So Small Implies So Large: For a Material Design

The minimum coupling of the form p-eA implies that the momentum p is equivalent to the gauge field A (vector potential), where e is the charge of an electron. Then, the effect of a magnetic field B on a phenomenon with a length scale a is roughly estimated by the ratio Φ/Φ₀, where Φ = Ba² is a magnetic flux passing through the typical area and Φ₀ = h/|e| is the flux quantum [1]. This is very small as Φ/Φ₀ ∼ 0.0006 ≪ 1 for a typical lattice scale of standard materials: a ∼ 5 [Å] even for a strong magnetic field B ∼ 10.0 [T]. Still, the magnetic field induces Landau levels, which results in diamagnetism for the itinerant electrons and quantum Hall effects at low temperatures.

The gauge transformation modifies p and A, and only the gauge invariant combination p-eA is physical. According to the principles of quantum mechanics, the momentum is given by a spatial derivative. It implies that an intrinsic phase factor of a wavefunction is also equivalent to the gauge field. Acquiring an additional phase may induce an effective gauge field even if the external magnetic field is zero. Quantum spin Hall effect is one such phenomenon where the atomic spin–orbit interaction induces a phase factor of electrons, and it is eventually transformed into the effective gauge field where the effective magnetic fields are reversed in direction for each spin [2,3]. Anomalous quantum Hall effect and valley Hall effect are other examples where the phase factors of electrons due to atomic origin induce gauge fields. In these materials, the Dirac cone like linear dispersion with a (small) gap is important. The gap closing in two dimensions implies the appearance of an effective massless Dirac fermion, which may be accidental or due to some symmetry protection. The linear dispersion of graphene is a typical example. Any gap opening perturbation implies a mass that breaks the “time reversal” effectively near the gap closing momentum [4]. In principle, an infinitesimal perturbation induces “band inversion” in a system with Dirac cones.

Noting that there is an equivalence between the momentum and the gauge field, it is not difficult to expect that a real space deformation couples to the gauge field as the deformation of the carbon nanotube induces a gauge field [5], and the strained graphene has pseudo Landau levels without breaking the time reversal symmetry [6]. The shifting of the Dirac cones implies a gauge field. The same mechanism is applied to three-dimensional Dirac/Weyl semi-metals, and there has been considerable interest regarding this pseudo gauge field. Although the real magnetic field available experimentally is rather weak for a lattice scale as commented above, the shift of the Dirac cone, Δk, say 10% of the Brillouin zone size, results in a gauge field far beyond the available magnetic field when interpreted as the pseudo magnetic field B_eff ∼ 1.7·10³ [T], where Φ_eff = a²B_eff ∼ Φ₀Δk/(2π/a) [7,8].

In a recent paper [7], Toshikaze Kariyado proposed an experimental setup for realizing the pseudo magnetic field without breaking the time reversal invariance (see Fig. 1 [7]). Two bulk materials with slightly different Dirac cone positions are separated by a buffer material, which smoothly connects the two. Then, the interpolation of the Dirac cone position implies the pseudo gauge field that results in the pseudo Landau levels localized in the buffer region. Since the shift of the Dirac cone is finite, the Landau level structure is only allowed near zero energy. In addition, a compact formula for the number of available Landau levels is also given. The idea is confirmed numerically using a tight binding model, and an anti-perovskite material Ca_3(1−x)Sr_3xSnO is discussed based on the semi ab-initio calculation as a possible material with a mobile Dirac cone.

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Fig. 1. (a) Proposed setup for the realization of pseudo Landau levels without a real magnetic field, where B is a pseudo magnetic field. (b) Bulk band structure as a function of k_x. Figure is taken from Fig. 1 of Ref. 7.

References

• 1) |e|A/p ∼ |e|A/(h/a) ∼ Φ/Φ₀, B ∼ A/a. Google Scholar
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• 8) Δp ∼ |e|A_eff, ∼ |e|aB_eff, Φ_eff = a²B_eff ∼ aΔp/|e| = a(h/2π)Δk/|e| = Φ₀(a/2π)Δk = Φ₀Δk/(2π/a). Google Scholar

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