Spin-chain multichannel Kondo model via image impurity boundary condition

Abstract: One of the signature observables for the electronic multichannel Kondo model is the impurity entropy, which was claimed to be found in $J_1$-$J_2$ Heisenberg chain with open boundary condition (OBC) and periodic boundary condition (PBC), respectively for the one-channel and two-channel case. 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 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 spin both match the finite-temperature correction in the electronic multichannel Kondo model. Additionally, we show that the XXZ anisotropy reduces the impurity entropy through a power law of the effective Luttinger liquid parameter.