# Graph-theory treatment of one-dimensional strongly repulsive fermions

## TL;DR

This paper uses spectral graph theory to analyze symmetry properties and predict energy gaps in one-dimensional strongly interacting fermionic systems, aiding experimental and quantum control efforts.

## Contribution

It introduces a novel graph-theoretic approach to characterize symmetries and compute Tan contacts in strongly correlated fermion mixtures, surpassing brute force methods.

## Key findings

- Complete symmetry characterization of 1D fermionic mixtures
- Efficient prediction of energy gaps in complex spin systems
- Guidance for experimental design and adiabatic control

## Abstract

One-dimensional atomic mixtures of fermions can effectively realize spin chains and thus constitute a clean and controllable platform to study quantum magnetism. Such strongly correlated quantum systems are also of sustained interest to quantum simulation and quantum computation due to their computational complexity. In this article, we exploit spectral graph theory to completely characterize the symmetry properties of one-dimensional fermionic mixtures in the strong interaction limit. We also develop a powerful method to obtain the so-called Tan contacts associated with certain symmetry classes. In particular, compared to brute force diagonalization that is already virtually impossible for a moderate number of fermions, our analysis enables us to make unprecedented efficient predictions about the energy gap of complex spin mixtures. Our theoretical results are not only of direct experimental interest, but also provide important guidance for the design of adiabatic control protocols in strongly correlated fermion mixtures.