Type-III Weyl Semimetals: (TaSe4)2I
Xiao-Ping Li Ke Deng Botao Fu YongKai Li DaShuai Ma JunFeng Han Jianhui Zhou Shuyun Zhou Yugui Yao




These authors contributed equally to this work.
Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China


These authors contributed equally to this work.
State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China


These authors contributed equally to this work.
College of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University, Chengdu, 610068, China
Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China


These authors contributed equally to this work.
Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China


Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China


Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China


jhzhou@hmfl.ac.cn
Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences (CAS), Hefei 230031, Anhui, China


State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
Frontier Science Center for Quantum Information, Beijing, China


ygyao@bit.edu.cn
Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China




Weyl semimetals have been classified into type-I and type-II with respect to the geometry of their Fermi surfaces at the Weyl points.
Here, we propose a new class of Weyl semimetal, whose unique Fermi surface contains two electron or two hole pockets touching at a multi-Weyl point, dubbed as type-III Weyl semimetal.
Based on first-principles calculations, we first show that quasi-one-dimensional compound (TaSe_4)_2I is a type-III Weyl semimetal with larger chiral charges.
(TaSe_4)_2I can support four-fold helicoidal surface states with 
 long Fermi arcs on the (001) surface.
 Angle-resolved photoemission spectroscopy measurements are in agreement with the gapless nature of (TaSe4)2I at room temperature and reveal its characteristic dispersion. In addition, our calculations show that external strain could induce 
transitions in (TaSe_4)_2I among the type-III, type-II, and type-I Weyl semimetals, accompanied with the Lifshitz transitions of the Fermi surfaces.
Therefore, our work first experimentally indicates (TaSe_4)_2I as a type-III Weyl semimetal and provides a promising platform to further investigate the novel physics of type-III Weyl fermions.



Type-III Weyl Semimetals: (TaSe_4)_2I
    Yugui Yao
    December 30, 2023
=====================================



Introduction.–Recently, three-dimensional (3D) Weyl/Dirac semimetals possessing discrete and finite degenerate points, Weyl points (WPs), in the Brillouin zone (BZ), have attracted increasing attentions <cit.>.
Weyl/Dirac semimetals have been predicted in a large number of materials <cit.> and also exhibit various novel physical phenomena such as the chiral magnetic effect, ultrahigh mobility, negative longitudinal magnetoresistance and 3D quantum Hall effect, some of which have been recently confirmed experimentally <cit.>.
It is known that Weyl semimetals can be characterized by the chiral charge of the WP: single Weyl semimetals with chiral charge χ=±1 <cit.> and multi-Weyl (double or triple) semimetals with larger chiral charges χ=±2,±3 and the multi-fold Fermi arc states <cit.>.
The unique nonlinear energy dispersions of multi-Weyl semimetals may lead to striking non-Fermi-liquid behaviors <cit.>.
However, to date, only the single Weyl/Dirac semimetals have been observed in experiments, and it remains important to search for realistic materials for multi-Weyl semimetals.


The Fermi surface plays a crucial rule in understanding the fundamental physics of crystals, such as superconductivity, the charge/spin density wave <cit.> and anomalous transport <cit.>.
According to the geometry of the Fermi surfaces at the WPs, Weyl semimetals can also been classified into type-I and type-II <cit.>.
Unlike the point-like Fermi surface of type-I Weyl semimetals (Fig. <ref>(a)),
the type-II Weyl semimetals with over-tilted Weyl cones (Fig. <ref>(b)) <cit.>, whose Fermi surface consists of touched electron-hole pockets, break the Lorentz invariance and lead to many unusual electromagnetic responses <cit.>.
For example, magneto-transport experiment in WTe_2 shows that the tilting of Weyl cone makes the chiral anomaly (origin of various intriguing physical effects in Weyl semimetals), and the resultant negative longitudinal magnetoresistance exhibits strong orientation dependence and even disappearance <cit.>.
Thus, it is fundamentally important whether there exists new Weyl semimetal whose Fermi surface is distinctly different from the counterparts of type-I and -II Weyl semimetals, such as,
 two contacted electron or two hole pockets at the WP (Fig. <ref>(c)). If so, what are the relevant chiral charge and the unique physical properties?


In this work, we first discover a new type-III Weyl semimetal whose Fermi surface consists of two electron or two hole pockets touching at the WPs
in the quasi-one dimensional (1D) compound (TaSe_4)_2I.
(TaSe_4)_2I possesses two pairs of double WPs, thus supporting the four-fold helicoidal surface states on (001) surface with 
long Fermi arcs.
Meanwhile, strains that break C_4 symmetry could trigger transitions from type-III WPs to type-II and type-I ones, accompanied with the Lifshitz transitions of the Fermi surfaces. More importantly, our angle-resolved photoemission spectroscopy (ARPES) results suggest (TaSe_4)_2I is a type-III Weyl semimetal and provides the first material realization of multi-Weyl semimetals.





