A new scale anomaly in Dirac matter

Technical Summary: A New Scale Anomaly in Dirac Matter

Problem Statement
Dirac semimetals, including graphene and three-dimensional (3D) Dirac materials, are modeled at low energies by a massless Dirac Hamiltonian where the Fermi velocity ($v_F$) replaces the speed of light. Classically, the action for these systems is scale-invariant, implying a traceless energy-momentum tensor ($T^\mu_\mu = 0$). While scale anomalies arising from the renormalization of electric charge are well-established in Quantum Electrodynamics (QED) and have been adapted to 3D Dirac semimetals, the specific consequences of the running Fermi velocity in these materials remain to be fully characterized. In 2D graphene, the electric charge is not renormalized due to the specific nature of the Coulomb interaction (mediated by 3D photons), yet the Fermi velocity undergoes renormalization. The central question addressed is whether this velocity renormalization induces a distinct conformal anomaly and what macroscopic, observable consequences arise from it.

Methodology
The authors employ a Quantum Field Theory (QFT) approach to model graphene and Dirac semimetals.

1. Model Construction: The system is described by a QFT action where the Dirac matrices are redefined to incorporate the Fermi velocity ($\gamma^\mu = (\gamma^t, v_F \vec{\gamma})$). The Coulomb interaction in graphene is treated as a "brane-reduced QED," where 2D electrons interact via 3D photons, resulting in a static, non-local interaction with a $1/|k|$ propagator.
2. Renormalization Analysis: Using perturbation theory, the authors analyze the one-loop corrections. They establish that while the photon propagator remains finite (preventing charge renormalization), the electron self-energy diverges logarithmically. This leads to the renormalization of the Fermi velocity and the electron wave function. The beta function for the Fermi velocity, $\beta_v$, is derived as $\beta_v = -\alpha_G/4$, where $\alpha_G$ is the effective fine-structure constant.
3. Anomaly Derivation: The authors derive the conformal anomaly by considering the response of the quantum action to a global scale transformation. By relating the variation of the action to the change in the running Fermi velocity, they derive the trace of the energy-momentum tensor. This derivation follows standard procedures used for charge renormalization anomalies but applies them to the velocity parameter.

Key Contributions and Results
The primary contribution of this work is the identification and characterization of a new conformal anomaly induced specifically by the running of the Fermi velocity.

• The Anomaly Equation: The authors derive that the trace of the energy-momentum tensor is non-zero and proportional to the beta function of the Fermi velocity:
  $$\langle T^\mu_\mu \rangle = \beta_v \langle \bar{\Psi} i \gamma \nabla \Psi \rangle$$
  This relation holds for matrix elements and implies that the scale symmetry is broken at the quantum level due to the velocity renormalization.

• Thermodynamic Consequences:

  • Equation of State: The anomaly modifies the standard conformal equation of state ($E = 2P$) for a 2D Dirac fluid. The modified relation is $(1 + \beta_v/v_F)E = 2P$.
  • Specific Heat: This modification leads to a logarithmic suppression in the electronic specific heat of graphene, particularly noticeable at intermediate to high temperatures.

• Hydrodynamic Consequences:

  • Speed of Sound: The speed of hydrodynamic sound waves, $v_s$, is altered by the anomaly. The authors derive a temperature-dependent sound speed:
    $$v_s(T) = \frac{v_F(T)}{\sqrt{2}} \left( 1 - \frac{\beta_v}{v_F(T)} \right)^{-1/2}$$
    The authors note that since $\beta_v \leq 0$, the sound speed remains below the "relativistic" barrier ($v_F/\sqrt{2}$).
  • Bulk Viscosity: In scale-invariant theories, bulk viscosity ($\zeta$) is typically zero. However, the presence of the anomaly implies a non-vanishing bulk viscosity proportional to the square of the beta function ($\zeta \propto \beta_v^2$). This provides a mechanism for sound absorption in graphene that would otherwise be absent in a strictly scale-invariant system.

Significance and Claims
The paper claims that the renormalization of the Fermi velocity constitutes a distinct source of conformal anomaly in Dirac matter, separate from the charge renormalization anomaly known in 3D QED.

• Observability: The authors argue that these effects are experimentally accessible. The modification of the sound speed can be measured via terahertz photocurrent nanoscopy, while the bulk viscosity can be probed using Brillouin spectroscopy, laser transient grating spectroscopy, or acoustic spectroscopy.
• Material Context: The work highlights that in graphene, the effective fine-structure constant is large ($\alpha_G \sim 2$), meaning the corrections due to the anomaly are not necessarily small perturbations. This contrasts with high-energy QED where the coupling is weak.
• Generalizability: The results are presented as extendable to 3D Dirac and Weyl semimetals. In these 3D systems, both charge and velocity renormalization occur, potentially offering multiple channels for observing quantum anomalies in transport phenomena.

The authors maintain a modest stance, framing the work as a theoretical description of a new anomaly and an exploration of its thermodynamic and hydrodynamic consequences, rather than a proposal for new experimental setups beyond those already established in the literature for measuring sound and viscosity.