# A new scale anomaly in Dirac matter

Matteo Baggioli, Maxim Chernodub, Karl Landsteiner, Alessandro Principi,
María A.H. Vozmediano

► To cite this version:

Matteo Baggioli, Maxim Chernodub, Karl Landsteiner, Alessandro Principi, María A.H. Vozmediano. A new scale anomaly in Dirac matter. Annals of Physics, 2025, 483, pp.170267. 10.1016/j.aop.2025.170267. hal-05096988

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<figure><img src="image_1.png" /></figure>
A new scale anomaly in Dirac matter

Matteo Baggioli$^{a,b,c}$, Maxim N. Chernodub$^{d,e}$, Karl Landsteiner$^f$, Alessandro Principi$^g$
, María A. H. Vozmediano$^{h,i}$

$^a$School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

$^b$Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

$^c$Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

$^d$Institut Denis Poisson UMR 7013, Université de Tours, 37200 France

$^e$Department of Physics, West University of Timişoara, Bd. Vasile Pârvan 4, Timişoara 300223, Romania

$^f$Instituto de Física Teórica UAM/CSIC, Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain

$^g$Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13
9PL, United Kingdom

$^h$Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco; 28049 Madrid, Spain.

$^i$Donostia International Physics Center, Paseo Manuel de Lardizábal 4, 20018 San Sebastián, Spain
E-mail: b.matteo@sjtu.edu.cn, maxim.chernodub@univ-tours.fr,
karl.landsteiner@csic.es, alessandro.principi@manchester.ac.uk,
mahvozmediano@gmail.com

ABSTRACT: The dynamics of Dirac semimetals are modeled at low energies by the massless Dirac Hamiltonian with the Fermi velocity replacing the velocity of light. The classical action is scale invariant. In 3D materials, Coulomb interactions induce a conformal anomaly associated to the charge renormalization already known in quantum field theory. In this work, we describe a new conformal anomaly induced by the running of the Fermi velocity that applies to Dirac semimetals in two and three dimensions. The case of graphene is particularly interesting. We analyze the anomaly and explore its thermodynamic and hydrodynamic consequences. The anomaly modifies the propagation speed of hydrodynamic sound waves, alters the thermodynamic equation of state, and induces a non-vanishing bulk viscosity proportional to the beta function of the Fermi velocity.
# Contents

<table><tr><td>1 Introduction</td><td>1</td></tr><tr><td>2 QFT model of graphene and renormalization of the Fermi velocity</td><td>2</td></tr><tr><td>3 Conformal anomaly from the velocity renormalization in graphene</td><td>5</td></tr><tr><td>4 Physical consequences of the anomaly</td><td>8</td></tr><tr><td>4.1 Thermodynamics</td><td>8</td></tr><tr><td>4.2 Speed of sound</td><td>8</td></tr><tr><td>4.3 Bulk viscosity</td><td>9</td></tr><tr><td>5 Discussion and future directions</td><td>10</td></tr></table>

## 1 Introduction

Conformal (or scale) invariance<fnref n="1" /> is a concept that permeates many branches of physics, from critical points in phase transitions to cosmology, relativity, particle physics, and condensed matter [2–4]. In general, any physical system with no dimensionful parameters (massless particles and dimensionless coupling constants) is scale invariant. The conservation of the associated Noether current implies the tracelessness of the energy-momentum tensor of the system $T^{\mu\nu}$,

$$T_{\mu}^{\mu} = \partial^{\mu} J_{\mu}, \qquad (1.1)$$

up to the divergence of a virial current $J^{\mu}$ [1].

Under a Quantum Field Theory (QFT) point of view, strictly renormalizable theories (dimensionless coupling constants) with massless constituent particles are scale invariant. Quantization of these theories typically gives rise to scale-dependent coupling constants that induce conformal anomalies. Despite the fact that there are no massless constituent particles, the conformal anomaly has played a major role in cosmology, string theory, the quark-gluon plasma and, more recently, in modern hydrodynamics.

In the field of condensed matter, scale invariance and scale anomalies have been discussed in quantum critical systems, superfluids, and the unitary Fermi gas in two and three dimensions [5–7]. A recurrent question in this context is whether a quantum anomaly can lead to observable consequences in the macroscopic world.

