# Quantum Phase Transition in the Coupled-Top Model: From Z2 to U(1) Symmetry Breaking

## TL;DR

The paper studies how symmetry breaking in a quantum model leads to different types of quantum phase transitions.

## Contribution

The study reveals a novel critical exponent of 1 for U(1) symmetry breaking, distinct from the usual 1/2 for Z2 symmetry.

## Key findings

- The anisotropic model shows a Z2 symmetry breaking leading to second-order quantum phase transitions.
- U(1) symmetry breaking results in a vanishing energy gap and a new critical exponent of 1.
- The model's rich symmetry structure makes it a key system for studying quantum phase transitions.

## Abstract

We investigate the coupled-top model, which describes two large spins interacting along both x and y directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete Z2 to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete Z2 symmetry, and the symmetry breaking induced by strong coupling drives a quantum phase transition from a disordered paramagnetic phase to an ordered ferromagnetic or antiferromagnetic phase. In particular, the isotropic coupled-top model possesses a continuous U(1) symmetry, whose breaking gives rise to the Goldstone mode. The phase boundary can be well captured by the mean-field approach, characterized by the distinct behaviors of the order parameter. Higher-order quantum effects beyond the mean-field contribution can be achieved by mapping the large spins to bosonic operators via the Holstein–Primakoff transformation. For the anisotropic coupled-top model with Z2 symmetry, the energy gap closes, and both quantum fluctuations and entanglement entropy diverge near the critical point, signaling the onset of second-order quantum phase transitions. Strikingly, when U(1) symmetry is broken, the energy gap vanishes beyond the critical point, yielding a novel critical exponent of 1, rather than 1/2 for Z2 symmetry breaking. The rich symmetry structure of the coupled-top model underpins its role as a paradigmatic model for studying quantum phase transitions and exploring associated physical phenomena.