Large Fermi Arcs in Unconventional Weyl Semimetal RhSi

TL;DR

This paper identifies RhSi as an ideal material for studying unconventional chiral fermions, revealing large Fermi arcs and unique topological features that enable exploration of exotic transport phenomena.

Contribution

It proposes RhSi as a prime candidate for experimental investigation of unconventional chiral fermions with large topological features and novel surface states.

Findings

RhSi hosts a chiral double six-fold-degenerate Weyl node at R.

RhSi contains a four-fold-degenerate chiral fermion at Γ.

RhSi exhibits the largest momentum separation and Fermi arcs among known materials.

Abstract

The theoretical proposal of chiral fermions in topological semimetals has led to a significant effort towards their experimental realization. In particular, the Fermi surfaces of chiral semimetals carry quantized Chern numbers, making them an attractive platform for the observation of exotic transport and optical phenomena. While the simplest example of a chiral fermion in condensed matter is a conventional $|C|=1$ Weyl fermion, recent theoretical works have proposed a number of unconventional chiral fermions beyond the Standard Model which are protected by unique combinations of topology and crystalline symmetries. However, materials candidates for experimentally probing the transport and response signatures of these unconventional fermions have thus far remained elusive. In this paper, we propose the RhSi family in space group (SG) $\#$198 as the ideal platform for the experimental examination of unconventional chiral fermions. We find that RhSi is a filling-enforced semimetal that features near its Fermi surface a chiral double six-fold-degenerate spin-1 Weyl node at $R$ and a previously uncharacterized four-fold-degenerate chiral fermion at $\Gamma$. Each unconventional fermion displays Chern number $\pm4$ at the Fermi level. We also show that RhSi displays the largest possible momentum separation of compensative chiral fermions, the largest proposed topologically nontrivial energy window, and the longest possible Fermi arcs on its surface. We conclude by proposing signatures of an exotic bulk photogalvanic response in RhSi.

Tables

Table 1. Table 1: Comparison between RhSi and previous works

Materials	$k$-separation (${\text{Å}}^{-1}$)	Number	Charge	Topological Energy Window	$\Delta E$ (eV)
RhSi	$\sim$2.33	2	4	$\sim$1.2eV; nontrivial at EF=0	$\sim$0.41
TaAs Huang2015 ; Weng2015	$\sim$0.07	24	1	$\sim$0.08eV; nontrivial at EF=0	0
TaP Huang2015 ; Weng2015	$\sim$0.06	24	1	$\sim$0.07eV; nontrivial at EF=0	0
NbAs Huang2015 ; Weng2015	$\sim$0.02	24	1	$\sim$0.02eV; nontrivial at EF=0	0
NbP Huang2015 ; Weng2015	$\sim$0.02	24	1	$\sim$0.02eV; Trivial at EF=0	0
WTe2 type2Weyl1	0.01	8	1	$\sim$0.01eV; Trivial at EF=0	0
LaAlGe LaAlGe	0.01	40	1	nontrivial at EF=0	0

Equations

$$\displaystyle s_{2x}=\left\{C_{2x}\bigg{|}\frac{1}{2}\frac{1}{2}0\right\},\$$

$$\displaystyle s_{2z}=\left\{C_{2z}\bigg{|}\frac{1}{2}0\frac{1}{2}\right\},\$$

$$\frac{dj}{dt}=\frac{2I{\beta }_0}{c{ϵ}_0}C,{\beta }_0=\frac{\pi e^3}{h^2},$$

$$P_S=\sqrt{\sum _{i=x,y,z}⟨S_i{⟩}^2}.$$

$$\displaystyle s_{2x}=\left\{C_{2x}\bigg{|}\frac{1}{2}\frac{1}{2}0\right\},\$$

$$\displaystyle s_{2z}=\left\{C_{2z}\bigg{|}\frac{1}{2}0\frac{1}{2}\right\},\$$

$$s_{2x}$$

$$s_{2z}$$

$$\tilde{\mathcal{T}}$$

$$k_x$$

$${\sigma }^x$$

$${\tau }^x$$

$${\tau }^y$$

$${\tau }^y{\mu }^z$$

$${\tau }^x{\mu }^z$$

$${\mathcal{H}}_{198}(\vec{k})=\tilde{\mathcal{H}}(\vec{k}{)}_{198}+\sum _{i=1,2,3}{V}_{r,i}(\vec{k})+V_{s,i}(\vec{k})$$

$${\tilde{\mathcal{H}}}_{198}(\vec{k})$$

$$V_{r1}(\vec{k})$$

$$V_{r2}(\vec{k})$$

$$V_{r3}(\vec{k})$$

$$V_{s1}(\vec{k})$$

$$V_{s2}(\vec{k})$$

$$V_{s3}(\vec{k})$$

$${\tilde{s}}_{2x}=i{\tau }^y{\mu }^z,{\tilde{s}}_{2z}=i{\mu }^y,\tilde{\mathcal{T}}=K,$$

$$\{{\tilde{s}}_{2i},{\tilde{s}}_{2j}\}=-2{\delta }_{ij},$$

$${\tilde{\mathcal{H}}}_{198,R}=v_R\left({\tau }^y{\mu }^z{k}_x+{\tau }^y{\mu }^x{k}_y+{\mu }^y{k}_z\right)$$

$$s_{2i}={\tilde{s}}_{2i}\otimes i{\sigma }^i,\mathcal{T}=i{\sigma }^{y}K.$$

$$[s_{2i},s_{2j}]=0$$

$$V_{m,R}=m_{s,R}\left[{\tau }^y{\mu }^z{\sigma }^x+{\tau }^y{\mu }^x{\sigma }^y+{\mu }^y{\sigma }^z\right]$$

$${\mathcal{H}}_{198,R}(\vec{k})$$

$$V_{v,R}(\vec{k})$$

$$\displaystyle\tilde{C}_{3,111}=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&1\\ 1&0&0\end{array}\right),\ \tilde{s}_{2x}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&-1\end{array}\right),\tilde{s}_{2z}=\left(\begin{array}[]{ccc}-1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right),$$

$$[{\tilde{s}}_{2i},{\tilde{s}}_{2j}]=0,$$

$${\tilde{\mathcal{H}}}_{198,\Gamma }(\vec{k})$$

$$V_{m,\Gamma }$$

$${\mathcal{H}}_{198,\Gamma }(\vec{k})={\tilde{\mathcal{H}}}_{198,\Gamma }(\vec{k})+V_{m,\Gamma }+V_{v,\Gamma }(\vec{k}),$$

$$V_{v,\Gamma }(\vec{k})$$

$$s_{2i}=i{\sigma }^i,C_{3,111}=-(i{\tau }^y{)}^{2/3}(i{\sigma }^{111}{)}^{2/3},\mathcal{T}=i{\sigma }^{y}K,$$

