# Phase transitions in the $\mathbb{Z}_p$ and U(1) clock models

## TL;DR

This paper investigates quantum phase transitions in $ ext{Z}_p$ and U(1) clock models, confirming BKT transitions for $p ext{≥}5$, mapping models in the large $p$ limit, and estimating critical temperatures with high accuracy.

## Contribution

It provides numerical evidence for BKT transitions in $ ext{Z}_p$-clock models, maps the models to the $O(2)$ rotor chain, and estimates critical temperatures using dualities, advancing understanding of phase transitions in discrete symmetry systems.

## Key findings

- BKT transitions occur for $p ext{≥}5$ in $ ext{Z}_p$ models.
- The large $p$ limit maps to the quantum $O(2)$ rotor chain.
- Critical temperatures estimated match Monte Carlo results.

## Abstract

Quantum phase transitions are studied in the non-chiral $p$-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ge 5$, the self-dual $\mathbb{Z}_p$-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets $p=5$ as the lowest $p$ supporting BKT transitions in $\mathbb{Z}_p$-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the $p$-clock model we find that the phase diagram depends strongly on the parity of $p$ and only one BKT transition survives for $p\geq 5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large $p$ limit, to the quantum $O(2)$ rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar $p$-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.