Universality and quantum criticality in quasiperiodic spin chains

Utkarsh Agrawal,Romain Vasseur,Sarang Gopalakrishnan

doi:10.1038/s41467-020-15760-5

Utkarsh Agrawal, Romain Vasseur + Show 1 more

Open Access

https://doi.org/10.1038/s41467-020-15760-5

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Abstract

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. 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 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.

Highlights

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well
We explain why substitution sequences are attractors, using the illustrative example of the Heisenberg spin chain; we extend our analysis to the Ising and Potts models, and to a toy model of the many-body localization (MBL) transition[32,33]
The simplest case is in the Heisenberg spin chain, where the fixed-point sequence is given by the simple Fibonacci deflation rule; many results were previously known for this sequence, but our work shows that these results apply far more generally than had been appreciated

Summary

Introduction

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. 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. For a wide variety of models of quantum magnetism, the resulting random critical points are of the infinite-randomness type, for which asymptotically exact renormalization-group (RG) methods exist[2,3,4,5] Many properties of these critical points can be understood by central-limiting arguments that rely on the uncorrelated nature of the disorder potential. The central result of this work is that, for a wide class of models, generic quasiperiodic modulations flow under renormalization to discrete substitution sequences These sequences are attractors in the space of quasiperiodic modulations, i.e., fixed points of a functional RG. We explain why (and under what conditions) substitution sequences are attractors, using the illustrative example of the Heisenberg spin chain; we extend our analysis to the Ising and Potts models, and to a toy model of the many-body localization (MBL) transition[32,33]

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