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On the lattice, a renormalization group (RG) flow for two-dimensionalpartition functions expressed as a tensor network can be obtained using thetensor network renormalization (TNR) algorithm [G. Evenbly, G. Vidal, Phys.Rev. Lett. 115 (18), 180405 (2015)]. In this work we explain how to extend TNRto field theories in the continuum. First, a short-distance length scale$1/\Lambda$ is introduced in the continuum partition function by smearing thefields. The resulting object is still defined in the continuum but has nofluctuations at distances shorter than $1/\Lambda$. An infinitesimalcoarse-graining step is then generated by the combined action of a $rescaling$operator $L$ and a $disentangling$ operator $K$ that implements a quasi-localfield redefinition. As demonstrated for a free boson in two dimensions,continuous TNR exactly preserves translation and rotation symmetries and cangenerate a proper RG flow. Moreover, from a critical fixed point of this RGflow one can then extract the conformal data of the underlying conformal fieldtheory.

Continuous tensor network renormalization for quantum fields