# Indicators of Conformal Field Theory: entanglement entropy and multiple-point correlators

## TL;DR

This paper demonstrates that entanglement entropy alone is insufficient to confirm conformal field theory descriptions of quantum systems, proposing correlation functions as more reliable indicators, with numerical tests on specific models.

## Contribution

It introduces alternative tests based on two- and four-point correlators to verify CFT descriptions, highlighting limitations of entanglement entropy as an indicator.

## Key findings

- Entanglement entropy can mimic CFT behavior without a true CFT description.
- Correlation functions reveal the absence of CFT structure in certain states.
- Numerical results confirm CFT behavior in well-understood models but not in amplitude-product states.

## Abstract

The entanglement entropy (EE) of quantum systems is often used as a test of low-energy descriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state expressed in the valence-bond basis (where the amplitudes decay as a power law of the bond length and the wave function is the product of all bond amplitudes), we show that even though the EE exhibits the expected CFT behavior, there is no CFT description of this state. We provide numerical tests of the behavior predicted by CFT for the correlation functions in the critical transverse-field Ising chain and the $J$-$Q$ spin chain, where the conformal structure is well understood. That behavior is not reproduced in the amplitude-product state.