<footnote n="1">¹Although there are subtle differences between conformal and scale invariance for systems lacking Lorentz invariance [1], we will use the two names indistinctly.</footnote>
The advent of graphene [8, 9], and Dirac semimetals [10] provides a new platform to study scale invariant systems and to address the above mentioned question.

Dirac semimetals have two important features: Their fermionic quasiparticles behave at low energies as massless Dirac particles, and the hydrodynamic Gurzhi regime is experimentally accessible and, in fact, it has been observed in clean samples both in graphene [11–18] and in the three dimensional (3D) materials [19, 20] (for a recent review, see [21]). In the 3D case, the clean, interacting system is well described by massless Quantum Electrodynamics (QED). The chiral and mixed chiral-gravitational anomalies arising from electromagnetic and gravitational interactions were predicted to induce novel transport phenomena in Dirac semimetals [22–24]. The experimental confirmation of these quantum anomalies [25, 26] has been one of the recent highlights of the field.

The conformal anomaly induced by the renormalization of the electric charge in conformal invariant QED(3+1) was explored in [27, 28] and new electromagnetic transport phenomena were predicted to occur in the universe hard to access experimentally. These results were adapted recently to 3D Dirac semimetals where these novel transport phenomena are experimentally accessible [29–31]. The anomaly associated to the charge renormalization is absent in (2+1) dimensions were the electric charge is not renormalized. But in the Lorentz breaking QED-like models associated to Dirac matter, there is another running parameter in the Lagrangian: the Fermi velocity [32–35]. The experimental confirmation of the running of the Fermi velocity in graphene almost twenty years after the theoretical prediction based on QED [36–39] is a hallmark of the success of the hybridization between high and low energies brought by the novel materials.

The question addressed in this work is what kind of scale anomaly arises from the Fermi velocity renormalization and what are the observable effects associated to it. We will show that there is indeed an anomaly, the trace of the stress tensor being proportional to the Fermi velocity beta function. The renormalization of $v_F$ modifies the hydrodynamic speed of sound in graphene and the anomaly affects the correlation functions – and observables – involving $T_\mu^\mu$. In particular, it implies that the bulk viscosity will not be zero in graphene.

## 2 QFT model of graphene and renormalization of the Fermi velocity

Dirac semimetals in two and three space dimensions have Fermi points at or near a two band crossing. The linear dispersion gives rise to an effective low energy continuum Dirac Hamiltonian

$$ \mathcal{H} = v_F \int d^n \mathbf{r} \bar{\psi}(\mathbf{r}) \gamma^i \partial_i \psi(\mathbf{r}), \qquad (2.1) $$

where $v_F$ is the Fermi velocity, a parameter related to the properties of the underlying lattice, and $\gamma^i$ an appropriate set of Dirac matrices<fnref n="2" />. Finally, n labels the number of space dimensions.

<footnote n="2">²The Dirac nature of the Hamiltonian is often due to a pseudospin degree of freedom coming from the lattice and is not linked to the spin of the electrons.</footnote>
This Hamiltonian can be promoted to a full QFT action

$$S_0 = \int dt \, d^n \boldsymbol{r} \, \bar{\Psi} i \gamma^{\mu} \partial_{\mu} \Psi, \qquad (2.2)$$

with the Dirac matrices redefined as

$$\gamma^{\mu} = (\gamma^{t}, v_{F} \gamma). \qquad (2.3)$$

Unless established otherwise, these will be our Dirac matrices from now on.

In (3+1) dimensions, Coulomb interactions are described by standard QED

$$S = \int dt \, d^3 \boldsymbol{r} \, \bar{\Psi} i \gamma^{\mu} \nabla_{\mu} \Psi, \qquad (2.4)$$

where $\nabla_{\mu}$ is the covariant derivative (called Peierls substitution in Condensed Matter)

$$\nabla_{\mu} = \partial_{\mu} + i e A_{\mu}. \qquad (2.5)$$

The gauge field (photon) has a free Lagrangian $L_0 = F_{\mu\nu}F^{\mu\nu}$ where $F_{\mu\nu}$ is the Maxwell electromagnetic tensor: $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$. This is the model for Dirac matter in (3+1) dimensions upon redefining the gamma matrices as in (2.3).

