Topologically Protected Wormholes in Type-III Weyl Semimetal Co$_3$In$_2$X$_2$ (X = S, Se)

The observation of wormholes has proven to be difficult in the field of astrophysics. However, with the discovery of novel topological quantum materials it is possible to observe astrophysical and particle physics effects in condensed matter physics. It is proposed in this work that wormholes can exist in a type-III Weyl phase. In addition, these wormholes are topologically protected, making them feasible to create and measure in condensed matter systems. Finally, Co$_3$In$_2$X$_2$ (X = S, Se) are identified as ideal type-III Weyl semimetals and experiments are put forward to confirm the existence of a type-III Weyl phase.


INTRODUCTION
The discovery of topological quantum materials has opened a large path in experimental condensed matter physics. Initially, the first material discovered was the topological insulator which has an insulating bulk and a topologically non trivial surface state which derived from the Dirac cone. Interestingly these Dirac Fermions do not obey expected physics and behave relativistically [1][2][3][4][5][6][7][8]. Dirac fermions have later been seen to violate Lorentz symmetry and form tilted type-II and critically tilted type-III Dirac cones [9][10][11]. Tilted Dirac cones have been predicted to have the same physics as black holes in certain cases [12][13][14][15].
In the presence of additional perturbations, the formation of Weyl Fermions [16] and their surface Fermi arcs can be formed. These condensed matter excitations were first observed in condensed matter physics and have yet to be observed in high energy particle physics. Similarly to the Dirac cone, the edge states of the Weyl cone can be tilted to form type-II and type-III Weyl cones [17][18][19][20]. A plethora of type-I and type-II Weyl cones [21] have been discovered yet the discovery of a type-III Weyl Fermion phase has yet to be discovered conclusively [22,23].
Wormholes are a yet undiscovered but physically plausible object that can exist within the framework of general relativity [24,25]. Much work has been done in order to define wormhole topology, energy of formation, and experimental signatures. However, wormholes are largely considered improbable due to a need for a large amount of energy to form and maintain one within current models. The generation of a quasi-wormholes will prove valuable in understanding worm-arXiv:2008.11004v3 [cond-mat.str-el] 17 Mar 2022 2 hole physics [26][27][28].
The type-III Dirac cone is predicted to host a direct analogue to a black hole where similar physics can be observed and measured with respect to the Dirac quasiparticles that experience the effects of the critically tilted Dirac cone. Limited experiments have been conducted in order to confirm the effects of the type-III Dirac phase and little to no materials have been discovered [29][30][31][32][33][34]. The counterpart to the Dirac black hole is the Weyl type wormhole phase [35][36][37][38]. In this work, it is predicted that wormholes can be formed and are topologically robust in the type-III Weyl phase. In addition, Co 3 In 2 S 2 and Co 3 In 2 Se 2 are predicted to host an ideal type-III Weyl fermion.

