Weak antilocalization is a transport phenomenon in the quantum diffusion regime in disordered metals.^[1] The quantum diffusion in disordered metals can be defined by the mean free path ℓ and the phase coherence length ℓ_ϕ. The mean free path measures the average distance that an electron travels before its momentum is changed by elastic scattering from static scattering centers, while the phase coherence length measures the average distance that an electron can maintain its phase coherence. If the mean free path is much shorter than the system size and the phase coherence length, then the electrons suffer from scattering but can maintain their phase coherence. This is the quantum diffusive regime, in which the quantum interference between time-reversed scattering loops (see Fig. 1(a)) can give rise to a correction to the conductivity. The weak localization or weak antilocalization arises due to this correction in the conductivity in the quantum diffusive regime.

The phase coherence length is determined by inelastic scattering from electron–phonon coupling and interaction with other electrons. The inelastic scattering has to be suppressed significantly to make the phase coherence length much longer than the mean free path. Therefore, the quantum diffusion usually takes place at extremely low temperatures; e.g., below the liquid helium temperature. If the quantum interference correction is positive, then it gives a weak antilocalization correction to the conductivity and the conductivity goes up with decreasing temperature (see Fig. 1(b)). Because this correction requires time reversal symmetry, it can be suppressed by applying a magnetic field. One can then observe that the conductivity goes down with increasing magnetic field. This negative magnetoconductivity or positive magnetoresistivity is the more familiar signature of the weak antilocalization (see Fig. 1(c)).

Fig. 1.

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Fig. 1. (a) Schematic illustration of time-reversed scattering loops in the quantum diffusion regime in disordered metals. The open circles represent impurities and the arrows mark the trajectories that electrons travel. (b) and (c) The signatures of the weak antilocalization (WAL) in (b) temperature (T) dependence of the conductivity σ and (c) magnetoconductivity [defined as δσ ≡ σ(B) − σ(0)], where B is the magnetic field.

In contrast, the quantum interference can be negative, leading to a weak localization effect and totally opposite temperate and magnetic dependencies of conductivity. Whether a system has weak localization or weak antilocalization depends on the symmetry (see Table 1). According to the classification of the ensembles of random matrix,^[2] there are three symmetry classes. If a system has no time-reversal symmetry, then it is in the unitary class, in which there is no weak localization or antilocalization. If a system has time-reversal symmetry but no spin-rotational symmetry, then it is in the symplectic class, in which the weak antilocalization is expected. If a system has both time-reversal and spin-rotational symmetries, then it is in the orthogonal class, in which the weak localization is expected. These symmetry arguments have been well known since the studies on the conventional 2D electron gases.^[3]

Table 1.

Table 1.

Table 1. The relation between the symmetry classes (orthogonal, symplectic, and unitary)[2] and the weak localization (WL) and antilocalization (WAL).[3] . Orthogonal Symplectic Unitary Time-reversal ✓ ✓ × Spin-rotational ✓ × × WL/WAL WL WAL ×

Table 1. The relation between the symmetry classes (orthogonal, symplectic, and unitary)[2] and the weak localization (WL) and antilocalization (WAL).[3] .

In the last decade, weak antilocalization has been widely observed in topological materials, such as topological insulators^[4–9] and topological semimetals,^[10–13] in which the quasiparticles are described not by the Schrödinger equation but as Dirac fermions in the topological insulators and Weyl fermions in the topological semimetals. For Dirac fermions, the weak antilocalization has an alternative understanding based on the Berry phase argument. The Berry phase is a geometric phase collected in an adiabatic cyclic process.^[14,15] Since studies on carbon nanotubes have begun, it has been found that massless Dirac fermions can collect a π Berry phase after circulating around the Fermi surface.^[16] The π Berry phase induces a destructive quantum interference between time-reversed loops formed by scattering trajectories. The destructive interference can suppress backscattering of electrons, the conductivity is then enhanced with decreasing temperature because the decoherence mechanisms are suppressed at low temperatures.^[17,18]

One of the powerful theoretical approaches to study weak localization and antilocalization is the Feynman diagram techniques. Figure 2 summarizes the Feynman diagrams we have used to study the weak localization and antilocalization arising from the quantum interference and interaction. Simply speaking, it is based on the linear response theory of the conductivity, with disorder and interaction taken as perturbations. In the formulism, there are three main contributions to the conductivity. The leading order is the semiclassical Drude conductivity (Fig. 2(a)), and then the quantum interference correction (Fig. 2(b)) and the interaction correction (Altshuler–Aronov effect) (Fig. 2(d)).