Type-III Weyl fermions.–To illustrate the essential feature of the type-III Weyl fermions, we consider a general Hamiltonian near the WP,


    ℋ_n(k)=w_zk_z+w_∥k_∥^2+v_zk_zσ_z+(ak_±^nσ_++h.c.),


and the corresponding eigenvalues are E_±(k)=w_zk_z+w_∥k_∥^2±√(v_z ^2 k_z^2+a^2 k_∥^2 n).
Here k_∥=√(k_x^2+k_y^2), k_±=k_x± ik_y, σ_±=1/2(σ_x± iσ_y), σ_x,y,z are the Pauli matrices.
The subscript n=1,2,3 denotes the single-, double- and triple-WPs, respectively.
The first two terms refer to the linear and quadratic tilting terms, which are crucial for the emergence of type-II and type-III Weyl fermions, respectively.
To be specific, the linear tilting term w_z k_z (e.g. w_z>0) pulls down the band for k_z<0 and pushes up the band for k_z>0.
Once |w_z|>|v_z|, the Weyl cone (Fig. <ref>(a)) is over tilted and the Fermi surface changes from a point to a coexistence of electron and hole pockets, leading to a type-II WP (Fig. <ref>(b)) <cit.>.
On the other hand, the quadratic tilting term w_∥ k_∥^2 (w_∥>0) always pushes up energy bands for any in-plane k-path (k_z=0).
It is worth noting that the quadratic tilting term plays an important role in multi-Weyl semimetals. Let us first focus on the double Weyl semimetals.
When w_∥ k_∥^2 dominates the energy dispersion in the k_x-k_y plane (|w_∥|>|a|), the Weyl cone can be over-tilted, giving rise to a type-III WP.
The resulting Fermi surface consists of two contacted electron or two hole pockets (Fig. <ref>(c)).
The sign of the quadratic tilting term w_∥ determines the types of the two contacted pockets.
Our classification of Weyl fermions can be well understood in terms of the general titling mechanism. Specifically, the dominant nth order tilting term would lead to the type-(n+1) Weyl fermions <cit.>. It should be emphasized that the type-III WP can emerge in triple Weyl semimetals as well.







Material realization.–(TaSe_4)_2I is a typical quasi-1D compound that has been synthesized more than thirty years ago <cit.>. It forms a body-centred tetragonal lattice with the chiral space group I422 at room temperature <cit.>. As shown in Figs. <ref>(a)-<ref>(b), crystal of (TaSe_4)_2I contains right-handed TaSe_4 chains along the c-axis with halogen atoms filling inter-chain interstitial regions.
Very recently, the charge density wave (CDW) phase <cit.> and possible axion physics in (TaSe_4)_2I have generated a lot of interest <cit.>.

The electronic structure of (TaSe_4)_2I in Fig. <ref>(c) displays several notable features <cit.>.
First, we find that a band crossing (W_1) appears along the C_4 invariant Γ-Z path with linear dispersion in a broad energy range from -0.5 eV to 0.5 eV.
Another band crossing emerges at the corner of the BZ (W_2), which also has linear dispersion along the P-X path.
Second, two bands along the P-N path exhibit weak dispersions close to the Fermi level, which is attributed to the weak inter-chain coupling in (TaSe_4)_2I.
Third, the zoom region in Fig. <ref>(d) shows that these two bands stemming from P point will switch their orders and create a WP (W_4) with over-tilted linear dispersion.
Similar band switching process happens near W_1 point, forming another WP at W_3 in the k_y=0 plane.
That is, there exist four independent WPs marked as W_1,2,3,4 in the irreducible BZ.
Combining crystal symmetries with time reversal symmetry, we obtain two W_1,2 points and eight W_3,4 points in total in the first BZ.


Figs. <ref>(e)-<ref>(f) show that the WPs W_1 and W_2 have unusual quadratic dispersion along the k_x path and linear dispersion along the k_z path, indicating larger chiral charges.
To verify this point, we calculate their chiral charge and find χ=-2(2) for W_1(W_2), χ=1(-1) for W_3(W_4), respectively.
Moreover, the quadratic tilting terms make the valence band over-titled along the k_x direction for W_1 and the conduction band over-titled around the k_x direction for W_2, leading to unique Fermi surfaces. As shown in Figs. <ref>(g)-<ref>(h), the resulting Fermi surface near W_1 (W_2) contains two contacted hole (electron) pockets.
This is the remarkable signature of the aforementioned type-III Weyl fermions.