From the structure of the Maxwell kinetic term, we can see that the wavevector ($k$) dependence of the gauge field propagator in QFT is $1/k^2$ in any number of dimensions giving rise to the standard $1/r$ dependence of the Coulomb potential in 3D. The interacting term has the same scale dimension as the free kinetic term and hence the coupling constant is dimensionless. The full model is scale invariant.

The two dimensional QFT case is different. In planar QED the 2D photon field (Fourier transform of $1/k^2$) depends logarithmically on the distance. As a consequence, the coupling constant $e$ has dimension of mass ($\sqrt{M}$) and the theory is not scale invariant (it is called super-renomalizable; in fact, there are no ultraviolet infinities in QED(2+1)). There is no scale anomaly associated to the running coupling constant in 2D.

The case of (2+1) dimensional graphene is more subtle and very interesting. While the free action is given by (2.2) [40–42], the Coulomb interaction is not described by QED(2+1). The reason is that the charges confined to a plane can interact exchanging photons in the three dimensions (see Fig.1 for a cartoon). Hence we face the problem of coupling the 3D photon (with a $1/k^2$ propagator) to 2D electrons. This problem was already addressed in Ref. [32] as follows. The unscreened Coulomb interaction considered in solid state physics is

$$\mathcal{H}_{\text{int}} = \frac{e^2}{2\pi} \int d^2\mathbf{r}_1 \int d^2\mathbf{r}_2 \frac{\bar{\Psi}(\mathbf{r}_1)\Psi(\mathbf{r}_1)\bar{\Psi}(\mathbf{r}_2)\Psi(\mathbf{r}_2)}{|\mathbf{r}_1 - \mathbf{r}_2|}. \qquad (2.6)$$

This static, non-local, non relativistic interaction can be obtained as a one loop effective action from (2.4) by integrating out the vector field with a $1/|k|$ propagator and taking the
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Sketch of the Coulomb interaction in graphene. The electrons confined to the plane can exchange photons in the three spacial dimensions. As a consequence, the interaction between electrons is not captured by QED (2 + 1).</figcaption></figure>

limit $v_f/c \to 0$. This is equivalent to integrating the usual photon propagator over the component transverse to the plane [32] and taking the static limit:

$$ \begin{aligned} \Pi_{\mu\nu} &\equiv \langle T A_{\mu}(t, \mathbf{r}), A_{\nu}(t', \mathbf{r}') \rangle \tag{2.7} \\ &\approx -i\delta_{\mu\nu} \int \frac{d\omega}{(2\pi)^3} \int \frac{dk_z}{(2\pi)} \frac{e^{ik \cdot (\mathbf{r}-\mathbf{r}')}}{-\omega^2 + \mathbf{k}^2 + k_z^2} \\ &= -i\delta_{\mu\nu}\delta(t-t') \frac{1}{2} \int \frac{d^2\mathbf{k}}{(2\pi)^2} \frac{1}{|\mathbf{k}|} \exp[i\mathbf{k} \cdot (\mathbf{r}-\mathbf{r}')]. \end{aligned} $$

With this effective $1/|\mathbf{k}|$ propagator, the Coulomb interaction is marginal and the full action (2.4) is scale invariant. The resulting model was called “brain reduced QED” in [43] (see also [44]).

Comparing the Hamiltonians (2.1) and (2.6), and using a perturbative scheme, it can be seen that the effective two-dimensional coupling constant

$$ \alpha_G = e^2/4\pi v_F \qquad (2.8) $$

is identical to the fine structure constant of QED with the speed of light replaced by the Fermi velocity.

