Holographic encoding of universality in corner spectra

In numerical simulations of classical and quantum lattice systems, 2D corner transfer matrices (CTMs) and 3D corner tensors (CTs) are a useful tool to compute approximate contractions of infinite-size tensor networks. In this paper we show how the numerical CTMs and CTs can be used, additionally, to extract universal information from their spectra. We provide examples of this for classical and quantum systems, in 1D, 2D, and 3D. Our results provide, in particular, practical evidence for a wide variety of models of the correspondence between d-dimensional quantum and (d+1)-dimensional classical spin systems. We show also how corner properties can be used to pinpoint quantum phase transitions, topological or not, without the need for observables. Moreover, for a chiral topological PEPS we show by examples that corner tensors can be used to extract the entanglement spectrum of half a system, with the expected symmetries of the SU(2)k Wess-Zumino-Witten model describing its gapless edge for k=1,2. We also review the theory behind the quantum-classical correspondence for spin systems and provide a numerical scheme for quantum state renormalization in 2D using CTs. Our results show that bulk information of a lattice system is encoded holographically in efficiently-computable properties of its corners.