Effective k· p models.–To gain more insights into Weyl fermions in (TaSe_4)_2I, we would like to build the effective k· p model based on symmetry analysis.
Along the Γ-Z path, the corresponding little group is C_4 group.
Two bands that cross the Fermi level host two distinctive 1D irreducible representations, Λ_1 and Λ_2 <cit.>, the corresponding eigenvalues of C_4 are ±1.
It implies that a double WP (W_1) can appear once band inversion happens <cit.>, as shown in Fig. <ref>(c).


The P point at the corner of BZ has the little group of D_2, which possesses four non-equivalent 1D representations that do not guarantee two-fold degeneracy.
In reality, the P point is invariant under a joint operation of four-fold rotation and time reversal operation (C_4T).
According to Herring rules <cit.>, two bands around the Fermi level with representations P_3 and P_4 can stick together, forming a WP at the high symmetry point.
By further taking into account the constraints from other crystal symmetries, the energy dispersion of WP at P (W_2) point is quadratic in the k_x-k_y plane, forming a double WP.
Finally, we obtain the effective k· p Hamiltonian of double WPs, W_1 and W_2 as follows (see Supplemental Material <cit.>),


    H_III^(i)=H_DW^(i)+w_t^(i),


where H_DW ^(i) describes a type-I double Weyl fermion that has the form


    H_DW^(i)   	=   w_x^(i)σ_x+w_y^(i)σ_y+w_z^(i)σ_z,


and w_t^(i) is the tilting term. i=1,2 refers to W_i.
The explicit expressions w_x,y,z^(i) and w_t^(i) are given in Table-I.




One can see that the w_t^(1) includes two parts: the linear and quadratic tilting terms.
Based on our theoretical calculations, the linear term along the k_z direction is tiny but the quadratic term along the k_x or k_y direction is dominating.
The w_t^(2) only contains the quadratic term due to the symmetry constraints.
It has been pointed out that the over-tilted linear term can induce type-II Weyl semimetals with touching electron-hole pockets, while the over-tilted quadratic term plays a key role in forming type-III Weyl semimetals.
Specifically, w_t^(1) pushes the valence band around W_1 upward along the k_x or k_y path.
When |d_2|>√(a^2+b^2+c_2^2), we obtain a type-III WP with two contacted hole pockets (Fig. <ref>(g)).
Similarly, w_t^(2) pulls the conduction band around W_2 downward along the k_x or k_y path when |d_1|>|c_1| and gives birth to another type-III Weyl fermion with two contacted electron pockets (Fig. <ref>(f)).
These coefficients of k· p Hamiltonian obtained from fitting with our calculations confirm these analysis (see Supplemental Material <cit.>).







ARPES spectra and helicoidal surface states.–Fig. <ref> shows the ARPES measurement of electronic structure of (TaSe_4)_2I at 285 K.
The single crystal is with in-situ reproducibly cleaved to expose (110) surface due to its quasi-1D nature <cit.>.

The stripe-like constant energy contour near Fermi level in Fig. <ref>(a) indicates a strong nesting, which contributes to the CDW transition below 263 K <cit.>.
The ARPES spectra along cuts (1, 2, 3), marked as black dashed lines in Fig. <ref>(a), are presented in Figs. <ref>(c)-<ref>(e), respectively.

A V-shaped dispersion in cut1 in Fig. <ref>(c) indicates a linearly dispersing band structure near W_1.
In addition, we find the dispersions along cut2 and cut3 exhibit a quadratic characteristic in the vicinity of W_2 and W_1, respectively.
The dashed red lines in Figs. <ref>(c)-<ref>(e) are the calculated band structures along certain direction, and the measured ARPES spectra show good agreement with our calculations, supporting that (TaSe_4)_2I is a type-III Weyl semimetal at room temperature.
In addition, we calculate the energy bands of three types of Weyl semimetals in the presence of homogeneous magnetic fields along the z direction and find that the type-III Weyl semimetals exhibit a unique Landau level's turn-up due to the dominant quadratic tilting term. This peculiar Landau level spectra can be directly probed by magneto-optic experiments and act as a smoking gun signature of type-III Weyl fermions <cit.>.