The renormalization of the graphene model has been studied throughly [32, 33, 45, 46]. Of the three one loop graphs shown in Fig. 2, only the electron self-energy diverges and, in the case that we are considering, with a static photon propagator, only the spatial part has a logarithmic divergence. It can be shown to all orders in perturbation theory that the photon
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Primitively divergent Feynman graphs in QED (3+1). Fermion self-energy (a), vertex (b) and photon self-energy (c).</figcaption></figure>

propagator is finite and hence the electric charge is not renormalized. All the renormalization of the system comes from the electron propagator. The renormalization of the Fermi velocity and the electron wave function are obtained from the electron self-energy $\Sigma(\omega, \mathbf{k})$ computed from figure 2 (a) and from the relations:

$$ \Sigma(\omega, \mathbf{k}) = Z_{\psi}(\omega, \mathbf{k}) [\omega\gamma^0 - Z_v(\omega, \mathbf{k})v\gamma \cdot \mathbf{k}] . \qquad (2.9) $$

The beta function of the Fermi velocity turns out to be

$$ \beta_v \equiv \frac{\partial v}{\partial l} = \frac{\partial Z_v}{\partial l} = -\frac{e^2}{16\pi v_F} = -\frac{\alpha_G}{4}, \qquad (2.10) $$

where $l = \log \Lambda$ and $\Lambda$ is the ultraviolet cutoff. The Fermi velocity $v_F$ at the energy $\Lambda$ is then given by the renormalization group equation [32, 46, 47]:

$$ v(\Lambda) = v_{F,0} \left[ 1 - \frac{\alpha_G}{4} \log \frac{\Lambda}{\Lambda_0} \right], \qquad (2.11) $$

where the initial value $v_{F,0}$ can be taken from experiments. More specifically, at the RG normalization point $\Lambda_0 = 125$ MeV, the Fermi velocity equals to [48], $v_F(\Lambda_0) = v_{F,0} = 1.093 \times 10^6$ m/s.

The Fermi velocity renormalization of graphene was experimentally verified [36–39, 49, 50], more than a decade after the first theoretical prediction [32].

### 3 Conformal anomaly from the velocity renormalization in graphene

The graphene action discussed in the previous section

$$ S_0 = \int dt d^2x \bar{\Psi} i\gamma^\mu \nabla_\mu \Psi, \qquad (3.1) $$

is invariant under the global scale transformation:

$$ x \to \lambda x, \quad t \to \lambda t, \quad \Psi \to \lambda^{-1} \Psi. \qquad (3.2) $$
Notice that these transformations do not depend on the Fermi velocity which is included only into the definition of the gamma functions (2.3).

The transformations (3.2) are generated by the dilatation current [51]:

$$j_D^\mu = T^{\mu\nu}x_\nu. \qquad (3.3)$$

On the other hand, the energy-momentum tensor for the action (3.1) is:

$$T^{\mu\nu} = \frac{i}{2} \bar{\Psi} (\gamma^{\mu} \nabla^{\nu} + \gamma^{\nu} \nabla^{\mu}) \Psi - i \eta^{\mu\nu} \bar{\Psi} (\gamma^{t} \partial_{t} + v_{F} \gamma \cdot \nabla) \Psi, \qquad (3.4)$$

where, in the second row, we have made explicit the Fermi velocity presence in the spacial part. The metric tensor

$$\eta^{\mu\nu} = \mathrm{diag}(+1, -1, -1), \qquad (3.5)$$

is a flat spacetime metric which does not depend on the Fermi velocity. Here, the indices run over both time and space coordinates $\mu, \nu = 0, 1, 2$ similarly to the scale transformations (3.2). We notice that the energy density operator is as follows:

$$\epsilon \equiv T^{00} = -v_F \bar{\Psi} i \gamma \cdot \nabla \Psi, \qquad (3.6)$$

where again the Fermi velocity factor is made explicit.

The fermionic energy-momentum tensor (3.4) is given by the variation of the fermionic action $S_0$ with respect to the background metric $g_{\mu\nu}$:

$$T^{\mu\nu}(x) = 2 \frac{\delta S_0[g]}{\delta g_{\mu\nu}(x)}, \qquad S_0[g] = \int dt \, d^2x \sqrt{g} \mathcal{L}_{\psi}(g), \qquad (3.7)$$

with $g = \det(g_{\mu\nu})$, and then calculated in the flat-space limit $g^{\mu\nu} \to \eta^{\mu\nu}$.

