Direct measurement of a beta function and an indirect check of the Schwinger effect near the boundary in Dirac-Weyl semimetals

The electric field inside typical conductors drops down exponentially with the screening length determined by an intrinsic length scale of the system such as the density of mobile carriers. We show that in a classically conformal system with boundaries, where the intrinsic length scale is absent, the screening of an external electric field is governed by the quantum conformal anomaly associated with the renormalization of the electric charge. The electric field decays algebraically with a fractional power determined by the beta function of the system. We argue that this `anomalous conformal screening effect' is an indirect manifestation of the Schwinger pair production in relativistic field theory. We discuss the experimental feasibility of the proposed phenomenon in Dirac and Weyl semimetals what would allow direct experimental access to the beta function.

The electric field inside typical conductors drops down exponentially with the screening length determined by an intrinsic length scale of the system such as the density of mobile carriers. We show that in a classically conformal system with boundaries, where the intrinsic length scale is absent, the screening of an external electric field is governed by the quantum conformal anomaly associated with the renormalization of the electric charge. The electric field decays algebraically with a fractional power determined by the beta function of the system. We argue that this "anomalous conformal screening effect" is an indirect manifestation of the Schwinger pair production in relativistic field theory. We discuss the experimental feasibility of the proposed phenomenon in Dirac and Weyl semimetals what would allow direct experimental access to the beta function.
Introduction.-There is no electrostatic field in the bulk of an ideal conductor. The screening of the electric field occurs due to the presence of mobile charge carriers which, under the external electric field, redistribute themselves inside the conductor and generate an excess of the electric charge density at its boundary. The redistributed charges create its own electric field which compensates the external field inside the conductor [1].
The static electric field falls down with an exponential law, E(x) ∼ E(0)e −x/λ , as one moves from the conductor boundary at x = 0 towards its bulk, x > 0. The screening length λ determines the width of a layer of the redistributed mobile charges near the boundary.
For example, at high enough temperature T the charge carriers form a classical thermal plasma characterized by the Debye screening length λ D . At low temperature the system enters a quantum regime of a nonrelativistic Fermi gas the Fermi-Thomas screening length λ FT , which is produced by density (rather than thermal) effects. Both length scales are fixed by the dimensionful quantity, the density of the charge carriers n in a solid: where ε 0 is the vacuum permittivity, k B is the Boltzmann constant, and n is the density of the mobile carriers with the electric charge e and mass m and p F = (2π 2 n) 1/3 is the Fermi momentum. The exponential screening of a static electric field is supported by the fact that the conventional conductor possesses a characteristic length scale such as the Thomas-Fermi wavevector [1]. In the examples given above, the width of the surface layer of the displaced charge carriers is given by the screening scales of the system in the bulk (1). In the conformal limit of vanishing density and temperature, the Fermi liquid phenomenology can not be used and these expressions do not apply.
A physical system is conformal (or scale) invariant at the classical level when all its parameters are dimensionless quantities. The question of the electrostatic screening in a classically conformal system is relevant to a wide class of recently discovered Dirac and Weyl semimetals [2][3][4][5][6] whose low-energy properties are described by relativistic massless fermions [7,8]. High interest to these materials is motivated by the fact that they exhibit a plethora of exotic quantum effects restricted, until very recently, to fundamental high-energy systems such as extremely hot quark-gluon plasma [9]. In particular, the Dirac and Weyl semimetals manifest a diversity of quantum anomalies [10] which lead to various anomaly-related transport phenomena [11]. The axial anomaly [12][13][14][15] generates -via the chiral magnetic effect [16] -the experimentally accessible electric current parallel to the axis of a background magnetic field [17]. The mixed axial-gravitational anomaly [18] leads to a positive magneto-thermoelectric conductance for collinear temperature gradients and magnetic fields [19,20], while the conformal anomaly is suggested to generate -via the scale magnetic effect [21] -an anomalous thermoelectric current perpendicular to a temperature gradient and the direction of a background magnetic field [22,23]. In these material systems, as well as in massless QED, the photon polarization function that encodes the screening properties acquires a logarithmic dependence on the renormalization scale [25][26][27][28][29].
