Branches in the spectral flow of the inhomogeneous transfer matrix for the XXZ spin chain

Branches such as bifurcations or multiple branches are observed in the flow of eigenvalues for the inhomogeneous transfer matrix of the XXZ spin chain. The spectral flow for the XXZ anisotropic parameter Δ is obtained by directly diagonalizing the transfer matrix in the regime: -1 ≤ Δ ≤ 1, where the inhomogeneous transfer matrix is real and symmetric. The branches are considered as novel spectral behaviors, which are not decomposed into simple superpositions of level crossings. We may call them level bifurcations. We also observe that no branches appear in the homogeneous case. The appearance of some of the branches are confirmed through the Bethe ansatz method.