Chiral Anomaly from Strain-Induced Gauge Fields in Dirac and Weyl Semimetals

Dirac and Weyl semimetals form an ideal platform for testing ideas developed in high-energy physics to describe massless relativistic particles. One such quintessentially field-theoretic idea of the chiral anomaly already resulted in the prediction and subsequent observation of the pronounced negative magnetoresistance in these novel materials for parallel electric and magnetic fields. Here, we predict that the chiral anomaly occurs—and has experimentally observable consequences—when real electromagnetic fields $\mathit{E}$ and $\mathit{B}$ are replaced by strain-induced pseudo-electromagnetic fields $\mathit{e}$ and $\mathit{b}$. For example, a uniform pseudomagnetic field $\mathit{b}$ is generated when a Weyl semimetal nanowire is put under torsion. In accordance with the chiral anomaly equation, we predict a negative contribution to the wire resistance proportional to the square of the torsion strength. Remarkably, left- and right-moving chiral modes are then spatially segregated to the bulk and surface of the wire forming a “topological coaxial cable.” This produces hydrodynamic flow with potentially very long relaxation time. Another effect we predict is the ultrasonic attenuation and electromagnetic emission due to a time-periodic mechanical deformation causing pseudoelectric field $\mathit{e}$. These novel manifestations of the chiral anomaly are most striking in the semimetals with a single pair of Weyl nodes but also occur in Dirac semimetals such as ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and ${\mathrm{Na}}_3\mathrm{Bi}$ and Weyl semimetals with unbroken time-reversal symmetry.

Popular Summary

Dirac and Weyl semimetals can be thought of as three-dimensional analogs of graphene. These materials (e.g., ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and ${\mathrm{Na}}_3\mathrm{Bi}$) contain electrons that behave, in many respects, as massless, relativistic particles. Dirac and Weyl semimetals exhibit a variety of exotic behaviors, including the so-called “chiral anomaly” predicted more than 40 years ago in the context of high-energy physics. This anomaly occurs when both electric and magnetic fields are applied to the material, and it has measurable consequences in electric transport. Using a combination of analytical tools and numerical techniques, we predict that the chiral anomaly can occur in Dirac and Weyl semimetals when real electromagnetic fields ($\mathit{E}$ and $\mathit{B}$) are replaced by strain-induced pseudoelectromagnetic fields ($\mathit{e}$ and $\mathit{b}$).

Strain can be induced by stretching and compressing crystals or by propagating sound waves. In graphene, a quintessential two-dimensional material with Dirac fermions, previous research has demonstrated the presence of strain-induced fields. Here, we explore the physics of strain-induced fields in three spatial dimensions in Dirac and Weyl semimetals shaped into very small wires (diameters on the order of hundreds of nanometers) or films. We consider both torsional and unidirectional strain. Using model calculations and numerical simulations, we find that these fields have several unusual experimentally observable consequences, including the chiral anomaly in the complete absence of real electromagnetic fields.

We expect that our findings will pave the way for better understanding the effects of electric transport and sound propagation in crystals.

Figure 1

Electron excitation spectra in a Weyl semimetal in the presence of (a) magnetic field $\mathit{B}$ and (b) pseudomagnetic field $\mathit{b}$ generated by a torsional deformation. Parallel electric field $\mathit{E}$ produces a charge-density imbalance in case (a), while it appears to produce excess total charge density in case (b). Panel (c) illustrates the displacement field $\mathit{u}$ in the presence of torsion. Consecutive layers of the crystal are rotated by relative angle ${\phi }_0=\mathrm{\Omega }(L/a)$.

Figure 2

The effect of strain on the hopping amplitudes in the tight-binding model. (a) Unidirectional strain along the $z$ axis simply changes the distance between the neighboring orbitals leading to the modification of the hopping amplitude $t_1$ that is linear in $u_{33}$ to leading order in small displacement. (b) Torsional strain changes the relative orientation of the orbitals and brings about hopping amplitudes that are disallowed by symmetry in the unstrained crystal, such as $t_{sp}$. The corresponding mathematical expression encodes the expectation that $t_{sp}$ would become equal to $\mathrm{\Lambda }$ if the $p$ orbital were displaced all the way to the horizontal position. In the real material, one of course expects Eq. (2.8) to be valid only for displacements that are small compared to the lattice parameter $a$.

Figure 3

Tight-binding model simulations of a Weyl semimetal wire under torsional strain and applied magnetic field $\mathit{B}=\widehat{z}B$. The top row of figures shows the band structure of the lattice Hamiltonian defined by Eqs. (2.4) and (2.9) computed for $\frac{1}{2}\text{−}{\mathrm{Cd}}_3{\mathrm{As}}_2$ model parameters, for a wire with a rectangular cross section of $30\times 30$ sites and a lattice constant $a=40\AA$. (We use a larger lattice constant here and in subsequent simulations than in real ${\mathrm{Cd}}_3{\mathrm{As}}_2$ in order to be able to model nanowires and films of realistic cross sections with available computational resources. Note that this does not affect the physics at low energies because the lattice Hamiltonian is designed to reproduce the relevant $\mathit{k}·\mathit{p}$ theory independent of $a$.) Open boundary conditions are imposed along $x$ and $y$, periodic along $z$. Parameters appropriate for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ are used. The middle and bottom rows show spectral functions $A^{\mathrm{bulk}}(\mathit{k},\omega )$ and $A^{\mathrm{surf}}(\mathit{k},\omega )$. The former is obtained by averaging the full spectral function $A_j(\mathit{k},\omega )$ over sites $j$ in the central $10\times 10$ portion of the wire, while the latter averages over the sites located at the perimeter of the wire. The torsion applied in columns c and d corresponds to the maximum displacement at the perimeter of $0.5a$, or ${\phi }_0\simeq 2^{\mathrm{o}}$ between consecutive layers.