In this Letter we show that the screening of the electrostatic field in a classically conformal conductor with a boundary, such as Dirac or Weyl semimetal, may occur via the quantum conformal anomaly. The screening may be understood as a boundary effect, with the external electrostatic field screened in the interior of a spatially bounded semimetal. To get rid of potentially non-conformal contributions from thermal and density effects we will consider a zero-temperature system in a point with a particle-hole symmetry, where chemical po-arXiv:1902.02694v1 [cond-mat.str-el] 7 Feb 2019 tentials of quasiparticles vanishes.
Conformal electromagnetic edge effects.-In a recent publication [30,31], it has been shown that a classical electromagnetic field F µν = ∂ µ A ν − ∂ ν A µ acting on a bounded quantum system of charged particles, generates an electric current at the boundary which, in relativistic notation, J µ = (cρ, J ) is given by: where n µ = (0, n) is the inner normal vector to the edge of the system, and x > 0 is the spatial distance to the boundary along n. The current (2) is associated with the conformal anomaly via the beta function [30,31]: which is relevant to the renormalization of the electric charge e = e(µ) at the energy scale µ. The anomalous effect at the boundary (2) does not depend on the particular choice of the reflective boundary conditions [31].
It is instructive to rewrite Eq. (2) in components. The background magnetic field B induces the electric current which is normal to the axis of the magnetic field and tangential to the edge of the system. The emergence of the quantum current (4) may be interpreted as a result of skipping orbits of particles and antiparticles created by the quantum fluctuations near the edge of the system in the background magnetic field [30,31]. Despite the presence of the divergent 1/x current might seem unlikely, the existence of the "conformal magnetic edge effect" (4) was recently demonstrated in first-principles numerical simulations of scalar quantum electrodynamics [32].
In the presence of a static electric field E, the conformal anomaly (2) leads to the accumulation of the electric charge density at the boundary [30]: Similarly to the scale electromagnetic effects [21], the conformal magnetic (4) and electric (5) edge effects arise at zero temperature and vanishing chemical potentials. Conformal electric edge effect (5) and the Schwinger pair production.-We suggest that the charge accumulation near the boundary (5) may qualitatively be understood as a consequence of the Schwinger pair production near a reflective boundary, as shown schematically in Fig. 1. In the field theory language, the background electric field leads to the quantum production of particleantiparticle pairs. The analogous effect in WSMs, the creation of electron-hole pairs in the presence of a uniform electric field (the Zener effect) has been described in a different context in [33][34][35]. Since the charge carriers are massless, the process of pair production in semimetals proceeds without prohibitive energy barrier contrary to the usual massive QED where the Schwinger pair production is exponentially suppressed by the electron mass.  (4) as an accumulation of electric charge due to the Schwinger pair production near reflective boundary (placed at x = 0).
As a pair is created near the boundary, the background electric field accelerates one of the particles towards the boundary and pushes the anti-particle towards the bulk. When the first particle reaches the reflective boundary, it scatters back into the bulk. Certain part of the reflected particles is subsequently annihilated with particles of an opposite charge that are created in the bulk in later times. Another part of the reflected particles comes back to the reflective boundary which scatters them again into the bulk. As a result of this repetitive processes, the boundary accumulates the electrically charged particles of certain sign which depends on the sign of the product nE in agreement with Eq. (4). An opposite boundary will accumulate the charges of the other sign. The system reaches an equilibrium when the produced particles form a charged layer which partially screens the external electric field thus stabilizing the vacuum in the bulk. This is the mechanism which unifies the Schwinger pair production and the screening of the electrostatic field by the conformal anomaly.
Anomalous conformal screening effect.-As an example, consider first the massless QED with N f species of the Dirac fermions described by the Lagrangian: where D µ = ∂ µ − ieA µ is the covariant derivative expressed via the gauge field A µ = (φ/c, A). Since in our stationary problem the magnetic field is absent, B = 0, we may safely set A = 0 and use the electrostatic potential φ to describe the electric field E = −∇φ.