MATERIALS AND METHODS
The band structure calculations were carried out using the density functional theory (DFT) program Quantum Espresso (QE) [39], with the generalized gradient approximation (GGA) [40] as the exchange correlation functional. Projector augmented wave (PAW) pseudo-potentials were generated utilizing PSlibrary [41]. The relaxed crystal structure was obtained from materials project [42,43] (for Co 3 In 2 S 2 ). The relaxed crystal for Co 3 In 2 Se 2 was calculated with QE. Crystal parameters that are calculated with DFT [ Table I] are compared to Co 3 In 2 S 2 [ Table II]. The energy cutoff was set to 60 Ry (816 eV) and the charge density cutoff was set to 270 Ry (3673 eV) for the plane wave basis, with a k-mesh of 25 × 25 × 25. High symmetry point K-path was generated with SSSP-SEEK path generator [44,45]. The bulk band structure [ Fig. 5] was calculated from the 'SCF' calculation by utilizing the 'BANDS' flag in Quantum Espresso. As oppose to utilizing plotband.x included in the QE package, a custom python code is used to plot the band structure with the matplotlib package.
Single crystals of Co 3 In 2 S 2 and Co 3 In 2 Se 2 (SG: R3M [166]) are grown via the Indium flux method [46,47]. Stoichiometric quantities of Co (99.9%, Alfa Aesar) and Se (∼200 mesh, 99.9%, Alfa Aesar)/ S (∼325 mesh, 99.5%, Alfa Aesar) were mixed and ground together with a mortar and pestle. Indium (99.99% RotoMetals) was added in excess (50%) in order to allow for a flux growth. All precursor materials were sealed in a quartz tube under vacuum and placed inside a high temperature furnace. The sample was heated up to 1000 • C over 1440 min, kept at 1000     This response is typically called butterfly magnetoresistance because of how the anisotropy typically looks when plotted on a polar plot [48][49][50]. The magnetoresistance is a function of the electron (hole) mobility in the sample. The mobility is also correlated with the Fermi velocity. We know from previous work that electrons (holes) that exist in the flat band will have zero Fermi velocity, this will lead to no mangetoresponse at the angle where the DOS of the type-III Weyl cone lines up with the magnetic field [Fig 4(A)]. In order to simulate this response we construct a toy model of the variable Fermi velocity as a function of angle for different chemical potentials by using trigonometric functions. The actual magnetoresisistance can be measured by magnetotransport. In a type-III Weyl semimetal which is composed of two pairs of Weyl points we expect to see typical butterfly magnetoresponse at the Weyl line, but as the chemical potential moves away from the Weyl line level we expect to see that response to decrease [Fig 4(B,C)] and show less of an intense mangetoresponse. The Fermi level can be adjusted by backgating, top gating, or a combination of the two in order to access the Weyl line states and to measure the electrical respose (e.g. R xx vs V topgate , R xx vs V backgate ). In the case of magnetic type-III WSMs (Co 3 In 2 Se 2 , Co 3 In 2 S 2 ), the Weyl line contribution can be convoluted with the normal magnetic response of the material. Type-III Weyl states exist above the Fermi level in both Co 3 In 2 Se 2 and Co 3 In 2 S 2 . In order for these states to be more accesible, the chemical potential can be tuned by methods such as potassium doping K x Co 3 In 2 Se 2 (X ≤ 0.1) or impurity doping Co 3 In 2 Se 2−x I x (X ≤ 0.1).

DISCUSSION
In order to form a type-III Weyl phase in a crystal lattice it is necessary to satisfy several conditions. Firstly, perfectly flat bands must exist with a large enough momentum dispersion to connect two bands (or a band must be flat for a period between these two bands). The band must be chiral and connect a hole-like band to an electron-like band, this condition allows for inversion symmetry to be preserved (from band inversion). Materials that satisfy this condition only satisfy a type-III Dirac semimetal phase, therefore in order to break time reversal symmetry and turn the Dirac cone into two Weyl nodes magnetism (or spin orbit interactions) must also exist in these materials in order for there to be a type-III Weyl phase.  Fig. 5 (A,B)]. It is well understood that Weyl fermions can be discribed as two entagled Dirac fermions of opposite chirality [16]. There have been several ongoing expriements in order to discover the existence of a quasi-black hole in the type-III Dirac phase [29,34,56]. If a black hole exitation is found in a type-III Dirac semimetal and a wormhole is discovered in a type-III Weyl  First, the type-III Weyl phase must be confirmed to be physically possible. The simple Hamiltonian that describes a 3D Weyl point can be described as: From this formula, a Dirac point can be constructed with two 3D Weyl points of opposite chirality, thus forming the Dirac cone.
From this formulation, the prototypical type-I Weyl semimetals can be easily constructed and formed. However, magnetism must be introduced in order to tilt the Hamiltonian (and therefore the Weyl cone) so that a critically tilted weyl cone can be formed which leads to a type-III Weyl phase. (where magnetism is added in theẑ direction in order to allow for the Weyl cone to tilt).
This realizes a Weyl point with +1 or -1 Chern number. The type of Weyl cone depends on the value of the parameter C where C > |1| is a type-II Weyl semimetal , C < |1| is a type-I Weyl semimetal, and C = −1 is a type-III Weyl Semimetal. In the case where magnetic field is applied in theẑ direction, the Hamiltonian can be rewritten as [? ]: