# Extracting the Central Charge of Conformal Field Theory by Central Spin Decoherence

## TL;DR

This paper proposes a novel experimental method to determine the central charge of 2D conformal field theories by measuring the decoherence of a probe spin coupled to the system, validated on the 2D Ising model.

## Contribution

It introduces a new approach using central spin decoherence to extract the central charge of conformal field theories in two-dimensional lattice models.

## Key findings

- Successfully extracted the central charge of the 2D Ising model.
- Demonstrated the method's precision in determining the central charge.
- Provided a practical experimental technique for conformal field theory analysis.

## Abstract

Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly $c$. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.