Classically, the dilatation current (3.3) has zero divergence because the classical equations of motion imply

$$\partial_{\mu} j_{D}^{\mu} = T_{\alpha}^{\alpha}, \qquad (3.8)$$

and the trace of the energy-momentum tensor (3.4) vanishes at the classical level,

$$T_{\alpha}^{\alpha} = -2 \bar{\psi} i \gamma^{\mu} \nabla_{\mu} \psi \equiv 0, \qquad (3.9)$$

as a consequence of the classical equation of motion of the theory (3.1),

$$\gamma^{\mu} \nabla_{\mu} \psi = 0. \qquad (3.10)$$

Therefore, the classical theory is invariant under the scale transformations and the dilatation current (3.3) vanishes classically: $\partial_{\mu} j_{D}^{\mu} = 0$.
However, scale invariance is broken by quantum fluctuations. Therefore, the quantum expectation value of the right-hand side of Eq. (3.8) is nonzero and, consequently, we expect that on a quantum level the dilatation current (3.3) is no longer conserved. Below, we will prove this statement and find the expression for the trace of the energy-momentum tensor in the presence of a running Fermi velocity.

Here, we present a heuristic derivation, that follows a pretty standard procedure for the well-known type of trace anomaly that is associated with charge renormalization in QED.

Consider a small global scale transformation:

$$ \eta_{\mu\nu} \rightarrow g_{\mu\nu} = e^{2\xi} \eta_{\mu\nu}, \quad \delta g_{\mu\nu} = 2\xi \eta_{\mu\nu}, \qquad (3.11) $$

where the last relation is written for the small scale factor $\xi$ with $|\xi| \ll 1$. Equation (3.7) implies:

$$ S \rightarrow S_{\xi} = S + \xi \int dt \, d^2x \, T_{\mu}^{\mu}(x) + O(\xi^2). \qquad (3.12) $$

Thus, the variation of the quantum action with respect to the local dilatation factor gives us the trace of the energy momentum tensor:

$$ \frac{\partial S_{\psi}}{\partial \xi} = \left\langle \int dt \, d^2x \, T_{\mu}^{\mu} \right\rangle. \qquad (3.13) $$

On the other hand, under the dilatation (3.11), the Fermi velocity changes as follows:

$$ v_F \rightarrow v_F + \xi \beta_v, \qquad (3.14) $$

where $\beta_v$ is the beta function associated with the running of the Fermi velocity (2.10).

By combining Eqs. (3.12) and (3.14), we obtain

$$ \frac{\partial S_{\psi}}{\partial \xi} = \beta_v \int dt \, d^2x \, \langle \bar{\Psi} i \gamma \mathbf{D} \Psi \rangle, \qquad (3.15) $$

that implies – via Eq. (3.13) – the following equation for the conformal anomaly:

$$ \langle T_{\mu}^{\mu} \rangle = \beta_v \langle \bar{\Psi} i \gamma \nabla \Psi \rangle. \qquad (3.16) $$

This is the main result of this work.

Repeating the derivation with a coordinate-dependent factor $\xi = \xi(x)$, using a variation instead of the differential operator in Eq. (3.15), we also get a relation for a matrix element with arbitrary insertions of $T_{\mu}^{\mu}$'s:

$$ \langle T_{\mu}^{\mu}(x_1) \dots T_{\mu}^{\mu}(x_n) \rangle = \beta_v^n \langle \bar{\Psi}(x_1) i \gamma \nabla \Psi(x_1) \dots \bar{\Psi}(x_n) i \gamma \nabla \Psi(x_n) \rangle. \qquad (3.17) $$

This formula implies the validity of the local relation:

$$ T_{\mu}^{\mu}(x) = \beta_v \bar{\Psi}(x) i \gamma \nabla \Psi(x). \qquad (3.18) $$

We notice that this relation is valid only for the matrix elements (expectation values) and it should not be considered as an operator relation.

Exactly the same situation holds in 3+1 dimensional QED, where the trace of the energy-momentum tensor is locally associated with a (gauge) field with a beta-function as a pre-factor.
# 4 Physical consequences of the anomaly

## 4.1 Thermodynamics

The standard anomaly in QED, associated with the renormalization of the electric charge, reveals itself in the classical conformal-violating background field [52]. Similarly, the conformal anomaly will reveal itself at finite density and/or finite temperature, where the right-hand-side of Eq. (3.16) becomes nonzero. The right-hand side of Eq. (3.16) is, up to a proportionality coefficient, the energy density of the Dirac gas (3.6):

$$E \equiv \langle T^{00} \rangle = -v_F \langle \bar{\Psi} i \gamma \nabla \Psi \rangle. \qquad (4.1)$$

