Field theory approach to the quantum transport in Weyl semimetals

We analyze the structure of the surface states and Fermi arcs of Weyl semimetals as a function of the boundary conditions parametrizing the Hamiltonian self-adjoint extensions of a minimal model with two Weyl points. These boundary conditions determine both the pseudospin polarization of the system on the surface and the shape of the associated Fermi arcs. We analytically derive the expectation values of the density profile of the surface current, we evaluate the anomalous Hall conductivity as a function of temperature and chemical potential, and we discuss the surface current correlation functions and their contribution to the thermal noise. Based on a lattice variant of the model, we numerically study the surface states at zero temperature and we show that their polarization and, consequently, their transport properties, can be varied by suitable Zeeman terms localized on the surface. We also provide an estimate of the bulk conductance of the system based on the Landauer-Büttiker approach. Finally, we analyze the surface anomalous thermal Hall conductivity and we show that the boundary properties lead to a correction of the expected universal thermal Hall conductivity, thus violating the Wiedemann-Franz law.

Figure 1

The Fermi arc in the $xy$ surface is depicted for several values of $\gamma$ and $\mu =0$. The momenta are in units of $p_0$.

Figure 2

Anomalous Hall conductivity ${\sigma }_H$ as a function of the chemical potential ${\mu }_s/v$ and the boundary polarization $\gamma$ for $p_0=\pi /6$. (a) Zero temperature case defined by Eq. (3.12); for ${\mu }_s=0$ or $\gamma =\pi /2$, ${\sigma }_H$ assumes the universal value $e^2{p}_0/\pi h=(1/6)e^2/h$. (b) Conductivity at temperature $k_B{T}_s/v=\pi /3$ calculated from Eq. (3.11); for $\gamma =\pi /2$, ${\sigma }_H$ assumes the universal value $e^2{p}_0/\pi h=(1/6)e^2/h$ also at finite temperature.

Figure 3

Anomalous Hall conductivity ${\sigma }_H$ for a system with $p_0=\pi /6$ and ${\mu }_s=0$, calculated with Eq. (3.11) as a function of the parameter $|cos\gamma |k_B{T}_s/v$. For $T=0$ or $\gamma =\pi /2$, the universal Hall conductivity $e^2{p}_0/\pi h=(1/6)e^2/h$ is retrieved. The behavior of ${\sigma }_H$ is, in general, nonmonotonic in $T$.

Figure 4

Density of the anomalous Hall conductance in Eq. (3.15) for $\gamma =\pi /2$ and $p_0=\pi /5$ as a function of the distance $z$ from the surface and the momentum $p_x$ along the Fermi arc.

Figure 5

Density of the anomalous Hall conductance in Eq. (3.19) for ${\mu }_s=0$ and $p_0=\pi /5$ as a function of the distance $z$ from the surface and the boundary condition parameter $\gamma$.

Figure 6

Properties of the surface states for a surface Zeeman term (5.3) with amplitude $B=0.2v$ for $p_0=\pi /6$. The numerical data are obtained for a system of size $300\times 300\times 90$ from the lattice model in Eq. (5.1) with periodic boundary conditions along $\widehat{x}$ and $\widehat{y}$. In all panels the black lines delimit the domain of the surface states based on the constraint ${\tilde{p}}_z>0$ [see Eq. (2.10)] for $\gamma =1.906$, the red lines depict the shape of the Fermi arc in Eq. (2.13) for the same value of $\gamma$, and the green dots correspond to the projection of the two Weyl points on the surface Brillouin zone. (a) Squared amplitude $A_{1/4}$ of the wave functions of the surface states evaluated for $z\le 20$; only states with $A_{1/4}>0.4$ and $|{\varepsilon }_{\mathrm{s}}|<0.5v$ have been considered. (b) Surface polarization of a subset of the surface states with $A_{1/4}>0.4$. $\gamma$ varies weakly in the surface Brillouin zone with values typically in the range (1.85,1.97) and average $\gamma =1.906$ (see inset). (c) Energy of the surface states evaluated from Eq. (2.11). (d) Energy of the surface states calculated numerically; the analytical description in (c) matches well the numerical results in (d) for energies close to 0.

