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Spin-Chain Multichannel Kondo Model via Image Impurity Boundary Condition

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 paper, 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 $\ln\sqrt{3}$ 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.