Therefore, the conformal anomaly equation (3.16) may also be represented in the form:

$$\langle T_{\mu}^{\mu} \rangle = - \frac{\beta_{v}}{v_{F}} E. \qquad (4.2)$$

Taking into account that the trace of the energy-momentum tensor is related to the pressure $P$ and energy $E$ by, $\langle (T_{\psi})^{\mu}_{\mu} \rangle = E - 2P$, we get:

$$\left(1 + \frac{\beta_v}{v_F}\right) E = 2P. \qquad (4.3)$$

Thus, the presence of the beta function $\beta_v$, associated with the renormalization of the Fermi velocity, modifies the conformal equation of state $E = 2P$. As a consequence, the density of state and all the thermodynamic quantities get also modified. In particular, the electronic specific heat of graphene, a quantity that is now accurately measured [53] will suffer a logarithmic suppression most noticeable at intermediate to high temperatures (this suppression was noticed in the early work [54]).

## 4.2 Speed of sound

The barotropic equation of state of a perfect fluid or a relativistic gas in special relativity reads:

$$\frac{\partial P}{\partial E} = \frac{v_s^2}{c^2}, \qquad (4.4)$$

where $c_s$ is the speed of sound and $c$ is the speed of light (the latter is played by the Fermi velocity in graphene, $c \leftrightarrow v_F$).

Due to the modification of the equation of state by the conformal anomaly (4.3), the speed of hydrodynamical sound gets modified:

$$v_s(T) = \frac{v_F(T)}{\sqrt{2}} \left(1 - \frac{\beta_v}{v_F(T)}\right)^{-1/2} \qquad (4.5)$$

where temperature $T$ plays of the energy scale $\Lambda$. The speed of sound of a free relativistic Dirac fluid in $d$ dimensions obeys the rule $v_s < c/\sqrt{d}$. The renormalization of the Fermi
velocity does not lead to the breaking of the “relativistic sound barrier” in Eq. (4.5) because $\beta_v \le 0$ and

$$v_s(T) \le \frac{v_F(T)}{\sqrt{2}}. \qquad (4.6)$$

The sound velocity can be accurately measured by terahertz photocurrent nanoscopy [55].

## 4.3 Bulk viscosity

The bulk viscosity (or sometimes “second viscosity”) $\zeta$ characterizes how a system of particles deviates from thermodynamic equilibrium in the course of uniform expansion (a nice introduction can be found in Ref. [56]). This parameter, assumed to be very small or zero in many classical fluids, has proven to be relevant in condensed matter systems [57, 58]. A finite bulk viscosity is necessary to eliminate the well-known discrepancy between theory and observation of sound absorption in liquids at very high frequencies [59]. In scale-invariant theories, the bulk viscosity is identically zero since, upon a uniform expansion, an equilibrium distribution function remains in equilibrium (albeit at an adapted temperature) if the dispersion relation exhibits scale invariance. More technically, the Ward identity associated to scale invariance implies that $\zeta$ identically vanishes at any temperature and frequency (see e.g. appendix B in [60] or [61]).

In a rotationally invariant two-dimensional system, the shear $\eta$ and the bulk $\zeta$ viscosities are related to the spatial components $i, j, l, m = 1, 2$ of the correlation function of the energy-momentum tensor $T_{ij}(x)$ via Kubo's formula [62],

$$\begin{aligned} & \eta(\omega) (\delta_{il}\delta_{jm} + \delta_{im}\delta_{jl} - \delta_{ij}\delta_{lm}) + \zeta(\omega)\delta_{ij}\delta_{lm} \\ & = \frac{1}{\omega} \lim_{k \to 0} \int d^2x \int_0^\infty dt e^{i(\omega t - k \mathbf{x})} \langle [T_{ij}(t, \mathbf{x}), T_{lm}(0, \mathbf{0})] \rangle, \end{aligned} \qquad (4.7)$$

where the vanishing of the two-dimensional wavevector $k$ corresponds to the uniform limit with the frequency $\omega$ kept finite.

