# Non-equilibrium Renormalization Group Fixed-Points of the Quantum Clock Chain and the Quantum Potts chain

## TL;DR

This paper develops an exact renormalization group approach for quantum clock and Potts chains, revealing complex fixed points that govern dynamical phases and quantum phase transitions in these models.

## Contribution

It introduces an exact RG recursion relation for the Loschmidt amplitude in quantum clock and Potts chains, analyzing their fixed points and dynamical phase transitions.

## Key findings

- Dynamical quantum phase transitions occur in the clock model for all Q.
- In the Potts model, transitions occur only for Q<4.
- Fixed points are generally complex and influence non-analyticities in the Loschmidt rate.

## Abstract

We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed-points of the renormalization group flow are found to be complex in general. These fixed-points control the dynamical phases of the two models, giving rise to non-analyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the clock model for all $Q$s considered, while in the Potts model, they only occur when $Q$ < 4.