Full text

The allowed band crossings in condensed matter have, until recently, been considered closely linked to elementary particles in high-energy physics Weyl ; Wilczek ; Volovik2003 ; NewFermion . In 3D systems without spatial inversion ($\mathcal{I}$) or time-reversal ($\mathcal{T}$) symmetry, two-fold-degenerate band crossings are permitted, resulting in condensed matter realizations of Weyl fermions with quantized Chern numbers Weyl ; Wilczek ; Volovik2003 ; NewFermion ; Kane05p226801 ; Kane05p146802 ; Hasan2010 ; Qi2011 ; BernevigHgTe ; RevBansil ; Hasan_TaAs ; TaAs_Ding ; MIT_Weyl ; Wan2011 ; Burkov2011 ; Murakami2007 ; Huang2015 ; Weng2015 ; type2Weyl1 ; type2Weyl2 ; LaAlGe ; KramersWeyl . Since the experimental realization of the Weyl semimetal state in TaAs Hasan_TaAs ; TaAs_Ding , recent theoretical efforts have become focused on finding unconventional condensed matter quasiparticle excitations beyond the Dirac and Weyl paradigm described by the Standard Model DoubleDirac ; NewFermion ; trip1 ; trip2 ; Nexus ; Phonon . These efforts have rapidly expanded the set of known nodal features, which now additionally include symmorphic three-fold nexus fermions trip1 ; trip2 ; Nexus , eight-fold-degenerate double Dirac fermions DoubleDirac , and, as detailed by Bradlyn, Cano, Wang, et al. (BCW) in Ref. NewFermion, , three-fold-degenerate single and six-fold-degenerate double spin-1 Weyl points. Unconventional chiral fermions, in particular, hold great promise for experimental applications, as they broaden beyond conventional Weyl semimetals the search for materials candidates for the observation of topological surface states, bulk chiral transport, and exotic circular photogalvanic effects MMonFS ; chiral_mag ; photocurrentweyl ; photoTaAs ; photogalvanic ; arc1 ; arc2 ; Riemann .

Although the fundamental theory for these unconventional fermions has been established, one outstanding issue has been the relative lack of ideal material candidates for their experimental examination. In the band structures of previously proposed materials, the unconventional fermions have typically sat away from the Fermi energy, or have in the cases of unconventional chiral fermions coexisted with additional, trivial bands. In these systems, while the unconventional fermions may be experimentally observable by photoemission, their topological properties are still prohibitively difficult to detect and utilize for transport and optical response. For example in MoP, a three-fold nexus fermion is observed 1 eV below the Fermi level, but the Fermi surface itself is unrelated to the three-fold fermion and carries no net Chern number Nexus_exp .

In this paper, we identify the RhSi materials family of structurally chiral cubic crystals in space group (SG) 198 $P2_{1}3$ RhSi (Fig. 1(a)) as the first ideal materials candidates for the experimental study of the novel transport and response effects of unconventional chiral fermions. Using first-principles calculations detailed in Section A of the Supplemental Material (SM A), we find that the Fermi surface of RhSi consists of only two well-isolated pieces which carry equal and opposite quantized Chern number. The bulk bands near the Fermi energy feature a chiral six-fold-degenerate double spin-1 Weyl at the Brillouin zone (BZ) corner $R$ and a previously uncharacterized four-fold-degenerate chiral fermion at the zone center $\Gamma$ (Fig. 1(c,d), Fig. 2(a,b)). RhSi therefore displays the largest possible separation of chiral fermions allowed in crystals. With an otherwise large bandgap, RhSi also therefore features the largest topologically nontrivial energy window proposed thus far (Fig. 1(d) and SM E). Furthermore, as these two chiral fermions lie at time-reversal-invariant momenta (TRIMs), they are unrelated by symmetry and free to exhibit an energy offset; here, the four-fold fermion at $\Gamma$ lies roughly 400 meV above the six-fold fermion at $R$. This offset allows for the possibility of unique optical transport, such as the quantized circular photogalvanic effect photogalvanic . Among all known chiral semimetals, both conventional Weyl and unconventional higher-fold fermion, RhSi therefore stands as possibly the most electronically ideal material yet proposed.

To understand the unusual high-fold-degenerate nodes displayed in the minimal band connectivity of SG 198, we construct an eight-band tight-binding (TB) model (SM C). SG 198 is characterized by three nonintersecting two-fold screw rotations $s_{2x,y,z}$, related by diagonal cubic three-fold rotation $C_{3,111}$ BigBook :

[TABLE]

Without the three-fold rotation, this combination of screws and $\mathcal{T}$-symmetry characterizes orthorhombic SG 19, and has been shown to force groups of eight or more bands to tangle together fillingconstraint ; Layer_group ; QuantumChemistry ; Adrian19 ; BalatskySG19 . The additional cubic three-fold rotation $C_{3,111}$ in SG 198 serves to increase the band degeneracy at TRIMs while still preserving this eight-band connectivity. At an electron filling of $\nu\in 8\mathbb{Z}+4$, RhSi is gapless due to the combination of time-reversal and nonsymmorphic symmetries, and is therefore a “filling-enforced” semimetal (SM B) fillingconstraint ; Layer_group ; magDirac2 . We find that our minimal TB model of SG 198 captures all of the degeneracy structure and topological character of RhSi. We describe our results for the full BZ in detail in SM C, and here focus on the chiral node structure at $\Gamma$ and $R$.

