Abstract
The spacetime light cone is central to the definition of causality in the theory of relativity. Recently, links between relativistic and condensed matter physics have been uncovered, where relativistic particles can emerge as quasiparticles in the energymomentum space of matter. Here, we unveil an energymomentum analogue of the spacetime light cone by mapping time to energy, space to momentum, and the light cone to the Weyl cone. We show that two Weyl quasiparticles can only interact to open a global energy gap if they lie in each other’s energymomentum dispersion cones–analogous to two events that can only have a causal connection if they lie in each other’s light cones. Moreover, we demonstrate that the causality of surface chiral modes in quantum matter is entangled with the causality of bulk Weyl fermions. Furthermore, we identify a unique quantum horizon region and an associated ‘thick horizon’ in the emergent causal structure.
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Introduction
The intriguing connections between high energy and condensed matter physics have led to a deeper understanding of quantum matter^{1,2,3,4,5,6,7,8,9,10,11,12,13,14}. One such connection manifests itself in topological materials where relativistic particles can emerge as quasiparticles. A familiar example is the Weyl fermion, a massless spin1/2 particle proposed in 1929, which has been realized in many condensed matter systems^{10,11,12,13}. Weyl fermions have also attracted recent attention due to their unique quantum responses, such as the quantized circular photogalvanic effect^{15,16,17,18,19,20,21}. Another frontier concerns strongly interacting systems which host unusual effects driven by the interplay of correlations, topology, and geometry^{2,3,4,5,6,7,8}.
Here, correlated Weyl semimetals provide a perfect platform for exploring interaction effects on singleparticle physics. The separation of the individual Weyl nodes with opposite topological charges in momentum space makes it impossible to hybridize these nodes and produce a fully gapped insulating state without violating symmetries. In correlated systems, however, interactions can in principle open a global gap in the system with Weyl fermions. For instance, it has been reported that a Weyl semimetal can be gapped out into an axion insulator by the chargedensitywave (CDW) pairing interaction^{22,23,24,25}, although the general mechanism of this metalinsulator transition remains elusive. Even though Weyl fermions are rooted in quantum field theory, how causal physics^{1} enters the interaction dynamics of Weyl semimetals remains unexplored. With this motivation, we discuss a topological phase transition mechanism for a CDWcorrelated Weyl semimetal. We focus on a more likely scenario in real materials, wherein the CDW arises in a system where the Weyl nodes exist but are not caused by the Weyl nodes and hence it can possess a different periodicity (Fig. 1a).
Results
We start from an inversionsymmetry breaking model with four Weyl quasiparticles in the first Brillouin zone (BZ):
where A and k_{1} are two constants with k_{1} ≠ π, and the σ are the Pauli matrices. We first consider the case with θ = 0 which preserves the timereversal symmetry of the system and makes the Fermivelocity of each Weyl quasiparticle roughly the same. The positions of four Weyl nodes with ∓ 1 chirality are at k_{W} = ± (k_{1,}0, ± π/2) with black (white) dots representing positive (negative) chirality. The Weyl nodes of different chirality are at energies \({E}_{W}=\pm A\sin {k}_{1}\) (Fig. 1b).
To discuss the dynamics of interacting Weyl fermions, we consider the CDW instability as a quasionedimensional Peierls instability such that there is only one unidirectional CDW Qvector (Q_{CDW}). For simplicity, but without losing generality, we fix Q_{CDW} and vary the separation between the Weyl nodes instead. We choose Q_{CDW} = (π, 0, 0) as a representative, which reflects the Peierls dimerization in a double supercell along the xdirection, see Methods section titled: CDW tightbinding model in real space.
When the CDW wavevector is equal to the momentum separation of the Weyl fermions (\({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}={{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\)) and the Weyl nodes lie at the same energy (\({E}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}={E}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\)), our calculations show that the CDW interaction will gap out the Weyl fermions, which is consistent with previous work^{24,25}. However, when the separation of the two Weyl nodes is not the source of the nesting vector of CDW (\({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}\ne {{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\)), the whole system remains a Weyl semimetal, see Supplementary Fig. 1. This indicates that the relationship between the Qvector and the separation of the Weyl nodes plays a key role in determining the topological phase transition in correlated Weyl semimetals.
To be more representative of real materials^{22,23,26,27,28}, we discuss the general case where the two interacting Weyl nodes lie at different energies (\({E}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}\ne {E}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\)). We consider an illustrative example using A = 0.3 and k_{1} = 1.3π/2, which yields four Weyl nodes at ± (1.3π/2, 0, ± π/2) with an energy difference of around 0.6 eV (Fig. 1c). To understand how the Weyl nodes are folded in the double supercell BZ (the reduced BZ), the folded band structure for the double supercell along the xdirection with CDW interaction strength δ set as 0 is plotted in Fig. 1d. The Weyl nodes are folded into the outside of the dispersion cone of each other. For nonzero CDW interaction strength δ = 0.1, we find that the Weyl nodes do not annihilate each other and the system remains in the metallic phase (Fig. 1e). To figure out the condition for two Weyl nodes to annihilate when \({E}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}\ne {E}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\) and \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}\ne {{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}\), we change the location of Weyl nodes to ± (1.1π/2, 0, ± π/2) (Fig. 1f). In this case, the Weyl nodes in the folded BZ are within the dispersion cone of each other (Fig. 1g). Surprisingly, after the inclusion of a nonzero CDW interaction strength δ, a global gap between the conduction and valence bands is seen to open up (Fig. 1h).
