# Detecting quantum critical points in the $t-t'$ Fermi-Hubbard model via complex network theory

## TL;DR

This paper introduces a novel approach using complex network theory to detect quantum critical points in the $t-t'$ Hubbard model, linking entanglement patterns to phase transitions in high-temperature superconductors.

## Contribution

It demonstrates that analyzing quantum mutual information graphs as complex networks can identify quantum critical points even on small lattices, overcoming finite size limitations.

## Key findings

- Quantum mutual information graphs reveal critical behavior.
- The method detects QCPs on small lattices.
- Complex network characteristics change near the transition.

## Abstract

A considerable success in phenomenological description of high-T$_{\rm c}$ superconductors has been achieved within the paradigm of Quantum Critical Point (QCP) - a parental state of a variety of exotic phases that is characterized by dense entanglement and absence of well-defined quasiparticles. However, the microscopic origin of the critical regime in real materials remains an open question. On the other hand, there is a popular view that a single-band $t-t'$ Hubbard model is the minimal model to catch the main relevant physics of superconducting compounds. Here, we suggest that emergence of the QCP is tightly connected with entanglement in real space and identify its location on the phase diagram of the hole-doped $t-t'$ Hubbard model. To detect the QCP we study a weighted graph of inter-site quantum mutual information within a four-by-four plaquette that is solved by exact diagonalization. We demonstrate that some quantitative characteristics of such a graph, viewed as a complex network, exhibit peculiar behavior around a certain submanifold in the parametric space of the model. This method allows us to overcome difficulties caused by finite size effects and to identify the transition point even on a small lattice, where long-range asymptotics of correlation functions cannot be accessed.