We begin by examining the band splitting and previously uncharacterized four-fold-degenerate unconventional chiral fermion at $\Gamma$. In the absence of SOC, our eight-band model permits only a single mass term at $\Gamma$ which splits bands into a $3\times 2$-fold-degenerate fermion and a doubly degenerate quadratic band, which in RhSi lies more than 2 eV above the Fermi energy (Fig. 1(c)). Upon the introduction of SOC, this quadratic crossing opens into a Kramers Weyl KramersWeyl , and the $3\times 2$-fold-degenerate node splits into a four-fold-degenerate unconventional fermion and a second Kramers Weyl (Fig. 2(a,d)). This four-fold-degenerate fermion is distinct from the spin-3/2 chiral fermion introduced in Ref. NewFermion, : whereas that fermion is described by a corepresentation equivalent to the four-dimensional irreducible representation $\bar{F}$ of chiral point group $432$ ($O$), the four-fold-degenerate fermion in RhSi is described by the $\mathcal{T}$-symmetric corepresentation formed by pairing the two-dimensional irreducible representations ${}^{1}\bar{F}$ and ${}^{2}\bar{F}$ of chiral point group $23$ ($T$) BigBook . In the language of atomic orbitals, this four-fold degeneracy can be understood by modeling the six degenerate states without SOC by three $p$ orbitals and an electron spin in the $111$ direction. Calling $z^{\prime}$ the $111$ direction and $x^{\prime},y^{\prime}$ as orthonormal axes spanning the plane normal to $z^{\prime}$, we group the $p$ orbitals into a $p_{z^{\prime}},\ m_{l}=0$ orbital and $p_{x^{\prime}}\pm ip_{y^{\prime}},\ m_{l}=\pm 1$ orbitals. When coupled to the spin-$1/2$ electron, the six total states split into four $J=1/2$ and two $J=3/2$ states. Time-reversal pairs states with the same $J$ and opposite $m_{j}$, and $s_{2x}$ flips $m_{s}$ without affecting $m_{l}$, such that under the SG 198 generators two $J=3/2$ states pair with two $J=1/2$ states and the remaining two $J=1/2$ states split off and form the second Kramers Weyl (SM C.2). By numerically calculating the eigenvalues of $C_{3,111}$ and considering the symmetry-allowed term $\vec{k}\cdot\vec{J}$, each band near $\Gamma$ can be assigned $J$ and $m_{j}$ eigenvalues, a structure we confirm explicitly with a symmetry-generated four-band $k\cdot p$ model in SM C.2. As the irreducible representations at $\Gamma$ are reflective of the position-space atomic orbitals, this analogy should also provide physical insight into the bonding character of RhSi QuantumChemistry . By the integrating the Berry curvature between bands with $J=1/2,\ m_{j}=\pm 1/2$ over a k-space sphere in the vicinity of $\Gamma$ Volovik2003 , we find that this unconventional fermion exhibits Chern number $+4$ at the Fermi level in RhSi (Fig. 2(d)).

Our examinations of the unconventional fermions at $R$ with and without SOC (SM C.2) confirm the results of previous analyses of SGs 19 and 198 Adrian19 ; BalatskySG19 ; DoubleDirac ; Layer_group ; NewFermion . When SOC is taken into consideration, RhSi displays at $R$ a six-fold-degenerate chiral double spin-1 Weyl at $\sim 0.4eV$ below the Fermi level, which at the finite-$q$ gap spanned by the Fermi energy exhibits Chern number $-4$ (Fig. 2(b,e)). Projecting out of our TB model the six-band subspace of this chiral fermion results in a $k\cdot p$ theory related by a unitary transformation to that presented in Ref. NewFermion, of two coupled spin-1 fermions with individual $\vec{k}\cdot\vec{S}$ dispersion.

Calculating the surface states of RhSi (Fig. 3(a)) through surface Green’s functions (SM A), we find that the $(001)$-surface displays four topological Fermi arcs connecting the projections of the bulk chiral fermions at $\bar{\Gamma}$ to those at $\bar{M}$ across the entire surface BZ. Unlike the recently observed trivial arcs in WTe2  trivialarc ; LaAlGe , the long Fermi arcs in RhSi are guaranteed by bulk topology, and should therefore be robust against changes in surface chemical potential and disorder (SM E). Though the arcs in our calculations demonstrate a particularly elaborate connectivity (Fig. 3(c)), a much simpler direct connectivity is also allowed (Fig. 3(d)). We also find the Fermi arcs to have $\sim 80\%$ spin polarization arcDetect2 (Fig. 3(b)). Therefore, RhSi is also an attractive platform for spintronic applications spintronics1 ; spintronics2 .

To summarize our analysis of the electronic structure of RhSi, we find that it is a remarkably ideal candidate for the observation of chiral transport and optical phenomena and for the direct examination of unconventional fermions. Bands within the $k\cdot p$ regime of the unconventional fermions at $\Gamma$ and $R$ cleanly characterize the entire Fermi surface, such that the separation between Fermi pockets of opposite Chern number is the entire length of the 3D diagonal of the BZ cube. The remaining bulk band manifolds are otherwise separated by a gap of $\sim$ 1.2 eV (Fig. 1(d)), such that RhSi has by far the largest topologically nontrivial energy window of any previously proposed or experimentally realized chiral semimetal (SM E). RhSi also therefore displays on its surface topologically-guaranteed Fermi arcs that span the entire surface BZ, and uniquely come in time-reversed pairs (Fig. 3). Finally, unlike in previous band-inversion Weyl semimetals where pairs of Weyl points have been related by mirror symmetry, the chiral fermions in RhSi are free to sit with an energy offset, enabling chiral photogalvanic transport photogalvanic .

We therefore conclude with a numerical prediction of quantized optical transport in RhSi. In Ref. photogalvanic, , the authors show that in a structurally chiral system for which only a single two-band Weyl fermion is partially unoccupied, such as a Kramers Weyl metal KramersWeyl , the difference in the rate of current density resulting from exciting electrons with left- and right-handed circularly polarized light is quantized in terms of fundamental constants:

[TABLE]

where $I$ is the intensity of applied light and $C$ is the Chern number of the Weyl point (Fig. 4(a,b)). In RhSi, the four-fold fermion at $\Gamma$ sits just above the Fermi energy while the chiral double spin-1 Weyl at $R$ sits below and is fully occupied; the location of the chiral fermions in its band structure (Fig. 1(d)) is practically identical to the ideal case proposed in Ref. photogalvanic, . We observe that the angular momentum selection rules for circularly polarized light appear to strongly constrain the allowed transitions in this four-fold fermion, such that only transitions between bands with $\Delta m_{j}=\pm 1$ contribute to the photocurrent selectionrules . Therefore, when weighting by Fermi occupation factors, the photocurrent rate calculated from the trace of the gyrotropic tensor (SM D), though initially fluctuating, still saturates at the quantized value $(2I\beta_{0}/c\epsilon_{0})\times 4$ with increasing incident photon energy $E_{p}$ in the vicinity of $\Gamma$, or four times the value predicted for a conventional Weyl fermion (Fig. 4(b,d)). Therefore, despite the multiband complexities of its unconventional chiral fermions, RhSi remains a plausible candidate for probing the quantized photogalvanic effect.

I Note Added

We notice a related work that reports similar chiral fermions CoSi_zhang .

I.1 SM A. Density Functional Theoretic Methods

First-principles calculations were performed within the density functional theory (DFT) framework using the OPENMX package and the full-potential augmented plane-wave method as implemented in the package WIEN2k openmx1 ; openmx2 ; DFT2 ; DFT3 . The generalized gradient approximation (GGA) DFT4 was used. Experimentally measured lattice constants were used in DFT calculations of material band structures CoSi ; CoGe ; RhSi ; RhGe ; AlPd ; AlPt ; GaPt ; GaPd ; FeSi ; BaPtP . A $\Gamma$-centered $k$-point $10\times 10\times 10$ mesh was used and spin-orbit coupling (SOC) was included in self-consistent cycles.

To generate the $(001)$-surface states of RhSi, Wannier functions were generated using the $p$ orbitals of Si and the $d$ orbitals of Rh. The surface states were calculated for a semi-infinite slab by the iterative Green’s function method. The spin polarization was calculated by the method of Ref. arcDetect2, , in which the authors calculate the summed expectation values squared of spin $S_{i}$ in three orthogonal directions $i=x,y,z$:

[TABLE]

I.2 SM B. Additional Materials Candidates in Space Group 198

Although unconventional fourfold-degenerate fermions at $\Gamma$ and sixfold-degenerate double spin-1 Weyl points at $R$ are quite common in space group (SG) 198, due to being featured in the minimal band connectivity (SM C.2) NewFermion ; QuantumChemistry , not every material featuring them is ideal for experimental applications. Specifically, both for the measurement of bulk transport effects, such as the quantized photogalvanic effect discussed in SM D, and for the observation of surface Fermi arcs (Fig. 3(a)), a material in SG 198 should have a simple, chiral Fermi surface. The Fermi surfaces of strong materials candidates should therefore intersect the unconventional fermions at $\Gamma$ and $R$ and be absent of electron and hole pockets.

The first criteria can be achieved by selecting materials with the appropriate electron count. As noted in Ref. fillingconstraint, , SG 198 is cubic supergroup of Bieberbach group SG 19, which has a minimal insulating filling of $\nu\in 8\mathbb{Z}$. Furthermore, as shown in Ref. QuantumChemistry, , SG 198 has only a single maximal Wyckoff position ($4a$) and band connectivity ($8$), such that bands in SG 198 can never be forced by representation compatibility to form nonminimal connectivities. Therefore, in the noninteracting limit, crystals in SG 198 with electron counts other than $\nu\in 8\mathbb{Z}$ (such as those, in particular, with $\nu\in 8\mathbb{Z}+4$) will be metals or semimetals Layer_group . In Fig. 5, we artificially adjust the chemical potential in RhSi as if it had an electron count of $\nu\in 8\mathbb{Z}$, and demonstrate that the resulting Fermi surface is insulating and topologically trivial (Fig. 5(a-d)). As an example of a material in SG 198 that features a trivial gap at the Fermi energy, we present FeSi FeSi (Fig. 5(e,f)), which due to its electron count hosts unconventional chiral fermions that fail to contribute to the low-energy physics.

The second criteria, that the Fermi surface be free of electron and hole pockets, is necessary to observe surface Fermi arcs. More specifically, if there is no projected gap at a given surface, Fermi arcs cannot be resolved. As discussed in the main text and in more detail in Ref. NewFermion, , a fully isolated N-fold-degenerate chiral fermion can have up to N-1 band gaps, each of which can host its own nontrivial Chern number. However, not all gaps correspond to observable surface states; in many cases, the projections of bulk bands cover possible surface gaps and obscure topological Fermi arcs. Specifically, when bands disperse from a chiral fermion in opposite directions, such as in a conventional $|C|=1$ Weyl that isn’t tilted, a projected bulk gap is allowed (Fig. 6(a-c)). However, in cases like bands split by Rashba coupling (Fig. 6(d-f)), all surface gaps will be filled by projected bulk bands and Fermi arcs will not be visible.

We have identified several materials candidates in SG 198 that satisfy the above conditions for experimental viability. In the main text, we highlight RhSi RhSi , which we found in our materials search to be the strongest candidate for observing large Fermi arcs and bulk photogalvanic effects (SM D). There are also several other promising materials candidates isostructural to RhSi, which include CoSi CoSi , CoGe CoGe , RhGe RhGe , AlPd AlPd , AlPt AlPt , GaPd GaPd , and GaPt GaPt . Of these materials, CoSi, AlPd, and AlPt were previously identified as hosting sixfold-degenerate unconventional fermions in Ref. NewFermion, . In Fig. 6, we plot band structures calculated through DFT for CoSi (a), CoGe (b), RhSi (c), RhGe (d), AlPd (e), and AlPt (f). We additionally identified promising materials candidates in the BaPtX (X=P,As,Sb) family of materials BaPtP , for which the unit cell structure and bands calculated through DFT are plotted in Fig. 7.

