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arXiv:1408.1757 · Physical Review Letters

Macroscopic Quantum Entanglement of a Kondo Cloud at Finite Temperature

Seung‐Sup B. Lee, Jinhong Park, H.-S. Sim

Published Feb 4, 2015• 40 citations
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Abstract

We propose a variational approach for computing the macroscopic entanglement in a many-body mixed state, based on entanglement witness operators, and compute the entanglement of formation (EoF), a mixed-state generalization of the entanglement entropy, in single- and two-channel Kondo systems at finite temperature. The thermal suppression of the EoF obeys power-law scaling at low temperature. The scaling exponent is halved from the single- to the two-channel system, which is attributed, using a bosonization method, to the non-Fermi liquid behavior of a Majorana fermion, a "half" of a complex fermion, emerging in the two-channel system. Moreover, the EoF characterizes the size and power-law tail of the Kondo screening cloud of the single-channel system.

DOI
10.1103/physrevlett.114.057203

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Quantum entanglement between an impurity and its environment is expected to be central in quantum impurity problems. We develop a method to compute the entanglement in spin-1/2 impurity problems, based on the entanglement negativity and the boundary conformal field theory (BCFT). Using the method, we study the thermal decay of the entanglement in the multichannel Kondo effects. At zero temperature, the entanglement has the maximal value independent of the number of the screening channels. At low temperature, the entanglement exhibits a power-law thermal decay. The power-law exponent equals two times of the scaling dimension of the BCFT boundary operator describing the impurity spin, and it is attributed to the energy-dependent scaling behavior of the entanglement in energy eigenstates. These agree with numerical renormalization group results, unveiling quantum coherence inside the Kondo screening length.
We develop a numerical method to compute the negativity, an entanglement measure for mixed states, between the impurity and the bath in quantum impurity systems at finite temperature. We construct a thermal density matrix by using the numerical renormalization group (NRG), and evaluate the negativity by implementing the NRG approximation that reduces computational cost exponentially. We apply the method to the single-impurity Kondo model and the single-impurity Anderson model. In the Kondo model, the negativity exhibits a power-law scaling at temperature much lower than the Kondo temperature and a sudden death at high temperature. In the Anderson model, the charge fluctuation of the impurity contributes to the negativity even at zero temperature when the on-site Coulomb repulsion of the impurity is finite, while at low temperature the negativity between the impurity spin and the bath exhibits the same power-law scaling behavior as in the Kondo model.
The entanglement entropy in Kondo impurity systems is studied analytically using conformal field theory. From the impurity contribution to the scaling corrections of the entanglement entropy we extract information about the screening cloud profile for general non-Fermi-liquid fixed points. By also considering the finite-temperature corrections to scaling of the von Neumann entropy we point out a direct connection between the long-distance screening cloud profile and thermodynamic observables such as the specific heat.
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We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local moment and strong-coupling ground states, and is related to a change in the Luttinger volume of the conduction bath. Our results offer fresh insight on the nature of the emergent many-particle entanglement within the Kondo cloud, and pave the way for further investigations in more exotic contexts such as the fixed point of the over-screened multi-channel Kondo problem.
Impurities or boundaries often impose nontrivial boundary conditions on a gapless bulk, resulting in distinct boundary universality classes for a given bulk, phase transitions, and non-Fermi liquids in diverse systems. The underlying boundary states however remain largely unexplored. This is related with a fundamental issue how a Kondo cloud spatially forms to screen a magnetic impurity in a metal. Here we predict the quantum-coherent spatial and energy structure of multichannel Kondo clouds, representative boundary states involving competing non-Fermi liquids, by studying quantum entanglement between the impurity and the channels. Entanglement shells of distinct non-Fermi liquids coexist in the structure, depending on the channels. As temperature increases, the shells become suppressed one by one from the outside, and the remaining outermost shell determines the thermal phase of each channel. Detection of the entanglement shells is experimentally feasible. Our findings suggest a guide to studying other boundary states and boundary-bulk entanglement.