Computing the renormalization group flow of two-dimensional ${\varphi }^4$ theory with tensor networks

Abstract

We study the renormalization group flow of ${\varphi }^4$ theory in two dimensions. Regularizing space into a fine-grained lattice and discretizing the scalar field in a controlled way, we rewrite the partition function of the theory as a tensor network. Combining local truncations and a standard coarse-graining scheme, we obtain the renormalization group flow of the theory as a map in a space of tensors. Aside from qualitative insights, we verify the scaling dimensions at criticality and extrapolate the critical coupling constant $f_{\mathrm{c}}=\lambda /{\mu }^2$ to the continuum to find $f_{\mathrm{c}}^{cont.}=11.0861(90)$, which favorably compares with alternative methods.