Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

We study the structure of the continuous matrix product operator (cMPO)^[1] for the transverse field Ising model (TFIM). We prove TFIM's cMPO is solvable and has the form $T=rm{e}^{-frac{1}{2}hat{H}_{rm F}}$. $hat{H}_{rm F}$ is a non-local free fermionic Hamiltonian on a ring with circumference $beta$, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of $hat{H}_{rm F}$ is determined analytically. At the critical point, our results verify the state-operator-correspondence^[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO's ground state.