Entanglement structure of the two-channel Kondo model

Abstract

Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels we confirm field-theory predictions for the boundary entropy difference, ln(g UV /g IR ) = ln(2)/2, between the ultraviolet and infrared limits and the leading ln(x)/x impurity correction to the block entropy. The impurity entanglement, S imp , is shown to scale with the characteristic length ξ 2CK . We show that both the Schmidt gap and the entanglement of the impurity with one of the channels − as measured by the negativity− faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, ν ≈ 2 and β ≈ 0.2. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.Introduction.-The Kondo effect is one of the most intriguing effects in quantum many-body physics. At low temperatures, a localized magnetic impurity is screened by the conduction electrons leading to the formation of many-body entanglement. A generalization of the Kondo model was introduced by Nozières and Blandin [1], where another channel of electrons is also coupled to the impurity. This is the wellknown two-channel Kondo (2CK) model, for which various results were obtained using Bethe ansatz [2-4], conformal field theory [5, 6] (CFT), bosonization [7][8][9] and entanglement of formation [10]. This model is very different from the one-channel Kondo (1CK) model as the two channels compete to screen the spin-1/2 impurity, leading to an "overscreened" residual spin interacting with the electrons [5]. This leads to non-trivial properties including a residual zero-temperature impurity entropy and a logarithmic behavior of magnetic susceptibility and specific heat. However, channel symmetry is crucial; even the smallest asymmetry leads to screening of the impurity by the channel with the stronger coupling [1], and as the channel asymmetry is varied, an impurity quantum phase transition (IQPT) occurs at the symmetric point, corresponding to the 2CK model.Intensive research has been carried out to investigate the thermodynamics and the transport properties of the 2CK model [1-3, 5-9, 11-25]. Experimentally, signatures of the 2CK model have been observed in mesoscopic structures [26][27][28][29]. Still, the real-space entanglement structure and the imprints of the two distinct length scales ξ 2CK ∼ u/T 2CK and ξ * ∼ u/T * with u the spin velocity-implied by the known crossover energy scales T 2CK (2CK temperature) and T * (critical crossover in the channel-asymmetric case) [5, 23] − have not yet been unraveled. A way forward is to use a spi...

Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class

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“…Several numerical efforts have been carried in order to identify this noninteger ground state degeneracy in the subleading term of the EE, and its RG flow. This was accomplished in various models, including Kondo models 20,21 and the boundary Ising chain 22 ; for a review see Ref. [23].…”

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