In order to illustrate the screening effect, let us consider the system (6) with a single boundary at x = 0 in a semi-infinite space x > 0. We apply the background electric field normal to the boundary, E = (E x , 0, 0) and do not restrict the system in the yz plane. Due to the translational invariance in the y and z directions, the problem becomes one-dimensional with all quantities dependent on the x coordinate only. The relevant Maxwell equation, divE = ρ, in one spatial dimension is as follows: where is the vacuum permittivity (we use SI units) related to the fine-structure constant α QED 1/137. In a linear-response regime, the electric charge density is determined by the anomalous contribution only (5): In addition to the anomalous term (9), we may also expect the appearance of the thermodynamic contribution to the local charge density Here we assumed that the system resides in a thermodynamic equilibrium with the effective local chemical potential µ(x) = φ(x) related, via the electrostatic potential φ(x), to the electric field: However, the first term in Eq. (10) is vanishing in the linear order while the second term is strictly zero at zero temperature. Therefore we omit the thermodynamic contribution and concentrate on the conformal part (9) only. The solutions of Eqs. (7), (9) and (11) for the electrostatic potential φ, the electric field E and the charge density ρ are, respectively, as follows: where C and φ 0 are the integration constants and x > 0. The conformal anomaly screens the electrostatic field in the interior of the semimetal as a polynomial 1/x ν with the "conformal screening exponent", determined by the beta function of the electric charge (3). The polynomial screening of the electric field (12b) is natural in the classically conformal regime because the theory has no length parameter to appear in the role of a width of the charge layer at the boundary. The one-loop beta function of the massless QED (6), implies that the conformal screening exponent (13) is proportional to the fine structure constant of QED: α QED = e 2 /(4π 0 c), a small quantity, ν 1.55N f × 10 −3 . Anomalous conformal screening in semimetals.-The low-energy physics of the chiral relativistic quasiparticles in Dirac and Weyl semimetal is well captured by the Lagrangian (we restore the constants and c): which is similar to the multi-species massless QED (6) albeit the appearance of the Fermi velocity v F in the place of the speed of light in a spatial part of Eq. (6). The derivation of the exponent (13) for the semimetal Lagrangian (15) follows the same steps. Due to the anisotropic dispersion relation originated by the Fermi velocity in (15), both the Fermi velocity and the velocity of light (permittivity) are renormalized [24,25,28,29].
The final expression for the conformal screening exponent ν amounts to replace α QED → α WSM : where we also put N f = 1. Since the Fermi velocity of typical WSMs is v F ∼ 10 −3 c and ε ∼ 10, the conformal screening exponent in WSMs is approximately a hundred times bigger than that in the vacuum. Consider now the semimetal in the form of a slab of a finite length L in the x direction (0 x L). We apply the electrostatic potential ∆φ ≡ φ(x = L) − φ(x = 0) to the opposite boundaries x = 0, L of the slab. Without loss of generality we take The charge density induced by the conformal anomaly (5) in between the two boundaries is as follows: The solutions of Eqs. (7), (11), (18) consistent with the boundary conditions (17) on the segment 0 < x < L are: is the Euler incomplete beta function and is the normalization coefficient expressed via the gamma function Γ(x) and the beta function B(a, b) ≡ B (1; a, b).
The static electric field inside the bulk of the semimetal is determined by the conformal screening exponent ν proportional to the beta function β e (16). Near the boundaries, x → 0, L, the bulk fields (19) reproduce the polynomial screening behaviour (12). We illustrate the conformal screening solutions (19) in Fig. 2. FIG. 2. The anomalous conformal screening inside a semimetal slab of the length L, characterized by a large (ν = 1/5) conformal screening exponent (13). The electrostatic potential φ, the electromagnetic field E, and the electric charge density ρ are given in Eq. (19) and shown above in units of potential difference ∆φ applied to the opposite boundaries (17).
Experimental verification of conformal screening.-Our results suggest that the anomalous conformal screening effect may be subjected to a direct experimental test. It is sufficient to apply a potential difference to the opposite boundaries of the semimetal slab and to measure the spatial profile of the electrostatic potential difference φ(x) with respect to one of the boundaries, Fig. 3.   FIG. 3. The experimental setup to probe the anomalous conformal screening. The DC voltage source creates the difference ∆φ = φ+ − φ− in the electrostatic potentials at the opposite boundaries of the semimetal crystal. The behavior of the local potential (19) inside the bulk, V ≡ φ(x), is determined by the conformal screening exponent (16).
The experiment is predicted to reveal the near-boundary polynomial screening behavior (12) associated with the conformal anomaly, Fig. 4. We expect that the experiment may shed light on the polynomial screening mechanism associated with the conformal anomaly and, indirectly, with the Schwinger pair production of the quasiparticles. In addition, the experiment may provide us with direct experimental access to the value of the beta function associated with the renormalization of the fine structure constant of the fermionic quasiparticles.