In this paper, we review our recent efforts^[19–22] on the weak antilocalization and interaction-induced localization of Dirac and Weyl fermions in topological insulators and topological semimetals.^[19–25] Part of the contents have been reviewed in Ref. [26], where only topological insulators are addressed. In Section 2, we discuss the Berry phase argument and the crossover between weak antilocalization and weak localization in magnetically modulated topological insulator and topological insulator thin films. In Section 3, we show the weak localization of Dirac fermions as a result of electron–electron interactions. In Section 4, we review the weak antilocalization and interaction-induced localization of Weyl fermions in 3D topological semimetals. Finally, remarks and perspective are given in Section 5.

Fig. 2.

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Fig. 2. (a) The diagrams for the semiclassical conductivity σsc. v and are the bare velocity and the velocity after the vertex correction, respectively. (b) The Hikami boxes of the maximally-crossed diagrams, from which the conductivity correction from the quantum interference σqi is calculated.[18] (c) The iteration equations for the Diffuson (Λ), Cooperon (Γ), and dynamically screened interaction. (d) The Fock and Hartree self-energies dressed by Diffuson and Cooperon, from which the conductivity correction from the electron–electron interaction σee is calculated. k and q stand for the wave vectors, and εn and ωm for the Matsubara frequencies. Adapted from Ref. [19].

Topological insulators are gapped band insulators with topologically protected gapless modes surrounding their boundaries.^[4–9] The surface of a three-dimensional topological insulator hosts an odd number of two-dimensional gapless Dirac cones. The Dirac cone has a helical spin structure in momentum space.^[27] The topological insulators have attracted tremendous interest in their transport properties.^[28,29] It turns out that the known topological insulator materials all have poor mobility. The mean free path is of the order of 10 nm, while the phase coherence length can reach up to 100–1000 nm below the liquid helium temperature. In other words, these materials are well in the quantum diffusion regime at low temperatures, where the weak (anti-)localization is expected. In experiments, the negative magnetoconductivity arising from the weak antilocalization has been observed in almost every topological insulator sample.^[30–39]

The surface states of a topological insulator can be described by a two-dimensional massless Dirac model

The Berry phase is a geometric phase collected in an adiabatic cyclic process,^[15,16] and it can be found as

The π Berry phase can give an explanation of the weak antilocalization of the two-dimensional massless Dirac fermions. The time-reversed scattering loops in Fig. 1(a) are equivalent to moving an electron on the Fermi surface by one cycle. The Berry phase then plays an important role. The π Berry phase give rises to a destructive quantum interference that suppresses the back scattering and enhances the conductivity, leading to the weak antilocalization.^[17,18] The π Berry phase can lead to the absence of backscattering,^[41] and further the delocalization of the surface electron.^[42,43]

To verify the role played by the Berry phase, one can alter the Berry phase by including a mass to the Dirac model^[7,44]

Now the conduction and valence bands are separated by a gap Δ. With the mass term, the Berry phase turns out to be^[20]

An alternative definition of a topological insulator is that its topologically protected surface states cannot be localized.^[4,5,42,43] The negative magnetoconductivity of the weak antilocalization has been regarded as a signature of this delocalization tendency. However, in most experiments, a suppression of the conductivity with decreasing temperature is observed (see Fig. 3),^{[35–37,60–62]} showing a tendency of localization.^[1,63] In other words, the experimental observation in magnetoconductivity and finite temperature conductivity presents a transport paradox in topological insulators.

The electron–electron interaction^[64,65] provides a possible way to understand the contradictory magnetic field and temperature dependences.^[19] The interplay of the interaction and disorder can lead to a temperature dependence much like the weak localization, known as the Altshulter–Aronov effect.^[64,65] The theory is established for conventional electrons. We show that Dirac fermions are not immune from the Altshuler–Aronov effect. With or without the magnetic field, the correction from the interaction to the conductivity decreases logarithmically with decreasing temperature. Although the weak antilocalization can enhance the conductivity with decreasing temperature, it is overwhelmed by the contribution from the interaction effect. Therefore, the overall temperature dependence of the conductivity shows a weak localization tendency. Our theoretical results agree well with the experiments, with comparable changes of the conductivity (several e²/h), temperatures (0.1 to 10 K), and magnetic fields (0 to 5 T).^{[35–37,60–62]}

Fig. 3.

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Fig. 3. (a) Measured logarithmic temperature-dependent conductivity (solid lines) at zero magnetic field for all samples with the edge-to-edge distance between antidots d indicated. The red dashed lines are guide for eyes. (b) Measured logarithmic temperature-dependent conductivity at indicating perpendicular magnetic fields (open circles) for sample d190. Solid lines are the linear fitting curves. (c) Measured temperature-dependent conductivity (solid lines) at B = 2 T for all samples with d indicated. The curves are offset for clarity and the red dashed lines are guide for eyes. (d) The slope obtained by linear fitting the curves in panel (b) is plotted (solid symbols) as a function of the magnetic field for sample d190. Solid lines are guide for eyes. Adapted from Ref. [59].

