Observing a scale anomaly and a universal quantum phase transition in graphene

O Ovdat,E Akkermans,E Y Andrei,Jinhai Mao,Yuhang Jiang

doi:10.1038/s41467-017-00591-8

Abstract

One of the most interesting predictions resulting from quantum physics, is the violation of classical symmetries, collectively referred to as anomalies. A remarkable class of anomalies occurs when the continuous scale symmetry of a scale-free quantum system is broken into a discrete scale symmetry for a critical value of a control parameter. This is an example of a (zero temperature) quantum phase transition. Such an anomaly takes place for the quantum inverse square potential known to describe ‘Efimov physics’. Broken continuous scale symmetry into discrete scale symmetry also appears for a charged and massless Dirac fermion in an attractive 1/r Coulomb potential. The purpose of this article is to demonstrate the universality of this quantum phase transition and to present convincing experimental evidence of its existence for a charged and massless fermion in an attractive Coulomb potential as realized in graphene.

Highlights

One of the most interesting predictions resulting from quantum physics, is the violation of classical symmetries, collectively referred to as anomalies
A well-studied example is provided by the problem of a particle of mass μ in an attractive inverse square potential[4, 5], which plays a role in various systems[6,7,8,9] and more importantly in Efimov physics[10, 11]
Building on the previous example, it can be anticipated that the problem of a massless Dirac fermion in an attractive Coulomb potential23–25, −Zα/r, is scale invariant (CS) and that the spectrum of resonant quasibound states presents similar features and a corresponding quantum phase transition (QPT)

Summary

Introduction

One of the most interesting predictions resulting from quantum physics, is the violation of classical symmetries, collectively referred to as anomalies. A remarkable class of anomalies occurs when the continuous scale symmetry of a scale-free quantum system is broken into a discrete scale symmetry for a critical value of a control parameter This is an example of a (zero temperature) quantum phase transition. Building on the previous example, it can be anticipated that the problem of a massless Dirac fermion in an attractive Coulomb potential23–25, −Zα/r, is scale invariant (CS) and that the spectrum of resonant quasibound states presents similar features and a corresponding QPT. The degeneracy of the overcritical fractal spectrum is removed and two intertwined geometric ladders of quasi-bound states appear in the s-wave channel All these features are experimentally demonstrated using a charged vacancy in graphene. We relate our findings to Efimov physics as measured in cold atomic gases

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