Spectroscopy with the tensor renormalization group method

We present a spectroscopy scheme for the lattice field theory by using the tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we cannot only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, the wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Second, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing appropriate matrix elements represented by the tensor network. As a demonstration of the new scheme, we show the spectroscopy of the $\left(1+1\right)d$ Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using Lüscher's formula.