We show that the Klein bottle entropy [H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017)] for conformal field theories perturbed by a relevant operator is a universal function of the dimensionless coupling constant. The universal scaling of the Klein bottle entropy near criticality provides an efficient approach to extract the scaling dimension of lattice operators via data collapse. As paradigmatic examples, we validate the universal scaling of the Klein bottle entropy for Ising and Z 3 parafermion conformal field theories with various perturbations using numerical simulation with continuous matrix product operator approach.
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 $T=e^{-\frac{1}{2}\hat{H}_F}$. $\hat{H}_F$ 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 $\hat{H}_F$ 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.
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