Two-dimensional $J_1$-$J_2$ clock model: Enhanced symmetries, emergent orders, and Landau-incompatible transitions

TL;DR

This study explores the complex phase diagram of the frustrated $J_1$-$J_2$ clock model on a 2D lattice, revealing emergent orders, unconventional transitions, and the role of frustration in stabilizing novel phases.

Contribution

It uncovers a non-standard emergence of $ ext{Z}_q$ order via relevant operators and maps the full phase diagram using advanced numerical methods.

Findings

Identification of five distinct phases in the frustrated regime.

Discovery of a $ ext{Z}_q$ order emerging from a relevant operator.

Observation of diverse phase transitions including BKT, Ising, and Landau-incompatible types.

Abstract

We present a comprehensive study on the frustrated $J_1$-$J_2$ classical $q$-state clock model with even $q>4$ on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime ($J_1>2J_2$), the model reproduces the standard clock model phenomenology: a low-temperature $\mathbb{Z}_q$-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered (QLRO) phase with emergent $U(1)$ symmetry, and a high-temperature paramagnet. For $J_1<2J_2$, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking $\mathbb{Z}_q\times\mathbb{Z}_2$ symmetry, two $\mathbb{Z}_2$-broken nematic phases (one with and one without QLRO), and an exotic stripe phase with emergent discrete $\mathbb{Z}_q$ spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden $\mathbb{Z}_q$ order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a non-standard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.