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arXiv:1701.04121

Observing a Scale Anomaly in Graphene: A Universal Quantum Phase Transition

O. Ovdat, Jinhai Mao, Yuhang Jiang, Eva Y. Andrei, Éric Akkermans

Published Jan 15, 2017• 1 citation
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Abstract

Classical symmetries broken at the quantum level are called anomalies. Since the discovery of this surprising phenomenon, it has become a very active field of research in physics. A remarkable class of anomalies occurs when the continuous scale symmetry (CS) of a scale free quantum system is broken into a discrete scale symmetry (DS) for a critical value of a control parameter. This is an example of a (zero temperature) quantum phase transition (QPT). Examples exist which can be cast in the framework of this anomaly, but surprisingly enough none, as yet, has been presented as a universal quantum phase transition. A physical realization of broken continuous scale symmetry is the quantum inverse square potential which plays a role in various systems and more importantly in Efimov physics. This broken CS also appears for a charged and massless Dirac fermion in an attractive 1/r Coulomb potential -- the so-called atomic collapse. The purpose of this letter is to exhibit the underlying universality of this particular QPT and to present convincing experimental evidence of its existence for a charged and massless fermion in an attractive Coulomb potential as realised in graphene.

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Continuous phase transitions where symmetry is spontaneously broken are ubiquitous in physics and often found between "Landau-compatible" phases where residual symmetries of one phase are a subset of the other. However, continuous "deconfined quantum critical" transitions between Landau-incompatible symmetry-breaking phases are known to exist in certain quantum systems, often with anomalous microscopic symmetries. In this Letter, we investigate the need for such special conditions. We show that Landau-incompatible transitions can be found in a family of well-known classical statistical mechanical models with anomaly-free symmetries, introduced by José et al. [Phys. Rev. B 16, 1217 (1977).PLRBAQ0556-280510.1103/PhysRevB.16.1217]. The models are anisotropic deformations of the classical 2D XY model labeled by a positive integer Q. For a range of temperatures, even Q models exhibit two Landau-incompatible partial symmetry-breaking phases and a direct transition between them for Q≥4. Characteristic features of deconfined quantum criticality, such as enhanced symmetries and melting of charged defects, are easily seen in a classical setting. For odd Q and corresponding temperature ranges, two regions of a single partial symmetry-breaking phase appear, split by a stable "unnecessary critical" line. We discuss experimental systems that realize these transitions and present anomaly-free quantum models that also exhibit similar phase diagrams.

arXiv:2404.19009•Physical Review Letters•Mar 6, 2025•7 citations
Classical symmetries broken at the quantum level are called anomalies. Since the discovery of this surprising phenomenon, it has become a very active field of research in physics.
A remarkable class of anomalies occurs when the continuous scale symmetry (CS) of a scale free quantum system is broken into a discrete scale symmetry (DS) for a critical value of a control parameter. This is an example of a (zero temperature) quantum phase transition (QPT). Examples exist which can be cast in the framework of this anomaly, but surprisingly enough none, as yet, has been presented as a universal quantum phase transition.
A physical realization of broken continuous scale symmetry is the quantum inverse square potential which plays a role in various systems and more importantly in Efimov physics. This broken CS also appears for a charged and massless Dirac fermion in an attractive 1/r Coulomb potential -- the so-called atomic collapse. The purpose of this letter is to exhibit the underlying universality of this particular QPT and to present convincing experimental evidence of its existence for a charged and massless fermion in an attractive Coulomb potential as realised in graphene.
The recently discovered fermion-induced quantum critical points of Dirac materials present a new opportunity to revisit and extend the standard paradigms of quantum criticality. The appearance of a second diverging length scale at a continuous quantum phase transition is one such extension. The authors present a functional renormalization group study of fermion-induced quantum critical points in the broken-symmetry phase, in which the presence of gapless fermions and discrete symmetry breaking provide the natural setup for this phenomenon to occur.
Critical points and their properties are one of the most active topics in condensed-matter physics.The activity in this field can be partially attributed to the abundance of theoretically and experimentally accessible phenomena in the vicinity of continuous phase transitions.In recent years quantum critical points have attracted a particular amount of interest.Although quantum phase transitions are observed only at absolute zero temperature, they also affect certain properties at nonzero temperatures, which can be studied experimentally.One remarkable feature of critical points is that in their vicinity macroscopic properties do not depend on the microscopic details of the sample and the underlying Hamiltonian.Therefore, the various transitions can be assigned to different universality classes, where all representatives of a class show exactly the same properties.Furthermore, the so-called scaling hypothesis implies that close to the critical point the observable quantities follow a universal scaling form.As a result, by properly rescaling all system-dependent parameters it is possible to collapse all data of a particular quantity onto a single curve, given by the respective universal scaling function.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
A unified theory of quantum critical points beyond the conventional Landau-Ginzburg-Wilson paradigm remains unknown. According to Landau cubic criterion, phase transitions should be first-order when cubic terms of order parameters are allowed by symmetry in the Landau-Ginzburg free energy. Here, from renormalization group analysis, we show that second-order quantum phase transitions can occur at such putatively first-order transitions in interacting two-dimensional Dirac semimetals. As such type of Landau-forbidden quantum critical points are induced by gapless fermions, we call them fermion-induced quantum critical points. We further introduce a microscopic model of SU(N) fermions on the honeycomb lattice featuring a transition between Dirac semimetals and Kekule valence bond solids. Remarkably, our large-scale sign-problem-free Majorana quantum Monte Carlo simulations show convincing evidences of a fermion-induced quantum critical points for N = 2, 3, 4, 5 and 6, consistent with the renormalization group analysis. We finally discuss possible experimental realizations of the fermion-induced quantum critical points in graphene and graphene-like materials.Quantum phase transitions are governed by Landau-Ginzburg theory and the exceptions are rare. Here, Li et al. propose a type of Landau-forbidden quantum critical points induced by gapless fermions in two-dimensional Dirac semimetals.
Continuous phase transitions where symmetry is spontaneously broken are ubiquitous in physics and often found between "Landau-compatible" phases where residual symmetries of one phase are a subset of the other. However, continuous "deconfined quantum critical" transitions between Landau-incompatible symmetry-breaking phases are known to exist in certain quantum systems, often with anomalous microscopic symmetries. In this Letter, we investigate the need for such special conditions. We show that Landau-incompatible transitions can be found in a family of well-known classical statistical mechanical models with anomaly-free symmetries, introduced by José et al. [Phys. Rev. B 16, 1217 (1977).PLRBAQ0556-280510.1103/PhysRevB.16.1217]. The models are anisotropic deformations of the classical 2D XY model labeled by a positive integer Q. For a range of temperatures, even Q models exhibit two Landau-incompatible partial symmetry-breaking phases and a direct transition between them for Q≥4. Characteristic features of deconfined quantum criticality, such as enhanced symmetries and melting of charged defects, are easily seen in a classical setting. For odd Q and corresponding temperature ranges, two regions of a single partial symmetry-breaking phase appear, split by a stable "unnecessary critical" line. We discuss experimental systems that realize these transitions and present anomaly-free quantum models that also exhibit similar phase diagrams.