Figure 4

Tight-binding model simulations of a Weyl semimetal under applied magnetic field $\mathit{B}=\widehat{z}B$ and unidirectional strain. Parameters for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ listed in Appendix pp1 are used in all panels. Only the spin-up sector of the model is considered with $B=10\mathrm{T}$. (a) Band structure of the system with periodic boundary conditions in all directions (no surfaces) projected onto the $z$ axis ($k$ denotes the crystal momentum along the $z$ direction). Solid (dashed) lines show occupied (empty) states. Occupation of the strained system is determined by adiabatically evolving the single-electron states of the unstrained system. (b) Band structure of a slab with thickness $d=1000\AA$ (50 lattice sites). Only positive values of $k$ are displayed, but the band structure is symmetric about $k=0$. Red (black) lines show occupied (empty) states. The central panel indicates the nonequilibrium occupancy of the strained system obtained by adiabatically evolving the single-electron states of the unstrained system. The right panel shows the occupancy of the strained system once the electrons relax back to equilibrium. All three panels correspond to the same total number of electrons $N$. (c) Change in the electron density in response to the applied strain as a function of coordinate $y$ perpendicular to the slab surfaces. Here, $\delta \rho$ refers to the nonequilibrium distribution, while $\delta {\rho }_{\mathrm{eq}}$ refers to the relaxed state. Note that density oscillations near the edges apparent in $\delta \rho$ average to zero: There is no net charge transfer between the bulk and the surface in the nonequilibrium state, as can also be deduced from the vanishing $\delta \rho$ in the bulk.

Figure 5

Numerically calculated change in the bulk charge density $\delta {\rho }^{\mathrm{bulk}}$ in response to unidirectional strain $\alpha =0.03$ as a function of the applied field $B$. Parameters for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ are used with $\mu =0$ and $d=1000\AA$ (50 lattice sites). Solid black symbols give result for the p-h symmetric version of the $\frac{1}{2}\text{−}{\mathrm{Cd}}_3{\mathrm{As}}_2$ model obtained by setting all $C_j$ parameters to zero.

Figure 6

Band structure of the spin-down sector of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ in magnetic field $B=10\mathrm{T}$. Two chiral branches are visible at low energies; however, they are now strongly distorted by p-h–symmetry breaking terms, and they no longer traverse the gap between the valence and the conduction band.

Figure 7

Equilibrium current density in the Weyl semimetal wire under torsion. (a) Schematic depiction of the bulk or surface current flow. (b) Ground-state current density computed from the lattice model, Eqs. (2.4) and (2.9), at chemical potential $\mu =5\mathrm{meV}$. Warm (cold) colors represent positive (negative) current density $\mathit{j}$. The ring-shaped inhomogeneity in $\mathit{j}$ apparent in the bulk of the wire reflects Friedel-like oscillations in electron wave functions caused by the presence of the surface.

Figure 8

Proposed geometry for the EM-field emission measurement in the limit when all the dimensions of the crystal are much larger than the sound wavelength ${\lambda }_s$. (a) A slab of thickness $d=2d^{\prime }$ is subjected to magnetic field $\mathit{B}$ and a longitudinal acoustic sound wave propagating along the $z$ direction. (b) A snapshot of the electric-field distribution near the surface calculated from Eq. (4.22). As a function of time, the entire pattern moves in the $z$ direction at the speed of sound $c_s$.

Figure 9

Dispersion relations for the spin-up sector of the lattice Hamiltonian (2.3) describing ${\mathrm{Cd}}_3{\mathrm{As}}_2$ (top row) and ${\mathrm{Na}}_3\mathrm{Bi}$ (bottom row). The parameters used in the simulations include the particle-hole symmetry-breaking terms and are summarized in Table 1. We used a lattice with $40\times 40$ sites and the magnetic fields shown in the green boxes for each of the materials. Notice the different magnitude of effective magnetic fields for different compounds—this is due to the different lattice constants and a different sign of the physical magnetic field between the two rows. A different sign of magnetic fields shows that the physical magnetic field compensates the torsional one in opposite Weyl points for opposite directions of the magnetic field in accordance with the interpretation in the main text.

Figure 10

Persistent currents in a ${\mathrm{Cd}}_3{\mathrm{As}}_2$ nanowire under torsion and magnetic field. (a) Band-structure detail for spin-up (blue) and spin-down (green) sectors in a $30\times 30$ lattice with $B=b=3.2\mathrm{T}$ and other parameters as in Fig. 9. (b) Calculated current density $j_z$ for $\mu =0$, including contributions from both spin sectors.

Figure 11

Ratio of conductance of a disordered $W\times W\times 20$ system to the conductance of the clean system averaged over 100 disorder realizations. The green line is the best fit to the data—parabolic; grey curves show the failure of the linear [with non-negative $G(0)$] and cubic fits.