Furthermore, to verify the interaction effect, we proposed an artificial nanostructure^[59] as shown in Fig. 4. The temperature dependence of the conductivity is characterized by the slope defined as

In strong magnetic fields, the slope is solely determined by the interaction effect. As an interaction effect, the slope is supposed to be changed by changing the dielectric constant ɛ_r. We calculated the slope from the interaction κ^ee as a function of ɛ_r for the surface electrons as plotted in Fig. 5(a). One can see that κ^ee decreases monotonically with decreasing ɛ_r.

To modulate the dielectric constant, we fabricated an array of antidots arranged in a periodic triangular lattice in thin films of Bi₂Te₃, as schematically indicated in Fig. 4(a). The diameter of each antidot is fixed at 200 nm. The edge-to-edge distances d of two neighboring antidots are 40 nm, 90 nm, 130 nm, 190 nm, and 250 nm, respectively. A smaller value of d represents a larger density of antidots. The dielectric constant of Bi₂Te₃ is of the order of 10–100,^[68] while that of the vacuum at the antidots is 1. According to the effective medium theory,^[67] with decreasing distance between the antidots, i.e., more antidots, the dielectric constant is supposed to be reduced. As a result, the slope is also expected to decrease with decreasing distant between the antidots. This is consistent with the observation in Figs. 3 and 5.

In this way, we show that two-dimensional massless Dirac fermions can show the localization tendency when both the electron–electron interaction and disorder scattering are taken into account.

Fig. 4.

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	Fig. 4. (a) Schematic illustration of a Hall bar sample with antidot array. (b) Scanning electron microscopy image of a Bi2Te3 thin film with antidot array, where the edge-to-edge distance between antidots d = 40 nm. The inset is the atomic force microscope image indicating that the antidots depth is about 25 nm. Adapted from Ref. [59].

Fig. 5.

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Fig. 5. (a) The calculated slope κee as a function of relative dielectric constant ɛr for gapless Dirac fermions. vħ is chosen to be 2.5 eV·Å, comparable with that in the experiment.[66] (b) The slope κ of the conductivity–ln T curve as a function of magnetic field B for different antidot distance d. The measured κ (scatters) are fitted by the theory[19] (solid curves). (c) The dielectric constant as a function of d, obtained by fitting the transport theory (ɛr) and calculated from the Bruggeman effective medium theory , respectively.[67] The intrinsic dielectric constant ɛb = 22.0 of the no-antidot sample is extracted by fitting the transport theory. Adapted from Ref. [59].

Weyl semimetal is a three-dimensional (3D) topological state of matter, in which the conduction and valence energy bands touch at a finite number of nodes.^[10] The nodes always appear in pairs, in each pair the quasiparticles (dubbed Weyl fermions) carry opposite chirality and linear dispersion, much like a 3D analog of graphene. In the past few years, a number of condensed matter systems have been suggested to host Weyl fermions.^{[11–13,70–78]} Recently, angle-resolved photoemission spectroscopy (ARPES) has identified the Dirac nodes (doubly-degenerate Weyl nodes)^[72] in (Bi_1−xIn_x)₂Se₃,^[79,80] Na₃Bi,^{[74,76,81,82]} and Cd₃As₂^[76,83–86] and Weyl nodes in the TaAs family.^[87–91] The negative magnetoconductivity arising from the weak antilocalization has been observed recently in Bi_0.97Sb_0.03,^[92,93] ZrTe₅,^[94] and TaAs.^[88,95]

Fig. 6.

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Fig. 6. Nontrivial band structure and Berry curvature of a topological semimetal. (a) A schematic of the energy spectrum of a topological semimetal. (kx, ky, kz) is the wave vector. . (b) The vector plot of the Berry curvature in momentum space. The conduction and valence bands of a topological semimetal touch at the Weyl nodes, at which a pair of monopoles are hosted. The arrows show that the flux of the Berry curvature flows from one monopole (red) to the other (blue), defining the nontrivial topological properties of the topological semimetal. Adapted from Ref. [69].