I.3 SM C. Tight-Binding Model

I.3.1 SM C.1 Symmetry-Generation of Hopping Terms

To construct a tight-binding model for SG 198, we begin by choosing a four-site unit cell for which each site carries only an $s$ orbital. Confirming the results of Ref. QuantumChemistry, for SG 198, we find that when SOC is on, the minimal band connectivity required to capture the insulating filling constraint $\nu\in 8\mathbb{Z}$ fillingconstraint is indeed sufficiently captured by placing $s_{1/2}$ orbitals at Wyckoff positions with the minimum multiplicity of $4$. Starting with a site designated $A$ at unit cell position $(0,0,0)$ in units of the orthogonal lattice constants $a_{x,y,z}$, we utilize an allowed permutation of the group generators as listed in Bradley and Cracknell BigBook :

[TABLE]

These generators, three nonintersecting screws related by cubic threefold rotation, generate three additional sites in the unit cell: $B$ located at $(\frac{1}{2},\frac{1}{2},0)$, $C$ located at $(\frac{1}{2},0,\frac{1}{2})$, and $D$ located at $(0,\frac{1}{2},\frac{1}{2})$. Graphically, the four sites comprising this model sit at the four alternating corners of a cube. A crucial realization here is that there are two distinct $C_{3}$ axes; each site is internally $C_{3}$-symmetric but is cyclically exchanged with two other sites by the $C_{3}$ of the fourth site. More concretely, under $C_{3,111}$ for example, the $A$ site is left invariant while the other three sites are cyclically rotated $B\rightarrow D\rightarrow C\rightarrow B$. From the $k\cdot p$ results in the next section (SM C.2), we find that this relationship between this cyclic operation and the minimum four-site construction of SG 198 is in fact the reason why its minimum band connectivity displays sixfold-degenerate unconventional fermions.

To construct a concrete model, we employ the method of Ref. Layer_group, . We construct the representations of the group generators at $\Gamma$ which, combined with the mappings they perform on the crystal momenta $\vec{k}$, allow us to enumerate all symmetry-allowed hopping terms up to a specified range in position space. Written in the mixed notation of the form of the unitary operation that transforms the sublattices Kronecker multiplied by the result of the $k$-space mapping under the operation, the three screw generators and spinful time-reversal $\mathcal{T}$ take the form:

[TABLE]

where $\tau^{x}$ is $s$-orbital-like hopping between the $A$ and $B$ and the $C$ and $D$ sites, $\mu^{x}$ is $s$-orbital-like hopping between the $A$ and $C$ and $B$ and $D$ sites, and $\sigma$ is the on-site $s_{1/2}$ spinorbital. The more complicated threefold diagonal rotation $C_{3,111}$ takes the form of a cyclic operation that enforces the mappings:

[TABLE]

where in this notation $\tau^{i}\mu^{j}\sigma^{k}$ stands for the Kronecker product $s^{i}\otimes s^{j}\otimes s^{j}$ of Pauli matrices $s^{x,y,z}$ and the $2\times 2$ identity $s^{0}$, and the identity terms $s^{0}$ are suppressed.

We enumerate symmetry-allowed hopping terms up to first-nearest-neighbor interactions, finding that there are $2$ SOC-free terms and $6$ terms with SOC. The overall tight-binding Hamiltonian $\mathcal{H}_{198}(\vec{k})$ can be expressed as the sum of its spinless part $\tilde{\mathcal{H}}_{198}(\vec{k})$ and the spin-orbit terms $V_{s/r,i}(\vec{k})$:

[TABLE]

where

[TABLE]

and

[TABLE]

I.3.2 SM C.2 $k\cdot p$ Theory of $R$ and $\Gamma$ Without and With SOC

To develop $k\cdot p$ models of the unconventional fermions at $R$ and $\Gamma$, we begin by keeping the full eight-band matrix structure at each TRIM from the tight-binding model in SM C.1 and note when algebra allows for higher degeneracy. For threefold- and sixfold-degenerate fermions, this approach is superior to projecting out three- and six-dimensional subspaces, as doing so removes our ability to exploit the Clifford algebra of $8\times 8$ matrices to prove that no other gaps may be opened at either TRIM. For the fourfold-degenerate unconventional fermion at $\Gamma$, we additionally exploit the fact that the Little group at $\Gamma$ is isomorphic to chiral point group $23$ ($T$) and use an understanding of the corepresentations of this point group to form a four-band $k\cdot p$ theory KramersWeyl ; BigBook . Therefore, procedurally, we begin by expanding the tight-binding model around each TRIM. We then use the eight-band forms of the group generators at the TRIMs to exhaustively check whether any additional mass terms are allowed. As we show below, all of the gaps at $\Gamma$ and $R$ are fully opened just by considering the symmetry-allowed first-nearest-neighbor hopping terms from SM C.1.

We begin at $R$ ($\vec{k}=(\pi,\pi,\pi)$). When SOC is weak, the system gains an additional $SU(2)$ invariance and a second set of spinless symmetries are additionally enforced, which we denote with tildes. At $R$ under weak SOC, the spinless screws and time-reversal take the forms:

[TABLE]

the screws mutually anticommute:

[TABLE]

and the spinless threefold rotation $\tilde{C}_{3,111}$ exchanges the screws by the same cyclic relations listed above. As the screws and $\tilde{\mathcal{T}}$, the generators of SG 19 BigBook , themselves enforce a $4\times 2$-fold degeneracy through this algebra Adrian19 ; BalatskySG19 ; Layer_group , our eight-band model of SG 198 also displays a spinless eightfold-degenerate chiral fermion DoubleDirac ; BalatskySG19 at $R$ under vanishing SOC:

[TABLE]

where minus signs have been suppressed in expanding $\tilde{\mathcal{H}}_{198}(R)$.

