Chiral anomaly without Landau levels: From the quantum to the classical regime

Junhyun Lee, J. H. Pixley, and Jay D. Sau

Phys. Rev. B 98, 245109 – Published 6 December 2018

Abstract

We study the chiral anomaly in disordered Weyl semimetals, where the broken translational symmetry prevents the direct application of Nielsen and Ninomiya's mechanism and disorder is strong enough that quantum effects are important. In the weak disorder regime, there exist rare regions of the random potential where the disorder strength is locally strong, which gives rise to quasilocalized resonances and their effect on the chiral anomaly is unknown. We numerically show that these resonant states do not affect the chiral anomaly only in the case of a single Weyl node. At energies away from the Weyl point, or with strong disorder where one is deep in the diffusive regime, the chiral Landau level itself is not well defined and the semiclassical treatment is not justified. In this limit, we analytically use the supersymmetry method and find that the Chern-Simons term in the effective action which is not present in nontopological systems gives rise to a nonzero average level velocity which implies chiral charge pumping. We numerically establish that the nonzero average level velocity serves as an indicator of the chiral anomaly in the diffusive limit.

Received 7 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.245109

Physics Subject Headings (PhySH)

Anomalies Electronic structure

Disordered systems Weyl semimetal

Chern-Simons gauge theory Nonlinear sigma model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Junhyun Lee¹, J. H. Pixley², and Jay D. Sau¹

¹Department of Physics, Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
²Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA

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Vol. 98, Iss. 24 — 15 December 2018

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Figure 1
$ϕ$ dispersion on the “minizone” for several parameters. (a) The clean limit without disorder and also without magnetic field $(W = 0, Φ = 0)$ . We can observe the two linearly dispersing Weyl nodes. (b) Again the clean limit, but now with one magnetic flux quanta $(W = 0, Φ = Φ_{0})$ . Landau levels develop due to the magnetic field and the $n = 0$ chiral Landau level (CLL) appears. The CLL with positive (negative) chirality is colored in green (blue). (c) The weakly disordered case with one flux quanta $(W = 0.5 t, Φ = Φ_{0})$ . The disorder is uncorrelated $(k_{0} \to \infty)$ . Due to the uncorrelated disorder the internode scattering is present and thus open gaps when the CLLs with different chiralities intersect (near $ϕ = 0, π)$ . (d) System with weak correlated disorder and one flux quanta $(W = 0.5 t, Φ = Φ_{0}, k_{0} = k_{N} / 2$ ; $k_{N}$ being the $z$ component of the crystal momentum of a Weyl node measured from the $Γ$ point). Apart from (c), the disorder correlation suppresses internode scattering, thus significantly reducing the gap between different chirality CLLs.
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Figure 2
$ϕ$ dispersions of the single Weyl node model. All four cases have an uncorrelated disorder potential of strength $W = 0.7 t$ , but with a different disorder realization. Each state is color coded with their effective IPR localization length $ξ_{L}$ . (a) $Φ = Φ_{0}$ case when the disorder realization does not contain any rare region. The positive chirality CLL, i.e., the linear band at low energy, can be clearly seen and the positive spectral flow is indicated with the arrow. (b) $Φ = Φ_{0}$ when the disorder realization contains one rare region. The flat band with very low $ξ_{L}$ is the quasilocalized rare state. Notice the hybridization of the rare state and CLL. (c) $Φ = 2 Φ_{0}$ case for the same disorder realization as (b). The twofold degeneracy of Landau levels is weakly broken due to the perturbative effect of the disorder potential. (d) $Φ = 2 Φ_{0}$ case when the disorder realization contains two rare regions. The two rare states both interact with the CLLs, opening gaps in all four intersections. Note that the total chiral spectral flow is intact despite rare states in all cases (b), (c), and (d).
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Figure 3
$ϕ$ dispersion for rare states with two Weyl nodes. The disorder strength is the same as in Fig. 2, $W = 0.7 t$ , and the specific disorder realization is chosen as the one with a single rare region. (a) $Φ = Φ_{0}$ case. One can observe the rare state (low $ξ_{L}$ state around $E / t \approx 0.02)$ hybridize with both CLLs. The CLL loses its chirality after hybridizing, which can also be checked from the zero spectral flow as $ϕ$ is increased from 0 to $2 π$ . (b) $Φ = 2 Φ_{0}$ case for the same disorder realization. The rare state mixes with all CLLs and the gap opens in four places. Note the difference from (a) that only one pair (with lower energy) of the CLLs has lost its chirality due to hybridization with the rare state. The other pair of CLLs still remain chiral and survive as a channel of charge pumping.
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Figure 4
Numerical data on verifying the relation (18). (a) $ϕ$ dispersion of a system with parameters $W = 6.0 t, k_{0} = k_{N} / 10$ , and $Φ = Φ_{0}$ . Notice in the energy range of $| E / t | > 0.5$ , the DOS does not have any structure in $ϕ$ and can be considered as a topological metal in the diffusive limit. (b) The $f (E)$ function (18), for the system in (a). The data are linear in energy with the coefficient of one, the number of flux, and this feature persists to the diffusive metal limit $(| E / t | > 0.5)$ . The red solid line is the guide to the eye with slope of one. (c) The same $ϕ$ dispersion as in Fig. 2, where there are rare states near the Weyl point and $Φ = 2 Φ_{0}$ . (d) The $f (E)$ function for the system in (c). The data match well that it is linear with the coefficient of two. The red solid line is the guide to the eye with slope of two.
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Figure 5
(a) The probability density projected to the $x y$ plane $(\sum_{z} {| ψ (x, y, z) |}^{2})$ of a rare state wave function with one flux quantum in the single-node Weyl model [here we are showing the rare state displayed in the dispersion in Fig. 2]. The rare state is chosen to be the eigenstate shown in Fig. 2, $ϕ = π$ (the state with low $ξ_{L}$ and $E / t \approx 0.06)$ . We can see the probability density is (quasi)localized about a rare region of the random potential. (b) The decay of this rare state wave function. To see the decay behavior, we made a “binned” wave function with equally spaced bins (in distance $r$ from its maximum), and assigned the average value of the wave functions in the bin. The dashed line is a linear fit from data $r < 6.4$ and has a slope of $- 2.15$ . This shows the power-law decay of the rare state wave function $ψ \sim 1 / r^{- 2.15}$ .
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Figure 6
The decay of the rare state wave function in the presence of a magnetic field. Here, the rare state is found in the two Weyl model and is the same state we show in the dispersion in Fig. 3, $ϕ = π$ . The dashed line is the linear fit for the $Φ = 0$ data, showing the power-law decay $ψ \sim 1 / r^{- 1.9}$ . The data show even in the presence of a strong magnetic field (note that $Φ = 20 Φ_{0}$ corresponds to $1 / 20$ flux per plaquette and the magnetic length $l_{B} / a \approx 1.78)$ the small- $r$ behavior is not drastically changed.
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