One of the low-energy descriptions of Weyl fermions in semimetals is

We calculate the magnetoconductivity arising from the quantum interference, as shown in Fig. 7. As B → 0, δσ^qi is proportional to for ℓ_ϕ ≫ ℓ_B or at low temperatures, and δσ^qi ∝ − B² for ℓ_ϕ ≪ ℓ_B at high temperatures. ℓ_B can be evaluated approximately as 12.8/ nm with B in units of Tesla. Usually, below the liquid helium temperature ℓ_ϕ can be as long as hundreds of nanometers to one micrometer, much longer than ℓ_B, which is tens of nanometers between 0.1 T and 1 T. Therefore, the magnetoconductivity at low temperatures and small fields serves as a signature for the weak antilocalization of 3D Weyl fermions. Figure 7(a) shows δσ^qi(B) of two valleys of Weyl fermions in the absence of intervalley scattering. For long ℓ_ϕ, δσ^qi(B) is negative and proportional to , showing the signature of the weak antilocalization of 3D Weyl fermions. This dependence agrees well with the experiment,^[92,93] and we emphasize that it is obtained from a complete diagram calculation with only two parameters ℓ and ℓ_ϕ of physical meanings. As ℓ_ϕ becomes shorter, a change from to −B² is evident. δ^qi(B) vanishes at ℓ_ϕ = ℓ as the system quits the quantum interference regime.

Fig. 7.

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Fig. 7. The magnetoconductivity δσqi(B) for different phase coherence length ℓϕ at ηI = η* = 0 (a), for different ηI at η* = 0 (b), and for different ηI at finite η* (c). The magnetic field B is applied along arbitrary directions. Parameters ℓ = 10 nm and ℓϕ = 1000 nm in panels (b) and (c). (d) The diagrams show the difference between ηI and η*, with ηI related to the intervalley scattering and η* measuring the intervalley correlation of intravalley scattering. The dashed lines represent the correlation of two scattering processes. ν = ± is the valley index. Adapted from Ref. [24].

Based on the theory in Ref. [24], we proposed a formula to fit the magnetoconductivity arising from the weak localization, and it has been applied in the experiment on TaAs^[95]

In the presence of the interaction, we find that the change of conductivity with temperature for one valley of Weyl fermions can be summarized as

Fig. 8.

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	Fig. 8. A schematic demonstration of the change of conductivity Δσ as a function of temperature T. We choose cee = cqi. Tc is the critical temperature below which the conductivity drops with decreasing temperature. Adapted from Ref. [24].

We find that the intervalley scattering and correlation can also lead to the weak localization. Two dimensionless parameters are defined for the inter- and intravalley scattering: measuring the correlation between intravalley scattering and measuring the weight of intervalley scattering, where is the scattering matrix element. Figure 7(d) schematically shows the difference between η_* and η_I. As shown in Fig. 7(b), with increasing η_I, the negative δσ^qi is suppressed, where η^I → 1 means strong intervalley scattering while η_I → 0 means vanishing intervalley scattering. Furthermore, figure 7(c) shows that the magnetoconductivity can turn to positive when η_I + η_* = 3/2. The positive δσ^qi(B) in Fig. 7(c) corresponds to a suppressed σ^qi with decreasing temperature, i.e., a localization tendency. This localization is attributed to the strong intervalley coupling which recovers spin-rotational symmetry (now the spin space is complete for a given momentum), the system then goes to the orthogonal class.^[2,3,17] Therefore, we show that the combination of strong intervalley scattering and correlation will strengthen the localization tendency in disordered Weyl semimetals.

In summary, we systematically studied the weak antilocalization and interaction-induced localization for Dirac and Weyl fermions in topological insulators and topological semimetals. With the help of Feynman diagram techniques, we considered the correction to the conductivity from the quantum interference correction and electron–electron interaction. We predicted the crossover between weak antilocalization and weak localization for the massive Dirac fermions. The theory can be applied to magnetically doped topological insulators and surface and bulk states in topological insulator thin films. With the help of an antidot nanostructure in topological insulator thin films, we verified the interaction induced localization tendency for Dirac fermions in topological insulators. We also studied the weak antilocalization and interaction effect in Weyl semimetals. For a single valley of Weyl fermions, we found that the magnetoconductivity is negative and proportional to the square root of the magnetic field at low temperatures, giving the signature for the weak antilocalization in three dimensions as well as for the Weyl fermion. In the presence of strong intervalley effects, we expected a crossover from the weak antilocalization to weak localization. In addition, we found that the interplay of electron–electron interaction and disorder scattering can also give rise to a tendency to localization for Weyl fermions.

Finally, we remark on the possible future works. On the weak localization induced by the interaction, the bulk and surface states still coexist on the Fermi surface in the latest experiment work. To show the interaction-induced localization of the surface states, further experiments are still needed to be performed in intrinsic topological insulators by doping^[96] or using ternary and quaternary compounds.^[97–103] Topological Weyl semimetals provide a new platform to study the weak antilocalization in three dimensions. Our formula of magnetoconductivity can be used for a systematic study of the transport experiments on topological semimetals. One of the interesting fields is the theories of the weak localization for semimetals with monopole charges higher than 1, as has recently been explored for the double-Weyl semimetals.^[25] Also, the weak (anti-)localization theories for nodal-line and drumhead semimetal could be interesting topics for further research.