Entanglement Renormalization of the class of Continuous Matrix Product States

Abstract: Continuous tensor network gives a variational ansatz for the ground state of the quantum field theories (QFTs). The notable examples are the continuous matrix product state (cMPS) and the continuous multiscale entanglement renormalization ansatz (cMERA). While cMPS is just adapted to the non-relativistic QFTs, only the Gaussian cMERA is well-understood which we can not use to approximate the ground state of the interacting relativistic QFTs. But instead, cMERA also corresponds to a real-space renormalization group flow in the context of the wave functions. In this letter, we investigate the backward Gaussian cMERA renormalization group flow of the class of cMPS by putting the standard cMPS at the IR scale. At the UV scale, for the bosonic systems in the thermodynamic limit, we achieve the variational class of states that has been proposed recently as the relativistic cMPS (RCMPS) is adapted to the relativistic QFTs without requiring to introduce of any additional IR or UV cut-off. We also extend the RCMPS to fermionic systems and theories on a finite circle.