# Holographic encoding of universality in corner spectra

## TL;DR

This paper demonstrates how corner transfer matrices and tensors can be used to extract universal and topological information from lattice systems, revealing holographic encoding of bulk properties and aiding in phase transition detection.

## Contribution

It introduces a novel method to extract universal data from corner spectra in classical and quantum lattice models, linking corner properties to phase transitions and entanglement spectra.

## Key findings

- Corner spectra reveal universal properties across dimensions
- Corner tensors can identify quantum phase transitions without traditional observables
- Holographic encoding links bulk information to corner properties

## Abstract

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, {\it 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 new 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.