# Universality and Quantum Criticality in Quasiperiodic Spin Chains

## TL;DR

This paper reveals that generic quasiperiodic spin chains flow to discrete fixed points under renormalization, enabling exact analysis of their critical behavior and uncovering universal properties across different models.

## Contribution

It demonstrates that generic quasiperiodic modulations in spin chains flow to discrete sequences, allowing for a functional renormalization group approach to analyze their criticality.

## Key findings

- Quasiperiodic spin chains flow to discrete fixed points.
- The approach applies to Heisenberg, Ising, and Potts models.
- Provides insights into quasiperiodic many-body localization transition.

## Abstract

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a \emph{functional} renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.