We present a continuous tensor-network construction for the states of quantum fields called continuous projected entangled pair state (cPEPS), which enjoys the same spatial and global symmetries of ground states of relativistic field theories. We explicitly show how such a state can approximate and eventually converge to the free field theory vacuum and suggest a regularization-independent way of estimating the convergence via a universal term in the fidelity per site. We also present a detailed bottom-up construction of the cPEPS as the continuum limit of the conventional lattice projected entangled pair state.
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