Figure 7

Values of $\gamma$ determined by the surface polarization (blue points) and the surface density of states (yellow points) as a function of the boundary term (5.3). The error bars correspond to the standard deviation and to the standard error in the fitted parameter, respectively.

Figure 8

Number of surface states $n\left({\varepsilon }_{\mathrm{s}}\right)$ [see Eq. (5.4)] with $A_{1/4}>0.4$ for a lattice system of dimension $300\times 300\times 90$ and surface Zeeman fields $B=0.3v$ (blue circles) and $B=-0.05v$ (brown squares). The numerical data correspond to energy intervals with $d{\varepsilon }_{\mathrm{s}}=0.02v$. The curves illustrate the result of fits with $\gamma$ as the only fitting parameter.

Figure 9

Surface conductivity (in units of $e^2/h$) of a system with $L=30$, and $p_0=\pi /6$, as a function of $W$. The numerical calculation is performed at energy ${\mu }_{\mathrm{bulk}}={10}^{-3}v$ and ${\mu }_{\mathrm{lead}}=0$, with periodic boundary conditions along $\widehat{z}$ and Weyl semimetallic leads. For large system sizes, the Hall conductance approaches the expected universal value $(1/6)e^2/h$. Inset: quantized Hall conductance $G_H$.

Figure 10

Bulk conductance in Eq. (3.19) for $W=30$, ${\mu }_{\mathrm{bulk}}={10}^{-3}$, and $p_0=\pi /6$, as a function of $L$ and in logarithmic scale. The yellow dots are the results obtained from Eq. (4.2), whereas the blue dots denote the numerical results for a scatterer described by the lattice Hamiltonian (5.1) with metallic leads and periodic boundary conditions along $\widehat{x}$ and $\widehat{z}$. For large $L$, the analytical results typically underestimate the numerical data by a factor $\sim 3$.

Figure 11

Lorentz ratio $\mathcal{L}$, normalized by the parameter ${\pi }^2{k}_B^2/3e^2$, calculated for ${\mu }_s=0$ and $p_0=\pi /6$ by a numerical estimation of Eq. (6.14). The curve has been obtained by estimating ${\kappa }_H$ for several values of the temperature (represented by the point colors) and the boundary angle $\gamma$ and plotting them as a function of the parameter $|cos\gamma |k_B{T}_s/v$, consistently with Eqs. (6.19) and (6.20) for ${\mu }_s=0$. The Lorentz ratio at ${\mu }_s=0$ depends nontrivially on the temperature for $\gamma \ne \pi /2$, and the Wiedemann-Franz law $\mathcal{L}={\pi }^2{k}_B^2/3e^2$ is recovered in the limit $cos\gamma \to 0$.

Figure 12

Normalized Lorentz ratio $3e^2\mathcal{L}/{\pi }^2{k}_B^2$, derived from Eq. (6.20), as a function of the parameter $\alpha$ defined in Eq. (6.19).

Figure 13

Lorentz ratio $\mathcal{L}$ as a function of the boundary polarization $\gamma$. The Lorentz ratio is normalized by the parameter ${\pi }^2{k}_B^2/3e^2$ and is calculated from Eq. (6.20) for values of $\alpha$ in Eq. (6.19) determined by ${\mu }_s/v=0.5$, $p_0=\pi /6$ and several values of the temperature. The blue curve at temperature $k_B{T}_s/v=0.01$ shows that, for large values of $\gamma$ (and positive chemical potential), the Wiedemann-Franz law is violated and the Lorentz ratio diverges. The crossing point of all the curves corresponds to $\alpha =0$ in Eq. (6.19).

Figure 14

Lorentz ratio $\mathcal{L}$ as a function of the chemical potential ${\mu }_s/v$. The Lorentz ratio is normalized by the parameter ${\pi }^2{k}_B^2/3e^2$ and is calculated from Eqs. (6.19) and (6.20) for $\gamma =\pi /3$, $p_0=\pi /6$ and several values of the temperature. For $0<\gamma <\pi /2$, the Wiedemann-Franz law is violated in the low-temperature regime for chemical potentials sufficiently negative, as exemplified by the curve at $k_B{T}_s/v=0.01$.