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arXiv:cond-mat/9804119 · Physical review. B, Condensed matter

Room-temperature Kondo effect in atom-surface scattering: Dynamical 1/N approach

Jaime Merino, John Bradley Marston

Published Sep 15, 1998• 30 citations
arXivHTMLPDF

Abstract

The Kondo effect may be observable in some atom-surface scattering experiments, in particular, those involving alkaline-earth atoms. By combining Keldysh techniques with the noncrossing (NCA) approximation to solve the time-dependent Newns-Anderson Hamiltonian in the $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ limit, Shao, Nordlander, and Langreth found an anomalously strong surface-temperature dependence of the outgoing charge state fractions. Here we employ a dynamical $1/N$ expansion with a finite Coulomb interaction $U$ to give a more realistic description of the scattering process. We test the accuracy of the $1/N$ expansion in the $N=1$ case of spinless fermions against the exact independent particle solution. We then compare results obtained in the $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ limit with the NCA at $N=4$ and recover qualitative features already found. Finally, we analyze the realistic situation of Ca atoms with $U=5.8$ eV scattered off Cu(001) surfaces. Although the presence of the doubly ionized ${\mathrm{Ca}}^{2+}$ species can change the absolute scattered ${\mathrm{Ca}}^{+}$ yields, the temperature dependence is qualitatively the same as that found in the $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ limit. One of the main difficulties that experimentalists face in attempting to detect this effect is that the atomic velocity must be kept small enough to limit kinematic smearing of the Fermi surface of the metal.

DOI
10.1103/physrevb.58.6982

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The Kondo effect may be observable in some atom-surface scattering experiments, in particular, those involving alkaline-earth atoms. By combining Keldysh techniques with the noncrossing (NCA) approximation to solve the time-dependent Newns-Anderson Hamiltonian in the U\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}U\ensuremath→​\ensuremath∞ limit, Shao, Nordlander, and Langreth found an anomalously strong surface-temperature dependence of the outgoing charge state fractions. Here we employ a dynamical 1/N1/N1/N expansion with a finite Coulomb interaction UUU to give a more realistic description of the scattering process. We test the accuracy of the 1/N1/N1/N expansion in the N=1N=1N=1 case of spinless fermions against the exact independent particle solution. We then compare results obtained in the U\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}U\ensuremath→​\ensuremath∞ limit with the NCA at N=4N=4N=4 and recover qualitative features already found. Finally, we analyze the realistic situation of Ca atoms with U=5.8U=5.8U=5.8 eV scattered off Cu(001) surfaces. Although the presence of the doubly ionized Ca2+{\mathrm{Ca}}^{2+}Ca2+ species can change the absolute scattered Ca+{\mathrm{Ca}}^{+}Ca+ yields, the temperature dependence is qualitatively the same as that found in the U\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}U\ensuremath→​\ensuremath∞ limit. One of the main difficulties that experimentalists face in attempting to detect this effect is that the atomic velocity must be kept small enough to limit kinematic smearing of the Fermi surface of the metal.
This work is mainly devoted to the description of processes that involve the interaction between an atom and a surface, in which a strong Coulomb repulsion on the atomic site (U)(U)(U) limits the charge exchange to one electron (infinite-UUU limit). In this limit, the Anderson Hamiltonian for a many-fold (N)(N)(N) of states localized on the atomic site can be represented in terms of auxiliary bosons and physical operators in the mixed boson-electron space can be defined. In this work the Hamiltonian is solved by defining appropriate Green's functions for physical operators. Then we solved the equations of motion of these Green's functions, up to a second order in the atom-surface coupling, either for the stationary case or for a real time-dependent problem. We show that our approach reproduces the known exact results in the nondegenerate (N=1)(N=1)(N=1) case, and for $N\ensuremath{\succ}1$ gives excellent agreement with exact calculations and approximations valid for large NNN (the 1∕N1∕N1∕N expansion). Finally, the accurate description of dynamical processes is shown by the comparison with the exact results available for a small four-level system. In this case we also compare with results obtained by using the noncrossing approximation and with the usual spinless model calculation.
A theory for the effect of a strong intra-atomic Coulomb repulsion U on the nonadiabatic transfer of charge between a metallic surface and a moving atomic species is presented. Using slave bosons and a nonequilibrium Green's-function technique, we solve the equations appropriate for the U=\ensuremath{\infty} problem in the case when either the atom-surface hopping matrix element is small, or the number of degenerate atomic states is large. We generalize the earlier treatment of Langreth and Nordlander (LN) to include off-diagonal self-energies and present a general numerical scheme for the exact solution of the Dyson equations. We verify that our scheme gives the correct answer in several limiting cases where an exact solution is known, and give quantitative predictions of when deviations from these limits become important. These limits include (1) the simple master equation limit for low velocities and weak coupling, (2) the generalized master equation of LN for larger velocities and atom-surface coupling, (3) the approach to thermal equilibrium when the time dependence is removed, and (4) the maintenance of local thermal equilibrium when the energy parameters vary sufficiently slowly. From a calculation of the instantaneous (nonequilibrium) spectral function of the level on the scattering atom, we are able to study the rate of formation of the Kondo and mixed valent resonances near the Fermi level. We find a slow formation rate for such resonances relative to that of the broader parts of the spectral density centered near the bare atom level positions.
Using a recently developd many-body theory, we investigate the velocity dependence of charge transfer in atom-surface scattering. We show that under certain conditions, intra-atomic correlation effects can introduce nontrivial sources of nonadiabaticity. For situations where a sharp Kondo resonance is formed near the Fermi energy, the system can exhibit strong nonadiabatic effects even at low velocities. This nonadiabatic behavior is caused by the relatively long-time scale for the formation or survival of the Kondo peak.
Real materials are extremely complicated, and any attempt to understand their bulk properties must begin with the appropriate choice of an idealized model, or Hamiltonian. There are many situations where such models have furnished a decisive understanding of complex quantum phenomena, such as BCS superconductivity and quantum magnetism. There are also cases, for instance the unconventional superconductivity of doped cuprates or heavy-fermion metals, where even the simplest conceivable models are intractable to current theoretical techniques. A promising route toward understanding the physics of such models is to simulate them directly with a highly controlled quantum system. Ultracold neutral atoms, polar molecules, and ions are in many ways ideally suited to this task.\nIn this thesis, we emphasize how the unique features of particular atomic and molecular sys- tems can be leveraged to access interesting physics in experimentally feasible temperature regimes. In chapter 3, we consider prospects for simulation of the Kondo lattice model using alkaline- earth atoms. In particular, we show how groundstate properties--for instance anomalous mass enhancement--can be probed by looking at far-from equilibrium dynamics, which are a standard diagnostic tool in ultracold atom experiments. Chapter 4 describes a realistic implementation of a bosonic version of the Kondo lattice model, and we show how the Kondo interaction qualitatively changes the superfluid to Mott insulator phase transition. Chapters 5, 6, and 7 are unified through an attempt to understand the effects of dissipation in many-body quantum systems. In chapter 5, our goal is mainly to understand the detrimental effects of two-body reactive collisions on dipolar molecules in a 3D optical lattice. Chapter 6 takes a rather different perspective, and shows that this type of loss naturally induces quantum correlations in the steady state of reactive fermionic molecules or alkaline earth atoms. In chapter 7, we develop an exact analytic solution for the non-equilibrium dynamics of long-ranged Ising models with Markovian decoherence. We apply our solution to the benchmarking of dynamics in an existing trapped-ion quantum simulator, which due to its large size and long-ranged, frustrated, interactions is well beyond the reach of a brute force numerical description.
The noncrossing approximation (NCA) is generalized to the multichannel Kondo-spin Hamiltonian with arbitrary anisotropic exchange couplings and an external magnetic field, and applied---in the framework of Matveev's mapping---to the charge fluctuations in a single-electron box at the Coulomb blockade. The temperature dependences of the charge step and the capacitance are calculated for a narrow point contact. At low temperatures and close to the degeneracy point, the capacitance line shape exhibits an approximate scaling with U/T,U/\sqrt{T},U/T​, where U is the deviation in gate voltage from the degeneracy point. This scaling relation is proposed as a sharp experimental diagnostic for the non-Fermi-liquid physics of the system at low temperatures. Both the reliability and shortcomings of the Kondo NCA are discussed in detail. Through comparison with poor-man's scaling, we are able to pinpoint the omission of particle-particle processes as the origin of the NCA flaws. An extended diagrammatic scheme is devised to amend the NCA flaws.
A recent experiment has observed the antiferromagnetic interaction between the ground state ^{1}S_{0} and the metastable state ^{3}P_{0} of ^{171}Yb atoms, which are fermionic. This observation combined with the use of state-dependent optical lattices allows for quantum simulation of the Kondo model. We propose that in this Kondo simulator the anomalous temperature dependence of transport, namely, the Kondo effect, can be detected through quench dynamics triggered by the shift of a trap potential. For this purpose, we improve the numerical efficiency of the minimally entangled typical thermal states (METTSs) algorithm by applying additional Trotter gates. Using the improved METTSs algorithm, we compute the quench dynamics of the one-dimensional Kondo model at finite temperatures quasiexactly. We find that the center-of-mass motion exhibits a logarithmic suppression with a decrease in the temperature, which is a characteristic feature of the Kondo effect.