Introducing SOC, we break $SU(2)$ invariance and are left at $R$ with spinful screws and time-reversal:

[TABLE]

The spinful threefold rotation $C_{3,111}$ is the same as the spinless operation $\tilde{C}_{3,111}$ times the additional cyclic rotation of the Pauli matrices described in Eq. (6). These spinful screws now commute instead of anticommute:

[TABLE]

and all square to $+1$. Under these conditions, our minimal eight-band model admits just a single mass term at $R$:

[TABLE]

which derives from $V_{r,3}(R)$. Though many linear terms are allowed near $R$, we can use our tight-binding model to constrain the choices to those originating from first-nearest-neighbor hopping:

[TABLE]

This $k\cdot p$ Hamiltonian describes a sixfold-degenerate chiral double spin-1 Weyl point and a twofold-degenerate quadratic band. These two band features can be considered as the minimal degeneracy required by the symmetry algebra (twofold degeneracy by Kramers theorem) and a higher-fold degeneracy permitted to be gapless by the algebra. Specifically, if one represents the spinless parts of the screws and threefold rotation using $3\times 3$ matrices in terms of coordinate transformations (or as if they acted on $p$ orbitals QuantumChemistry ):

[TABLE]

one finds that a threefold degeneracy is already required. The addition of spinful time-reversal $\mathcal{T}$ doubles this degeneracy, resulting in the algebraic description of a double spin-1 Weyl point. This confirms the results of Ref. NewFermion, , in which the authors conclude that the sixfold-degenerate spinful chiral fermion at $R$ in SG 198 can be described by coupling two threefold-degenerate fermions that individually have Hamiltonians of the form $\vec{k}\cdot\vec{S}$.

At $\Gamma$, the picture without SOC is nearly identical to that at $R$ with SOC. The spinless screws are cyclically exchanged under spinless threefold rotation $\tilde{C}_{3,111}$, mutually commute:

[TABLE]

and square to $+1$. This algebra can be satisfied with a scalar (a $1\times 1$ matrix), or with $3\times 3$ matrices of the form of Eq. (26), locally protecting $1\times 2$- and $3\times 2$-fold degeneracies, respectively. We find that our minimal eight-band model with weak SOC permits just a single mass term at $\Gamma$ such that the spinless $k\cdot p$ Hamiltonian takes the form:

[TABLE]

where $m_{\Gamma}$ originates from $v_{1}(\Gamma)$. The band structure, analogously to that $R$ under strong SOC, features a twofold-degenerate quadratic band and a spinless double spin-1 Weyl point NewFermion .

When SOC is introduced at $\Gamma$, the band features and gaps are significantly altered. The twofold-degenerate quadratic band opens up into a Kramers Weyl fermion KramersWeyl and two more mass terms are allowed:

[TABLE]

which originate from $V_{r1}(\Gamma)$ and $V_{r2}(\Gamma)$, respectively. Again choosing linear terms originating from the first-nearest-neighbor tight-binding model expanded around $\Gamma$, we develop a $k\cdot p$ Hamiltonian for all eight bands:

[TABLE]

where

[TABLE]

This $k\cdot p$ Hamiltonian describes two, twofold-degenerate Kramers Weyl fermions KramersWeyl split from a previously uncharacterized fourfold-degenerate chiral unconventional fermion. The Kramers Weyl points represent the minimum twofold degeneracy necessary to support the generator algebra at this point Layer_group ; KramersWeyl , and the fourfold degeneracy represents a subtle gapless $4\times 4$ algebra which is also permitted. To understand this fourfold degeneracy, we make the analogy to first describing the weak-SOC spinless double spin-1 Weyl point at $\Gamma$ by three $p$ orbitals and an uncoupled electron spin. We choose the $p$ orbitals to be oriented along the $111$ direction, and as such define new axes $z^{\prime}$ along the $111$ direction and $x^{\prime}$ and $y^{\prime}$ spanning the plane normal to $z^{\prime}$. Expressing the three $p$ orbitals as linear combinations with distinct angular momentum eigenvalues $(l,m_{l})$, we arrive at an $l=0,m_{l}=0,\ p_{z^{\prime}}$ orbital and $l=1,m_{l}=\pm 1,\ p_{x^{\prime}}\pm ip_{y^{\prime}}$ orbitals. When coupled to the electron spin, which we can choose to also be oriented along $z^{\prime}$, only total angular momentum $J$ and its components along the $z^{\prime}$ direction $m_{j}$ are good quantum numbers. We arrive instead at only a quartet of states related by the spinful symmetry generators: $J=3/2,m_{j}=\pm 3/2$ and $J=1/2,m_{j}=\pm 1/2$ related within pairs by $\mathcal{T}$ and related between pairs by $s_{2x}$. As it is along the $z^{\prime}$ direction, $C_{3,111}$ leaves the states invariant and its eigenvalues can be used to deduce $J$. We can choose a basis in which $s_{2x}$ only acts on the electron spin component in the $z^{\prime}$ direction $m_{s}$; under this choice $C_{3,111}$ takes the form of its spinless eigenvalues times a spin rotation about the $z^{\prime}=111$ axis and the screws just act in spin space. Up to a sign, the rotationally-invariant linear term $\vec{k}\cdot\vec{J}$ splits the states by $m_{j}$ and can be used to label bands in the $111$ direction as pictured in Fig. 2(d).

More concretely, we can exploit the methods of Refs. DoubleDirac, ; NewFermion, ; KramersWeyl, and utilize the group-theoretic description of this point in Ref. BigBook, to form an explicit four-band gapless theory of this previously undescribed fourfold-degenerate fermion. Utilizing the crystalline symmetry textbook Bradley and Cracknell (Ref. BigBook, ) we identify the fourfold-degenerate fermion as a corepresentation of two-dimensional double-valued irreducible representations $5$ and $6$ of character $G^{9}_{24}$. As the Little group of the $\Gamma$ point in a crystal is unaffected by projections of lattice translations, it is isomorphic to a point group KramersWeyl . Here, we find that the Little group at $\Gamma$ is isomorphic to point group $23$ ($T$), and that correspondingly, the fourfold-degenerate fermion is described by the corepresentation of the two-dimensional irreducible representations ${}^{1}\bar{F}_{23}$ and ${}^{2}\bar{F}_{23}$. Using the character table for this point group in Ref. BigBook, , we can form representations of the point group generators in a four-band space:

[TABLE]

where

[TABLE]

In this basis, we find that to quadratic order, the most general symmetry-allowed Hamiltonian takes the form:

[TABLE]

where $\mathds{1}_{\tau\sigma}$ is the $4\times 4$ identity and $\mathcal{H}_{4,\Gamma}(\vec{k})$ has the dispersion relation

[TABLE]

where

[TABLE]

We can make the association that $\tau^{y}$ indexes orbital angular momentum in the $111$-direction $m_{l}$ and that $\sigma$ represents electron spin $\vec{S}$, under which we recapture our earlier orbital-based description of the fourfold-degenerate fermion. When the splitting by $m_{l}$ is tuned to be twice the splitting by $m_{s}$,

[TABLE]

the Hamiltonian $\mathcal{H}_{4,\Gamma}(\vec{k})$ takes the form $\vec{k}\cdot\vec{J}$ and displays a band structure qualitatively similar to that of the $\Gamma$ point in RhSi (Fig. 2(d)).

I.3.3 SM C.3 Fitted Parameters for RhSi

Fitting the simple tight-binding model in Eq. (7) to the bands of RhSi near the Fermi energy, we find remarkably strong agreement given the simplicity of the model. Using the eight parameters in Eq. (7), we are able to reproduce for RhSi all of the gaps and all of the unconventional fermion Chern numbers within those gaps. In order to more adequately capture the energy offset between the fourfold fermion at $\Gamma$ and the double spin-1 Weyl at $R$, we add a ninth parameter, which represents an unsigned second-nearest-neighbor hopping between adjacent unit cells:

[TABLE]

where $\mathds{1}$ is the $8\times 8$ identity. We obtain the parameters $v_{1}=0.55$, $v_{p}=-0.76$, $v_{2}=0.16$, $v_{r1}=0$, $v_{r2}=-0.03$, $v_{r3}=0.01$, $v_{s1}=-0.04$, $v_{s2}=0$ and $v_{s3}=0$ (Fig. 9), which in addition to reproducing all of the topological and band connectivity features of RhSi, also qualitatively describe bands closely resembling those calculated from first-principles DFT (Fig. 1(d)).

I.4 SM D. Photocurrent Calculation Details

To obtain a measure of the circular photogalvanic effect photocurrent rate, we calculate the multiband gyrotropic tensor $\beta_{ij}(\omega)$ as described in Ref. photogalvanic, :

[TABLE]

where $E_{\mathbb{k},nm}$ is the difference between the energies of bands $n$ and $m$ at crystal momentum $\mathbb{k}$, $f^{\mathbb{k}}_{nm}$ is the difference between the Fermi-Dirac distribution functions for bands $n$ and $m$ at $\mathbb{k}$, $r_{\mathbb{k},nm}=i\langle n|\partial_{\mathbb{k}}|m\rangle$ is the cross-gap Berry connection, and $\Delta^{i}_{\mathbb{k},nm}=\partial_{\mathbb{k}_{i}}E_{\mathbb{k},nm}/\hbar$. We then trace over this tensor to find an effective photocurrent rate $dj/dt$; we note that the individual components of the photocurrent rate $dj_{i}/dt$ are still generically anisotropic for the fourfold-degenerate fermion near $\Gamma$ in RhSi. To calculate the rates in Fig. 4(b), we tune to half-filling the tight-binding-derived $k\cdot p$ theories for a Kramers Weyl and for the fourfold unconventional fermion at $\Gamma$ in SG 198 ($t_{i}^{1}=t_{i}^{s}=0.04$ in Ref. KramersWeyl, and Eq. (30) with the fitted tight-binding parameters from SM C.3, respectively). To produce the more realistic traced photocurrent rate in Fig. 4(d), we tune the Fermi energy to be slightly below half-filling in Eq. (30). This rate also appears to saturate in the $k\cdot p$ regime at a value proportional to the Chern number $+4$ characterizing the finite-$q$ gap at the Fermi energy, though the individual components of $\beta_{ij}$ are still quite anisotropic. By roughly calculating the interband matrix elements for circularly polarized light, we confirm that transitions are effectively disallowed between bands unless they are characterized by $\Delta m_{j}=\pm 1$, in accordance with the association of $m_{j}=\pm 1$ for right- or left-handed circularly polarized light selectionrules .

I.5 SM E. Comparison of RhSi With Previous Examples of Weyl Semimetals

In this section, we compare RhSi with previous proposed Weyl semimetal states from the following perspectives: energy window of nontrivial topology, energy offset between chiral fermions, and Weyl point momentum-space separation.

I.5.1 E.1 Topologically Nontrivial Energy Window

Weyl fermions and other topological chiral fermions are local singularities in momentum space that can be regarded as sources or sinks of Berry curvature. The nontrivial topology of these chiral fermions persists only in limited energy windows where Fermi pockets exhibit quantized non-zero chiral charge. We emphasize that the isolation of Fermi pockets of opposite chiral charge is a key prerequisite for observing any topological phenomena associated with chiral fermions (such as the chiral anomaly and the quantized circular photogalvanic current). If the Fermi pockets arising from topological fermions with opposite chiral charges merge at the Fermi level, or if large trivial Fermi pockets with zero Chern number dominate the low-energy physics, then the novel physics of unconventional topological fermions cannot be isolated and studied. In previous works, the nontrivial energy window is closely related to the degree of band inversion arcDetect1 , which in previous works has been quite narrow (around 10-100meV. See Table S1). For example, the topologically nontrivial energy window in NbP is around 20meV Huang2015 ; Weng2015 . The energy dispersion passing through $W_{2}$ in NbP is plotted in Fig. 10(a), and the cartoon of Fermi pockets at different energies are plotted in Fig. 10(b). Within the Lifshitz transitions, the two compensative $W_{2}$ are enclosed in two isolated pockets. In contrast, away from the Lifshitz transitions, the two pockets merge into one trivial pocket. Since the Fermi level of NbP is away from the Lifshitz transitions, NbP is topologically trivial at the Fermi energy. Similarly, as shown in Fig. 10(c), WTe2 is also topological trivial at the Fermi level.

