# Quantum Phase Transition in Fully-Connected Quantum Wajnflasz-Pick Model

## TL;DR

This paper introduces a generalized quantum Wajnflasz-Pick model with infinite-range interactions, revealing first-order quantum phase transitions and anomalous successive transitions, which impact quantum annealing and adiabatic quantum computation performance.

## Contribution

The study constructs a quantum Wajnflasz-Pick model with degenerate two-level systems and analyzes its phase transitions, highlighting differences from the standard quantum Ising model.

## Key findings

- The model exhibits first-order phase transitions unlike the quantum Curie-Weiss model.
- Successive first-order phase transitions can occur under certain conditions.
- Results suggest controllability of quantum annealing performance via system parameters.

## Abstract

We construct a quantum Wajnflasz-Pick model that is a generalized quantum Ising model, and investigate a nature of quantum phase transitions of the model with infinite-range interactions. Quantum phase transition phenomena have drawn attention in the field of quantum computing as well as condensed matter physics, since the phenomena are closely related to the performance of quantum annealing (QA) and adiabatic quantum computation (AQC). We add a quantum driver Hamiltonian to the Hamiltonian of classical Wajnflasz-Pick model. The classical Wajnflasz-Pick model consists of two-level systems as with the usual Ising model. Unlike the usual Ising spin, each of the upper and the lower levels of the system can be degenerate. The states in the upper level and the lower level are referred to as upper states and lower states, respectively. The quantum driver Hamiltonian we introduced causes spin flip between the upper and the lower states and state transitions within each of the upper and the lower states. Numerical analysis showed that the model undergoes first-order phase transitions whereas a corresponding quantum Ising model, quantum Curie-Weiss model, does not undergo first-order phase transitions. In particular, we observed an anomalous phenomenon that the system undergoes successive first-order phase transitions under certain conditions. The obtained results indicate that the performance of QA and AQC by using degenerate two-level systems can be controlled by the parameters in the systems.