One of the signature observables for the electronic multichannel Kondo model is the impurity entropy, which was found in J_{1}-J_{2} Heisenberg chains with the open boundary condition (OBC) and periodic boundary condition (PBC), for the one-channel and two-channel cases, respectively. However, it is not clear how to generalize OBC and PBC in Heisenberg chains to find the multichannel Kondo impurity entropy with more than two channels. In this Letter, we demonstrate that the correct boundary condition for realizing multichannel Kondo physics in Heisenberg chains is the image impurity boundary condition (IIBC) which preserves reflection symmetry and yields the expected impurity entropy, ln[(sqrt[5]+1)/2] for the three-channel case and lnsqrt[3] for the four-channel case. Moreover, the IIBC reduces to OBC for the one-channel case and to PBC for the two-channel case. With IIBC, the finite-size scaling of the impurity entropy and the total impurity spin match the finite-temperature corrections in the electronic multichannel Kondo model. Additionally, we show dependence of the impurity entropy, the total impurity spin, and their scaling behaviors on the XXZ anisotropy Δ (equivalently the Luttinger liquid parameter), revealing impurity physics in a multichannel Luttinger liquid.
One of the signature observables for the electronic multichannel Kondo model is the impurity entropy, which was found in J_{1}-J_{2} Heisenberg chains with the open boundary condition (OBC) and periodic boundary condition (PBC), for the one-channel and two-channel cases, respectively. However, it is not clear how to generalize OBC and PBC in Heisenberg chains to find the multichannel Kondo impurity entropy with more than two channels. In this Letter, we demonstrate that the correct boundary condition for realizing multichannel Kondo physics in Heisenberg chains is the image impurity boundary condition (IIBC) which preserves reflection symmetry and yields the expected impurity entropy, ln[(sqrt[5]+1)/2] for the three-channel case and lnsqrt[3] for the four-channel case. Moreover, the IIBC reduces to OBC for the one-channel case and to PBC for the two-channel case. With IIBC, the finite-size scaling of the impurity entropy and the total impurity spin match the finite-temperature corrections in the electronic multichannel Kondo model. Additionally, we show dependence of the impurity entropy, the total impurity spin, and their scaling behaviors on the XXZ anisotropy Δ (equivalently the Luttinger liquid parameter), revealing impurity physics in a multichannel Luttinger liquid.
A multichannel Kondo model, where two or more equivalent but independent channels of electrons compete to screen a spin-21 impurity, shows overcompensation of the impurity spin, leading to the non-Fermi-liquid behavior in various thermodynamic and transport properties. However, when the channel symmetry is broken, an impurity quantum phase transition can occur at zero temperature. Identification of an order parameter describing the impurity quantum phase transition is very difficult since it is beyond the conventional Landau-Ginzburg-Wilson theory. By employing the natural orbitals renormalization group method, we study both two-channel and three-channel Kondo models, from the perspective of spin correlation between the impurity and electrons in electronic channels. Here we demonstrate that by introducing the spin-correlation ratio as an order parameter we can characterize impurity quantum phase transitions driven by channel asymmetry. In particular, the universal critical exponents $\ensuremath{\beta}$ of the spin-correlation ratio and $\ensuremath{\nu}$ of the correlation length are explicitly determined by finite-size-scaling analysis, namely, $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.0(1)$, and $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.5(1)$ for the two-channel and three-channel Kondo models, respectively.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-S impurity, with the number of channels n exceeding the size of the impurity spin, n>2S, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-21 impurity coupled isotropically to n spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the SU(2)n Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is Simp(T→0)=ln[2cos(π/(n+2))]. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with Simp(T→0)=ln2.
A Coulomb blockaded M-Majorana island coupled to normal metal leads realizes a novel type of Kondo effect where the effective impurity "spin" transforms under the orthogonal group SO(M). The impurity spin stems from the nonlocal topological ground state degeneracy of the island and thus the effect is known as the topological Kondo effect. We introduce a physically motivated N-channel generalization of the topological Kondo model. Starting from the simplest case N=2, we conjecture a stable intermediate coupling fixed point and evaluate the resulting low-temperature impurity entropy. The impurity entropy indicates that an emergent Fibonacci anyon can be realized in the N=2 model. We also map the case N=2, M=4 to the conventional four-channel Kondo model and find the conductance at the intermediate fixed point. By using the perturbative renormalization group, we also analyze the large-N limit, where the fixed point moves to weak coupling. In the isotropic limit, we find an intermediate stable fixed point, which is stable to "exchange" coupling anisotropies, but unstable to channel anisotropy. We evaluate the fixed point impurity entropy and conductance to obtain experimentally observable signatures of our results. In the large-N limit, we evaluate the full crossover function describing the temperature-dependent conductance.
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 a 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(gUV/gIR)=ln(2)/2 between the ultraviolet and infrared limits and the leading ln(x)/x impurity correction to the block entropy. The impurity entanglement Simp is shown to scale with the characteristic length ${\ensuremath{\xi}}_{2\text{CK}}$. 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, $\ensuremath{\nu}\ensuremath{\approx}2$ and $\ensuremath{\beta}\ensuremath{\approx}0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.
We demonstrate that a boundary defect in the single spin-21 quantum XX chain exhibits two-channel Kondo physics. Due to the presence of the defect, the edge spin fractionalizes into two Majorana fermions, out of which one decouples, and one is overscreened by the free fermion in bulk, leading to non-trivial boundary behavior characteristic of the two-channel Kondo model. When the ratio of boundary to bulk coupling exceeds a critical value of 2, a massive boundary-bound mode is exponentially localized near the impurity site for strong impurity coupling. This leads to unusual behavior in physical quantities, such as the g-function not being monotonic. We compute the g−function of the impurity from both thermodynamic and entanglement entropy calculations and show that it takes a non-integer value of 2 just as in the two-channel Kondo problem.
The multichannel Kondo model supports effective anyons on the partially screened impurity, as suggested by its fractional impurity entropy. It was recently demonstrated for the multi-impurity chiral Kondo model, that scattering of an electron through the impurities depends on the anyon's total fusion channel. Here we study the correlation between impurity spins. We argue, based on a combination of conformal field theory, a perturbative limit with a large number of channels k, and the exactly solvable two-channel case, that the interimpurity spin correlation probes the anyon fusion of the pair of correlated impurities. This may allow, using measurement-only topological quantum computing protocols, to braid the multichannel Kondo anyons via consecutive measurements.