Designer Curved-Space Geometry for Relativistic Fermions in Weyl Metamaterials

Weyl semimetals are recently discovered materials supporting emergent relativistic fermions in the vicinity of band-crossing points known as Weyl nodes. The positions of the nodes and the low-energy spectrum depend sensitively on the time-reversal and inversion symmetry breaking in the system. We introduce the concept of Weyl metamaterials where the particles experience a 3D curved geometry and gauge fields emerging from smooth spatially varying time-reversal- and inversion-breaking fields. The Weyl metamaterials can be fabricated from semimetal or insulator parent states where the geometry can be tuned, for example, through inhomogeneous magnetization. We derive an explicit connection between the effective geometry and the local symmetry-breaking configuration. This result opens the door for a systematic study of 3D designer geometries and gauge fields for relativistic carriers. The Weyl metamaterials provide a route to novel electronic devices as highlighted by a remarkable 3D electron lens effect.

Popular Summary

Weyl semimetals are recently discovered materials where charge carriers behave as if they were massless particles moving at a slower speed of light. Therefore, the electric conduction in these materials reflects the physics of Einstein’s special theory of relativity. But special relativity also assumes an absence of gravity, which Einstein formulated as a geometry of spacetime. In this work, we propose a method on how to go beyond special relativity and simulate Einstein’s theory of general relativity. Our method provides a route to making the charge carriers move as if they were living in a curved space, providing a tabletop laboratory for simulating certain cosmological phenomena as well as the interplay between quantum physics and gravity.

By local manipulation of a Weyl semimetal, such as by strain or magnetization, it is possible to generate an artificial geometry and magnetic field experienced by the charge carriers. We dub these systems (where the low-energy dynamics is that of relativistic particles in curved space) Weyl metamaterials. We derive a mathematical connection between the local manipulation and the resulting geometry and show how the curved geometry and quantum effects lead to unique particle dynamics. As an example of the application potential, we introduce a structure called a Weyl electron lens, where the trajectories of charge carriers are focused much like beams of light in an optical lens.

Weyl metamaterials enable new types of electronic devices through geometry engineering and new possibilities in studying curved-space quantum physics in laboratories. The physics of Weyl metamaterials also has diverse connections to particle physics and cosmology.

Figure 1

(a) Weyl semimetals exhibit one or more band-crossing pairs which act as sources and sinks of the Berry curvature. The low-energy spectrum near the nodes is described by the Weyl Hamiltonian $H=\pm v\mathbf{k}·\sigma$ resulting in a conical relativistic dispersion. (b) Locations of the Weyl nodes and the conical envelopes depend on the magnitude of the TR and $I$ breaking in the system and are not fixed by symmetry. For example, by varying the magnetization $M_i$, one can widely tune the spectrum. (c) A smoothly varying magnetization in real space (bottom) gives rise to local Weyl cones that are deformed depending on the texture. This translates to an effective curved-space geometry experienced by relativistic carriers (top). (d) Semiclassical trajectories of carriers in the sample can be engineered by designing suitable TR- and $I$-breaking textures.

Figure 2

(a) LDOS of the local Weyl approximation (red) and the exact four-band Hamiltonian (blue) are in excellent agreement in the energy window $E_g$ around the Weyl node. Results are calculated for the lattice model specified in the text with a rotating TR-breaking texture $\mathbf{u}(z)=u_0(\cos k_{0}z,\sin k_{0}z,0)$. The parameters are $u_0=0.4t$, $k_0=0.05$, $m=0.2t$, and the energy $\delta$ functions are smoothed by Lorentzians with width $0.01t$. The inset shows the geometry of the system where the LDOS is calculated. With periodic boundary conditions in the $x$ and $y$ directions, the LDOS depends only on $z$. The curves are plotted for $z=30$ in a system with $N_z=60$ lattice cites. (b) Magnification of the low-energy region marked by the red dashed square in (a). The energy scale $E_{\tilde{B}}$ marks the crossover between two trends where LDOS is approximately constant, and where it is parabolic. (c) Comparison of the LDOS for the linearized Weyl approximation (black) including both chiralities and the exact four-band Hamiltonian for the same model as in (a). Inset: Dispersion of one chiral Weyl fermion in constant magnetic field as a function of parallel wave vector. The constant LDOS region corresponds to the bulk cone gap due to the local effective magnetic field $\tilde{\mathbf{B}}=\nabla \times {\mathbf{k}}^W$. (d) LDOS for the linear Weyl approximation (black) and the exact Hamiltonian (blue) for texture $\mathbf{u}(z)=[0,0,u(z)]$, where $u(z)=u_1+(u_2-u_1)z/N_z$, with $u_1=0.3t$ and $u_2=0.8t$. The other parameters are the same as in (a). For this system, $\tilde{\mathbf{B}}=0$. Inset: Otherwise the same, but $\mathbf{u}(z)=[u(z),0,0]$. Now $\tilde{\mathbf{B}}$ does not vanish and the LDOS is modified around $E=0$ by the effective magnetic field.

Figure 3

(a) Weyl metamaterial with rotating magnetic texture (red arrows) realizes a system where trajectories of particles with constant energy converge at focal points. (b) Semiclassical trajectories of carriers for the model specified in the main text with parameters $v_x=v_y=v_z=v$, $m/u=0.5$, and $\omega =0.05\pi /\xi$. The numerical values of the coordinates are given in the units of $\xi$, which parametrizes the magnetic rotation. The initial conditions for the trajectories are $y=z=0$, $k_x=k_y=0$, $k_z=1.25u/v$. Different curves correspond to different initial $x$ coordinates. (c) Trajectories of the same system but for initial conditions $x=y=z=0$, $k_z=1.25u/v$. Different curves correspond to $k_x=\pm 0.02$, $\pm 0.04$, $\pm 0.08$, $\pm 0.16u/v$. (d) The same as (c) but for the particles of opposite chirality.

Figure 4

(a) 3D lens effect can be realized also by a unidirectional $\mathbf{u}$ texture where modulus increases away from the lens axis. (b) Unidirectional texture which varies only in $x$ direction realizes 2D lens effect in $x\text{−}z$ plane. (c) Semiclassical trajectories of carriers with texture like in (b) with $\mathbf{u}=[0,0,u(r)]$, $u(r)=u_0+u_1(1-e^{-x^2/{\xi }^2})$, $v_x=v_y=v_z=v$, $m/u_0=0.5$, $u_0/u_1=3$, and $\xi =100$. The numerical values of the coordinates are given in the units of $\xi$. The initial conditions for the trajectories are $y=z=0$, $k_x=k_y=0$, $k_z=0.6u/v$. Different curves correspond to different initial $x$ coordinates. (d) Trajectories of the same system but for initial conditions $x=y=z=0$, $k_z=0.6u_0/v$. Different curves correspond to $k_x=\pm 0.016$, $\pm 0.03$, $\pm 0.07$, $\pm 0.13u_0/v$.