The static limit $\omega \to 0$ of the correlation function (4.7) with contracted $i, j$ and $l, m$ indices gives us the bulk viscosity $\zeta$ expressed via the spatial trace of the energy-momentum tensor:

$$\zeta = \lim_{\omega \to 0} \frac{1}{4\omega} \int_0^\infty dt \int d^2x e^{i\omega t} \langle [T_{ii}(x), T_{jj}(0)] \rangle, \qquad (4.8)$$

where we also took the limit $k \to 0$ explicitly, assumed the sum over the silent indices, and with $x = (t, \mathbf{x})$. Expression (3.18) implies that the bulk viscosity is finite in the graphene fluid and proportional to the square of the beta function for the Fermi velocity.

In the relaxation time approximation, the bulk viscosity in d spatial dimensions is directly linked to the shear viscosity $\eta$ as [63]

$$\zeta = \#\eta \left( \frac{1}{d} - \frac{v_s^2}{c^2} \right)^2, \qquad (4.9)$$
as first recognized by Weinberg for a photon gas coupled to matter [64] (where # ≈ 15). In the graphene case c = v$_{F}$ .

Eq. (4.9) is reminiscent of an analogous expression derived from AdS-CFT and valid in the strong coupling limit [65],

$$
\zeta > 2\eta \left( \frac{1}{d} - \frac{c_s^2}{c^2} \right). \qquad (4.10)
$$

The bulk viscosity can be measured using several experimental techniques including Bril- louin spectroscopy, Laser transient grating spectroscopy, and acoustic spectroscopy [66].

5 Discussion and future directions

The results on the conformal anomaly induced by the renormalization of the Fermi velocity in graphene, can be easily extended to the three dimensional Dirac and Weyl semimetals [67]. Their low energy dynamics is described by a Lorentz breaking QED(3+1) (2.4) with the inclusion of the Fermi velocity in the spacial Dirac matrices given in (2.3). In this model, the divergence of the photon polarization (diagram in Fig. 2c) gives rise to the renormalization of the electric charge what, in turn, induces the standard QED conformal anomaly [27, 28] mentioned earlier:

$$
\langle T_{\mu}^{\mu}(x) \rangle = \frac{\beta_{e}}{2e} F^{\mu\nu}(x) F_{\mu\nu}(x). \qquad (5.1)
$$

This anomaly has already been studied in the condensed matter domain and it has been shown to give rise to new transport phenomena experimentally accessible [29–31].

But in these 3D materials the Fermi velocity is also renormalized [34, 35, 68, 69], their beta function is also negative and proportional to the coupling constant. The analysis done in this work, in particular eq. (3.16), goes through straightforwardly, improving the chances for the experimental observation of the phenomenon.

In addition to the finite bulk viscosity, the conformal anomaly modifies the equation of state of and affects the thermodynamic and hydrodynamic properties giving rise to other observable quantities experimentally accessible. In particular, the specific heat ..

The corrections discussed in this work are proportional to the beta function of the Fermi velocity. In graphene as well as in the 3D Dirac semimetals the beta function is proportional to the fine structure constant of the materials α$_{v}$ = e $^{2}$/ 4 πϵv$_{F}$ (we have introduced the dielectric constant of the environment, an important parameter which is ϵ = 1 in vacuum but can be tuned in the material experiments. ) It is important to notice that, due to the relation v$_{F}$ /c ∼ 10 − $^{2}$, the fine structure constant both in graphene and in 3D Dirac semimetals is not a small parameter as in QED and the corrections need not be small. The 3D materials are affected by multiple factors (they usually are slightly doped, have various types of disorder, other trivial bands cross the Fermi surface), but graphene can be extremely clean and its coupling constant in vacuum is already large, i.e., α$_{G}$ ∼ 2. The fact that graphene does not show strong coupling behavior is one of the most interesting aspects still remaining in the field despite some attempts of an explanation [70–73]. The experiments associated to the scale
anomaly will provide direct access to the value of $\alpha$ at the scale of the given experimental setup.

## Acknowledgements

M.B. acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01) and the sponsorship from the Yangyang Development Fund. The work of M.N.C. has been partially supported by the French National Agency for Research (ANR) within the project PROCURPHY ANR-23-CE30-0051-02.

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