Symmetric Fermion Mass Generation as Deconfined Quantum Criticality

Massless $(2+1)\mathrm{D}$ Dirac fermions arise in a variety of systems from graphene to the surfaces of topological insulators, where generating a mass is typically associated with breaking a symmetry. However, with strong interactions, a symmetric gapped phase can arise for multiples of eight Dirac fermions. A continuous quantum phase transition from the massless Dirac phase to this massive phase, which we term symmetric mass generation, is necessarily beyond the Landau paradigm and is hard to describe even at the conceptual level. Nevertheless, such transition has been consistently observed in several numerical studies recently. Here, we propose a theory for the symmetric mass generation transition which is reminiscent of deconfined criticality and involves emergent non-Abelian gauge fields coupled both to Dirac fermions and to critical Higgs bosons. We motivate the theory using an explicit parton construction and discuss predictions for numerics. Additionally, we show that the fermion Green’s function is expected to undergo a zero-to-pole transition across the critical point.

Popular Summary

In graphene—a sheet of carbon atoms laid out in a honeycomb pattern—the electrons move as if they have no mass. These “massless Dirac fermions” appear in a number of contexts and are essential to understanding the optical and electrical properties of many materials. Of particular interest is the stability of these particles when they interact with one another, which controls whether certain materials remain as semimetals or transform into semiconductors. When the interaction grows strong, the massless Dirac fermion can become unstable and a fermion mass can be spontaneously generated. There are several mechanisms of mass generation among Dirac fermions, including the famous Higgs mechanism that involves spontaneous symmetry breaking. Here, we study an altogether novel mechanism for fermion mass generation, which differs from the Higgs mechanism by not breaking any symmetry.

Recent numerical evidence has pointed towards such an exotic scenario, called symmetric mass generation, where interacting Dirac fermions with a particular flavor number can acquire a many-body energy gap without breaking any symmetry through a continuous quantum phase transition. Here, we investigate such a transition between a Dirac semimetal and a featureless Mott insulator. We propose that the transition could be a new type of deconfined quantum criticality, where the Dirac fermions fractionalize into exotic collective excitations (known as partons) only at the critical point, which is in close analogy to the spin-charge separation that was previously proposed for high-temperature superconductors.

Our proposal may shed light on the future numerical and theoretical study of this new type of deconfined quantum critical phenomenon.

Figure 1

Possible scenarios of transitions from the Dirac semimetal (Dirac SM) to featureless gapped phase. (a) Landau paradigm: an intermediate spontaneous symmetry breaking (SSB) phase sandwiched between the Gross-Neveu and the Wilson-Fisher transitions. (b) A direct continuous transition: the symmetric mass generation (SMG) as a deconfined quantum critical point, with emergent gauge field and fractionalized partons. (c) More exotic (and less likely) scenario: an intermediate Bose semimetal (Bose SM) critical phase between the Higgs and confinement transitions.

Figure 2

(a) Catching-up energy scales of the bosonic parton gap ${\mathrm{\Delta }}_B$, the fermionic parton gap ${\mathrm{\Delta }}_f$, and the gauge gluon gap ${\mathrm{\Delta }}_a$ on the confinement side of the SMG. (b) Hierarchical length scales of ${\xi }_B$, ${\xi }_f$, and ${\xi }_a$ near the SMG transition.

Figure 3

The zero-pole transition of the fermion Green’s function. Take the one of the $G(k)$ eigenvalues $(\omega -|\mathit{k}|)/({\omega }^2-|\mathit{k}{|}^2-|\mathit{M}{|}^2)$, for example.

Figure 4

Diagrams that contributes the correction of the interaction vertex. Wavy lines are the gauge boson propagators $D(q{)}_{\mu \nu }^{mn}=16q^{-1}({\delta }_{\mu \nu }-\xi q_{\mu }{q}_{\nu }/q^2){\delta }_{mn}$ at the large-$N_f$ fixed point. The arrowed lines are the fermion propagator $G(k)=1/(k_{\mu }{\gamma }^{\mu })$.