In contrast, RhSi has a much larger topologically nontrivial energy window of around 1.2eV. Figure 11(a) illustrates the chiral charge distribution of RhSi in the Brillouin zone. The Fermi pockets in RhSi at a range of energies are plotted in Figs. 11(c-h), in which the red pockets enclosing $\Gamma$ contain charge $+4$, the blue pockets enclosing $R$ contain charge $-4$, and the black pockets are topologically trivial. The red and blue pockets remain clearly isolated at energies of -0.2eV, 0eV, 0.2eV and 0.5eV (Figs. 11(d-g)), and merge into one trivial pocket at -0.5eV and 0.8eV (Figs. 11(c,h)). This energy window is extremely large, and therefore suggests that the topological properties of RhSi should be quite robust to impurities, implying that RhSi is an ideal platform for topological and chiral experimental applications, even at room temperature.

I.5.2 E.2 Energy Difference between Compensative Charges

All of the previous examples of Weyl semimetals relied on band inversion in crystals with mirror symmetries. Consequently, their Weyl points always had partners related to each other by the mirrors, and they were forced to sit pairwise at the same energies. In RhSi, the chiral fermions sit at the time-reversal-invariant momenta, and therefore are not related to each other by any symmetries. Consequently, they are free to sit at different energies. The cartoon in Figure 12 shows the energy difference between compensative chiral charges in several Weyl semimetals.

In particular, this energy offset has been identified as a necessary ingredient for a range of topological and chiral transport effects. For example, an external magnetic field has been proposed to induce an equilibrium dissipationless current when opposite chiral fermions are at different energies Weylcurrent , with the amplitude of the induced current being proportional to the energy difference between compensative charges. The 0.41eV energy offset between oppositely charged chiral fermions in RhSi renders it an ideal platform for realizing this effect. A chiral-node energy offset also allows the realization of the quantized photogalvanic effect photogalvanic , discussed here in both the conclusion of the main text and in further detail in SM D.

I.5.3 E.3 Momentum-Space Weyl Node Separation and Fermi Arc Length

The separation of compensative Weyl fermions in momentum space is one criterion to evaluate the robustness of Weyl semimetals. In previous works, this separation has been quite small, on the order of $10^{-2}$ $\textrm{\AA}^{-1}$. In contrast, in RhSi, the compensative chiral fermions have a separation of 2.33 $\textrm{\AA}^{-1}$. The detailed data are shown in Table S1.

The surface consequences of these bulk Weyl nodes, topological surface Fermi arcs, span a distance constrained by the separation of the bulk nodes. The longest Fermi arcs thus far experimentally observed have been in TaAs Huang2015 ; Weng2015 , and are still significantly shorter than the zone-spanning ones we propose in RhSi. Large Fermi arcs crossing a quarter of the surface BZ have also been observed in WTe2; however they have been shown to be topologically trivial trivialarc ; LaAlGe , and thus highly sensitive to surface conditions. In particular, by changing the surface chemical potential, the arcs in WTe2 could be fully removed (Fig. 13(a-d)). In contrast, the long Fermi arcs in RhSi are topologically protected by the bulk chiral charge of unconventional Weyl fermions, and will therefore will be robust to surface conditions (Fig. 13(e-h)).

To summarize, RhSi is by far the most ideal candidate topological chiral fermion semimetal thus far identified. In Table S1, we provide a comparison of RhSi with previously proposed Weyl semimetals, the results of which can be summarized as:

(1) RhSi has at its Fermi energy the minimum number of isolated chiral Fermions (2) allowed in a BZ.

(2) The two topological fermions have the largest possible separation in momentum space.

(3) RhSi has the longest possible topological Fermi arcs.

(4) RhSi has the largest chiral charge ever featured in a realistic material (4 times that of previous works).

(5) RhSi has the largest topologically nontrivial energy window of any material thus far.

I.6 SM F. Numerical Calculations of the Chiral Charges of High-fold Fermions

A conventional Weyl fermion forms at the degeneracy of two singly degenerate bands with one band gap. The chirality of a Weyl fermion is calculated from the Berry curvatures of the occupied bands, and dictates the minimum number of nontrivial surface states in the gap between the conduction and valence bands around the Weyl node. A cartoon of a Weyl fermion is drawn in Fig. 14(a). The chiral charge associated with the gap is labeled as $C_{1}$. In general, an N-fold fermion is formed by N bands with N-1 band gaps. For example, a threefold fermion is formed by three bands with two band gaps around, as shown in Fig. 14(b). For a Weyl fermion with multiple band gaps, we can calculate the chiral charge around each gap. For example, if we set the top valence band as the blue band (band 1), we observe a Chern number $C_{1}$, which decides the number of Fermi arcs in the gap between band 1 (blue) and band 2 (red). Similarly, if we set band 1 and 2 as the valence bands, we observe a different Chern number associated with the gap between band 2 (black) and band 3 (red), which is labeled as $C_{2}$. Therefore, the chiral charges of a threefold fermion can be defined by two numbers ($C_{1}$,$C_{2}$). In general, for any N-fold isolated fermion, we can calculate N-1 chiral charges ($C_{1}$,$C_{2}$,…,$C_{N-1}$) associated with different band occupations. These values decide the number of Fermi arcs in each band gap around the high-fold chiral fermion.

I.7 SM G. The Effects of Exchange-correlation Pseudopotentials on RhSi

Here we take the effects of exchange-correlation pseudopotentials into consideration. Specifically, we have conducted generalized gradient approximation (GGA) calculations with the Hubbard energies U=2eV and U=4eV. We have also done the calculation with the Heyd–Scuseria–Ernzerhof (HSE) exchange-correlation functional. The electronic structures of RhSi under different exchange-correlation pseudopotentials are shown in Fig. 15. The fourfold- and sixfold-degenerate fermions are still preserved and their Fermi pockets are still remained well separated in momentum space. This is because the fourfold and sixfold fermions in our work are symmetry guaranteed. And the exchange-correlation potentials do not break the crystal symmetries.