorder. Nature, 453(7197). 891–894. , (Citation: & al., ) , , , , , , , & (). Anderson localization of a non-interacting BoseEinstein condensate. Nature, 453(7197). 895–898. , (Citation: & al., ) , , , , , , , , & (). Three-dimensional localization of ultracold atoms in an optical disordered potential. Nature Physics, 8(5). 398–403. , (Citation: & al., ) , , , , , , , & (). Measurement of the mobility edge for 3D Anderson localization. Nature Physics, 11(7). 554–559. .

In the slightly more general polaron models, the impurity can move, but has no internal degrees of freedom. The moving impurity is then dressed by a cloud of surrounding particles, leading to the formation of the polaron quasiparticles. Ultracold atomic mixtures are a remarkable tool to investigate the properties of such polarons (Citation: & al., ) & (). Ultra-cold polarized Fermi gases. Reports on Progress in Physics, 73(11). 112401. . In such experiments a small number of atoms of one species act as an impurity. They are immersed in a large Fermi sea or Bose-Einstein condensate, formed by a second species. One can then read out the effective mass (Citation: & al., ) , , , , , , & (). Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass. Physical Review Letters, 103(17). and the binding energy of the impurity (Citation: & al., ) , , & (). Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms. Physical Review Letters, 102(23). . With a bosonic bath, these studies have opened the (Citation: & al., ) , , , , & (). Observation of the Phononic Lamb Shift with a Synthetic Vacuum. Physical Review X, 6(4). , (Citation: & al., ) , , , , , , & (). Observation of Attractive and Repulsive Polarons in a Bose-Einstein Condensate. Physical Review Letters, 117(5). , (Citation: & al., ) , , , , & (). Bose Polarons in the Strongly Interacting Regime. Physical Review Letters, 117(5). path towards a synthetic quantum vacuum.

On a possible implementation

Despite this progress, the observation of the Kondo effect remains elusive in the field of ultracold atoms. The impurity should have an internal spin degree of freedom, but remain localized on its site in these systems. Given its fundamental importance, this topic received intense attention recently (Citation: & al., ) , & (). Realizing a Kondo-Correlated State with Ultracold Atoms. Physical Review Letters, 111(21). , (Citation: , ) (). SU(3) Orbital Kondo Effect with Ultracold Atoms. Physical Review Letters, 111(13). , (Citation: & al., ) , , & (). Model for overscreened Kondo effect in ultracold Fermi gas. Physical Review B, 91(16). , (Citation: & al., ) & (). Proposal to directly observe the Kondo effect through enhanced photoinduced scattering of cold fermionic and bosonic atoms. Physical Review A, 93(2). , (Citation: & al., ) , , , , , , , , & (). Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nature Physics, 6(4). 289–295. . While quantum magnetism of Fermi gases has seen tremendous progress in recent years (Citation: & al., ) , , , , , , , , & (). A cold-atom FermiHubbard antiferromagnet. Nature, 545(7655). 462–466. , the Kondo effect seems still out of enormously challenging in existing set-ups (Citation: & al., ) , , , , & (). Localized magnetic moments with tunable spin exchange in a gas of ultracold fermions. arXiv. Retrieved from .

It seems quite natural to implement the Kondo model with a mixture of ultracold atomic gases (Citation: & al., ) , , & (). Model for overscreened Kondo effect in ultracold Fermi gas. Physical Review B, 91(16). . A first species, e.g. $^{40}$K or $^6$Li, would form the Fermi sea. A second species, e.g. $^{23}$Na, would then be tightly trapped by species-selective optical potentials and from the spin impurity. The spin-exchange could then be mediated by spin-changing collisions, very much in the spirit of our work on dynamical gauge fields (Citation: & al., ) , , & (). Schwinger pair production with ultracold atoms. Physics Letters B, 760. 742–746. , (Citation: & al., ) , , , & (). Implementing quantum electrodynamics with ultracold atomic systems. New Journal of Physics, 19(2). 023030. , (Citation: & al.) , & (n.d.). What are dynamical gauge fields ? A simplistic introduction by an AMO experimentalist.. Authorea Inc.. . We will see where the journey goes from here.

Fred Jendrzejewski
Fred Jendrzejewski
Assistant Professor