Interestingly, our calculations reveal more exotic features of the Fermi arc surface states on (001) plane.
As shown in Fig. <ref>(a), two type-III double WPs (W_2) above the Fermi energy are projected into the corner (W_2) of the surface BZ, forming a hole pocket with χ=4, while the other type-III double WPs (W_1) below the Fermi energy are projected into the center (W_1) of surface BZ, forming an electron pocket with χ=-4.
Meanwhile, four pairs of type-I single WPs (W_3) related by C_4 rotation are projected into four W_3 (χ=2) surrounding the W_1 point.
Similarly, four pairs of type-II single WPs (W_4) are projected into four W_4 surrounding W_2 point, forming four hole pockets with χ=-2.
So the four hole pockets of W_4 merge together with the hole pocket of W_2, forming a larger square-like hole pocket with χ=-4.
One can see that four branches of Fermi arcs stem from W_1, pass through W_3, connect to W_4 and finally merge into W_2.
Those long Fermi arcs that connect the W_1 and W_2 demonstrate a four-fold helicoidal nature <cit.>.
To visualize the helicoidal surface states, we calculate the surface local density of states (LDOS) along two clockwise loops (C_1,2) centered at W_1, as shown in Figs. <ref>(b) and <ref>(c).
For the loop C_1, the four right-moving chiral edge modes appear insides the band gap in line with the chirality of W_1.
Since the loop C_2 encompass both W_3 and W_1, the four edge modes instead have opposite chirality.
The four-fold right-handed and left-handed spiral surface state along the loop C_1 and C_2 are schematically demonstrated in Fig. <ref>(d).
Note that since the naturally cleaved surface of (TaSe_4)_2I is the (110) surface, the such novel topological surface states that exist on the (001) surface are therefore absent from our ARPES measurements. Instead, the signature of topological surface state could be accessible to transport measurement of micro-fabricated nano-scaled samples <cit.>.





Strain induced transitions.–We stress that a pair of double WPs with χ=2 locate at the corner of BZ P and -P, whereas the other pair with χ=-2 locates on the C_4 invariant axis thus can move along the Γ-Z path.
Thus, there is no loop available for annihilation of the two WPs with opposite chirality as long as the crystal symmetry is preserved.
It implies that these type-III double WPs in (TaSe_4)_2I exhibit remarkable robustness against external perturbations that respect crystal symmetries.




Strains have proven to be a clear and effective way to tune the electronic structure and the relevant physical properties.
We now investigate the impacts of strains that break crystal symmetries on the electronic properties in (TaSe_4)_2I.
Since the type-III double WPs are protected by C_4 or C_4T, applying a symmetry-breaking uniaxial strain along (100)/(010) direction could split a type-III double WP with χ=2 into two type-II WPs with unit chiral charge, accompanied with Lifshitz transition of the Fermi surface as shown in Figs. <ref>(a)-<ref>(b).
We also estimate that about 0.1% (0.02 GPa) compressive strain along a axis could induce such topological transitions.
Furthermore, imposing an extra biaxial in-plane strain, the type-II single WP will further transform into type-I single WP along with a second Lifshitz transition of the Fermi surface, as shown in Fig. <ref>(c).
Therefore, these three types of Weyl fermions can be mutually transformed through strain engineering in (TaSe_4)_2I, which provides a tunable platform for studying exotic properties of various Weyl semimetals.
In experiments, these transitions among different Weyl fermions can be induced via proper external pressure and the relevant Lifshitz transitions of Fermi surface as well as the corresponding surface states offer a measurable signal that can be probed by magnetic oscillation experiments <cit.>, ARPES <cit.> and magneto-transport <cit.>.
It is worth noting that (TaSe_4)_2I could support a new solid state phase combining long-range ordered but short-range disordered structure under pressure (about 20 GPa) as well as the anomalous two-step superconducting transition <cit.>.



In summary, we propose the new type-III Weyl semimetal whose Fermi surface that consists of two touched electron or hole pockets and first provide experimental evidence of such novel fermion in (TaSe_4)_2I by means of first principle calculations and ARPES experiments.
Two pairs of type-III Weyl points emerge due to the over-tilted quadratic terms and are protected by C_4 and C_4T symmetries, respectively.
We find that the strains that break C_4 symmetry trigger the  transitions from type-III to type-II, and to type-I WPs along with Lifshitz transitions of the Fermi surface.
In addition, a four-fold helicoidal surface state is predicted on (001) surface with long Fermi arcs.
Therefore, our work reports the first material realization of multi-Weyl semimetal and also provides a promising platform to further study the novel physics of type-III Weyl semimetals.



We are grateful to Zhi-Ming Yu for fruitful discussions. The work at BIT is supported by the National Key R&D Program of China (Grant No. 2016YFA0300600), the National Natural Science Foundation of China (Grants Nos. 11734003), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB30000000).
K.D. and S.Z. are supported by the National Natural Science Foundation of China (Grants No. 11725418).
J.Z. is supported by the High Magnetic Field Laboratory of Anhui Province.



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