Heavy Weyl Fermion State in ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$

A new type of topological state in strongly correlated condensed matter systems, the heavy Weyl fermion state, has been found in a heavy fermion material, ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$, which has no inversion symmetry. Two different types of Weyl points, types I and II, can be found in the quasiparticle band structure obtained by the $\mathrm{LDA}+\mathrm{Gutzwiller}$ calculations, which can treat the strong correlation effects among the $f$ electrons from cerium atoms. The surface calculations indicate that the topologically protected Fermi arc states exist on the (010) but not on the (001) surface.

Popular Summary

A Weyl semimetal (WSM) is a type of solid in which the electrons move like photons in a vacuum—they are effectively massless and spin around themselves clockwise or counterclockwise while traveling forward. This makes the electrons, known as Weyl fermions, highly mobile and chiral (they have a preferred spin direction). An external magnetic field applied to a WSM enhances conductivity by the chiral anomaly mechanism, which pumps electrons of opposite chirality and thus reduces the electrical resistance. This chiral current dissipates nearly no energy, which makes WSMs an attractive material for low-power consumption and high-frequency electronics. Only a few WSM, such as the crystal tantalum arsenide (TaAs), have been experimentally identified. We propose that another substance, ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$, is a new type of material that exhibits WSM-like behavior.

The electrons in all previously proposed WSMs, including TaAs, are weakly correlated—interactions with other particles are weak, so each electron can be treated as a single noninteracting particle. However, in compounds containing transition-metal or rare-earth elements, such as ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$, the electrons are strongly correlated, so standard analysis tools do not work. We use an advanced computational technique known as the LDA+Gutzwiller method to reveal that ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$ contains both type I and type II Weyl points, descriptions of how energy and momentum of electrons relate to one another. This makes ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$ a new type of material, a heavy Weyl fermion state.

Properties of heavy Weyl fermion states such as conductivity, optical response, and positions of the Weyl points can be finely tuned by varying temperature, pressure, electromagnetic fields, and other environmental parameters. Future experiments with ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$ might not only reveal such tunability but also lead to new phenomena.

Figure 1

Crystal structure and Brillouin zone (BZ). (a) The crystal symmetry of ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$. (b) The bulk BZ and the projected surface BZ for both (001) and (010) surfaces.

Figure 2

(a) The band structure of ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$ calculated by $\mathrm{GGA}+\mathrm{SOC}$ (solid line) and $\mathrm{Wannier}+\mathrm{SOC}$ (dashed line). (b) The renormalized band structure of ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$ calculated by the $\mathrm{LDA}+\mathrm{Gutzwiller}$ method. The Fermi energy is set to 0 eV. The gap of the bands in the inset is about 1.5 meV, which indicates that a zero gap Weyl point exists nearby.

Figure 3

(a) The energy dispersion along the direction connecting $W_1$ and $W_2$. (b) Three-dimensional view of the eight pairs of WPs in the BZ. (c) Top view from [001] and (d) side view from [010] directions for the WPs. The hollow and solid dots represent WPs in the $W_1$ and $W_2$ groups, respectively. (e) The distribution of Berry curvature for the $k_z=0.139{\AA }^{-1}$ plane. (f) The distribution of Berry curvature for the $k_z=0.227{\AA }^{-1}$ plane. The black and red dots denote the Weyl points with negative and positive chirality, respectively.

Figure 4

A pair of Weyl points with opposite chirality projected onto the surface of the Brillouin zone along one direction and connected by the Fermi arc. If the Fermi surface of the bulk state and the Fermi arc are projected together, the Fermi arc will connect two bulk Fermi surfaces, each of which covers a Weyl point.

Figure 5

Fermi surface on (a) top and (b) bottom (001) surfaces of the sample. All the dots represent two WPs with the same chirality projecting on top of each other. (c,d) Fermi surface on the top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (e) and (g) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (f) and (h) are the same as (e) and (g) but for the bottom surface.

Figure 6

(a) The occupation of $f$ orbitals with $J=5/2$, $J_z=\pm 1/2$, $\pm 3/2$, $\pm 5/2$ characters; (b) the quasiparticle weights for the $4f$ orbitals with $J=5/2$; (c) phase diagram of ${\mathrm{CeRu}}_4{\mathrm{Sn}}_6$, with a change of the total $4f$ orbital occupation number. The violet and green regions are semiconductor (SC) and semimetal (SM) phases. The blue and yellow regions are WSM phases with 8 and 12 pairs of Weyl points, respectively. (d) The histogram of the atomic configurations for the case of $n_f=0.9$.

Figure 7

Two different cleavage surface layers in the (010) direction enclosed by the green and red dashed lines.

Figure 8

Fermi surface of the second cleaved cell and the identification of the chirality of the surface state. (a,b) Fermi surface on top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (c) and (e) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (d) and (f) are the same as (c) and (e) but for the bottom surface.

Figure 9

Surface state of the first cleaved cell with a surface potential energy of 0.4 eV. (a,b) Fermi surface on the top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (c) and (e) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (d) and (f) are the same as (c) and (e) but for the bottom surface.

Figure 10

Surface state of the first cleaved cell with surface potential energy of $-0.4\mathrm{eV}$. (a,b) Fermi surface on the top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (c) and (e) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (d) and (f) are the same as (c) and (e) but for the bottom surface.

Figure 11

Surface state of the second cleaved cell with a surface potential energy of 0.4 eV. (a,b) Fermi surface on the top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (c) and (e) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (d) and (f) are the same as (c) and (e) but for the bottom surface.

Figure 12

Surface state of the second cleaved cell with a surface potential energy of $-0.4\mathrm{eV}$. (a,b) Fermi surface on the top and bottom (010) surfaces, respectively. The small dots represent the single projected WP, and the large dots represent two WPs with the same chirality projecting on top of each other. Panels (c) and (e) are the surface states of the top surface along the closed paths of E-F-B-A-E and A-B-C-D-A, respectively. Panels (d) and (f) are the same as (c) and (e) but for the bottom surface.