Quantum kinetic theory of the chiral anomaly

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Quantum kinetic theory of the chiral anomaly

Akihiko Sekine, Dimitrie Culcer, and Allan H. MacDonald

Phys. Rev. B 96, 235134 – Published 21 December 2017

Abstract

We present a general quantum kinetic theory of low-field magnetotransport in weakly disordered crystals that accounts fully for the interplay between electric-field-induced interband coherence, Bloch-state scattering, and an external magnetic field. The quantum kinetic equation we derive for the Bloch-state density matrix naturally incorporates the momentum-space Berry phase effects whose influence on Bloch-state wave-packet dynamics is normally incorporated into transport theory in an ad hoc manner. The Berry phase correction to the momentum-space density of states in the presence of an external magnetic field implied by semiclassical wave-packet dynamics is captured by our theory as an intrinsic density-matrix response to a magnetic field. We propose a simple and general procedure for expanding the linear response of the Bloch-state density matrix to an electric field in powers of magnetic field. As an illustration, we apply our theory to magnetotransport in Weyl semimetals. We show that the chiral anomaly (positive magnetoconductivity quadratic in magnetic field) that appears when separate Fermi surface pockets surround distinct Weyl points survives only when intervalley scattering is very weak compared to intravalley scattering.

Received 16 June 2017
Revised 5 December 2017

DOI:https://doi.org/10.1103/PhysRevB.96.235134

Physics Subject Headings (PhySH)

Magnetotransport Quantum transport Topological phases of matter

Semiconductors

Methods in transport

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Akihiko Sekine¹, Dimitrie Culcer², and Allan H. MacDonald¹

¹Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
²School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies, UNSW Node, The University of New South Wales, Sydney 2052, Australia

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Vol. 96, Iss. 23 — 15 December 2017

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Images

Figure 1
(a) Pumping of valley population in a Weyl semimetal by parallel electric and magnetic fields oriented along the $z$ direction in the absence of disorder. When the electronic structure of a Weyl semimetal is approximated by Dirac cones with masses that depend on $k_{z}$ , a magnetic field induces an anomalous $(N = 0)$ Landau level branch that has only one sign of velocity $v_{z}$ in a given valley. It follows that in each valley the density of states is increased for states with one sign of velocity and decreased for states with the other sign of velocity, and that the total current summed over a valley is already nonzero in equilibrium. When an electric field drives states through momentum space (diagonal red arrows), the total number of states in a valley varies. (b) Scattering within valleys (curved red arrows) can relax the current in each valley to its equilibrium value, but cannot establish a steady state because the number of states in each valley still changes at a constant rate. A steady state can be established only when intervalley scattering processes are present.
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Figure 2
Schematic illustration of a Weyl semimetal with two Weyl points with chiralities $Q_{\pm} = \pm 1$ . $2 b$ and $2 μ_{5}$ are the momentum-space distance and the chemical potential difference between the Weyl nodes, respectively. The dashed line indicates the Fermi level of the system, which can have different values in different valleys when the system is driven from equilibrium, for example, by the combined effect of electric and magnetic fields.
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Figure 3
Schematic illustration of longitudinal quadratic magnetoconductivity contributions $σ_{z z} (B_{z}^{2})$ [Eq. (101)] for $E ∥ B$ in a Weyl metal described by the Hamiltonian (73). $〈 n 〉$ and $〈 ξ 〉$ indicate band-diagonal density-matrix components, and $〈 S 〉$ indicates band-off-diagonal density-matrix components. See Eqs. (55), (57), and (58), respectively, for the formal expressions for $〈 n 〉, 〈 S 〉$ , and $〈 ξ 〉$ in a magnetic field. $τ$ and $τ_{intra}$ are the intervalley and intravalley scattering times, respectively. $σ_{z z} (B_{z}^{2})$ is given by the sum of the contribution from the chiral anomaly $σ_{z z}^{CA} (B_{z}^{2})$ and that from the Lorentz force $σ_{z z}^{LF} (B_{z}^{2})$ as $σ_{z z} (B_{z}^{2}) = σ_{z z}^{CA} (B_{z}^{2}) + σ_{z z}^{LF} (B_{z}^{2})$ . The other contributions to the quadratic magnetoconductivity vanish. Note that the contribution from the Lorentz force vanishes when the energy bands are isotropic in the $(k_{x}, k_{y})$ plane [see the discussion below Eq. (120)].
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Figure 4
(a) $k_{0}$ dependence of $C (b, Δ, T)$ in Eq. (116) for $Δ = 0, v_{F} = 1 eV Å, T = 0.005 eV$ , and $μ = 0.1 eV$ . In our model Hamiltonian for a two-node Weyl metal (73), the distance between the two Weyl nodes in momentum space is given by $2 k_{0} = 2 \sqrt{b^{2} - Δ^{2}} / v_{F}$ . Here, $k_{0}$ is given in units of $eV / ℏ v_{F} (\sim Å^{- 1})$ . The value of $C$ with $b = Δ = 0$ (with large $b$ and $Δ = 0)$ approaches $C = \frac{1}{2} (C = 1)$ in the low-temperature limit $T / μ \to 0$ . Also, the $k_{z}$ dependencies of the energy bands $\pm ɛ_{- k}$ at $k_{x} = k_{y} = 0$ for $k_{0} = 0, k_{0} = 0.05$ , and $k_{0} = 0.15$ in (a) are shown in (b)–(d), respectively. The dashed lines indicate the Fermi level, $μ = 0.1 eV$ . There exists a single isotropic Weyl cone when $k_{0} = 0$ , and it splits into two isotropic Weyl cones when $k_{0} \neq 0$ .
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