Energymomentum analog of causal structure
We find that whether or not the WeylCDW pairing interaction will drive the topological semimetalinsulator phase transition depends on the relative location of the Weyl nodes in the reduced BZ. In analogy with the causal structure in the theory of relativity, we define the region in energymomentum space within (outside) the Weyl cone as the energylike (momentumlike) region (Fig. 2a). We can express these energylike and momentumlike regions (Figs. 2b and 2c, left panels) as
where \(\delta {{{{{{{\bf{k}}}}}}}}={{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}^{{\prime} }{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}^{{\prime} }\), and \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}^{{\prime} }\) and \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}^{{\prime} }\) are the new Weyl nodes positions in the reduced BZ, \(\delta E={E}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}^{{\prime} }{E}_{{{{{{{{{\rm{W}}}}}}}}}_{2}}^{{\prime} }\) is the energy difference of two Weyl nodes with the CDW interaction, and V_{F} is the Fermi velocity of the Weyl cone. Here we used the sign convention of the Minkowski metric η_{μν} = diag(1,−1,−1,−1). For simplicity, we first assume that the two Weyl fermions have the same Fermi velocity, and that the Fermi velocity is isotropic.
Within the aforementioned classification, we find that the Weyl semimetal can become an insulator only when the pair of Weyl nodes around the Fermi level with CDW interaction are energylike (Fig. 2b, right panel). Note that, in this scenario, the Weyl nodes may not be gapped, but the whole system has a semimetal to insulator transition. In contrast, if the two Weyl nodes are momentumlike, the Weyl system remains in the semimetal phase even after the CDW phase transition (Fig. 2c, right panel). Whether or not a system can undergo the metaltoinsulator transition is thus equivalent to examining whether the Weyl nodes with opposite charges are energylike in the CDW phase.
We turn now to briefly discuss the relationship between the theory of relativity and our WeylCDW physical picture. In Einstein’s theory of relativity, space and time are connected by the speed of light and cannot be described independently. Causality means that a cause cannot have a causal connection (effect) on an observer if it does not lie in the light cone of the observer. That is, two events can be causally related only when they are timelike. A horizon is a boundary in spacetime beyond which events cannot affect an observer. Similarly, for a system with Weyl fermions, energymomentum space can be viewed as the analog of spacetime with Fermi velocity playing the role of speed of light. In the energymomentum space, only when the two Weyl nodes are energylike can they have correlation (causal connection) and make the system undergo a phase transition, and the Weyl cone plays the role of the horizon in the energymomentum space. Note that, although there is some existing literature relating Einstein’s theory of relativity to Weyl materials^{29,30,31}, the earlier work considers only a single Weyl node, while our focus is on the causal structure of the energymomentum space and interacting Weyl systems, which requires at least two Weyl nodes.
For practical purposes, we further simplify Eq. (2) to consider the case when the CDW interaction can be treated as a perturbation. We can assume that the energy and momentum of the Weyl point only acquire a small correction from CDW interaction in its reduced BZ. We define a critical length in the energymomentum space based on the energy difference of Weyl points and their Fermi velocities as \({K}_{C}=({E}_{{{{{{{{{\rm{W}}}}}}}}}_{1}}{E}_{{{{{{{{{\rm{W}}}}}}}}}_{2}})/{V}_{F}\). Hence, we can ascertain the possibility of a metalinsulator transition in the WeylCDW system by simply comparing the critical length K_{C} and the length of momentum separation after bandfolding without the CDW (δ = 0)(Fig. 2d).
Application to real materials
Because Weyl fermions are quite common in inversionsymmetrybreaking systems^{32,33}, our theory can be widely applied to the large class of noncentrosymmetric CDW materials. As an example, we consider (TaSe_{4})_{2}I, which is a Weyl semimetal at room temperature that turns into an incommensurate CDW phase with \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}=(0.027(\frac{2\pi }{a}),0.027(\frac{2\pi }{a}),0.012(\frac{2\pi }{c}))\) for temperatures below to 263 K^{22,23}. The Weyl nodes of (TaSe_{4})_{2}I (without SOC) are shown in Fig. 2e: The energy difference is δE = 0.068 eV and the fermi velocity is V_{F} ~ 3.47 eV ⋅ Å. Thus K_{C} is estimated to be around 0.02 Å^{−1}. We take approximate the CDW supercell to the nearest rational number as a commensurate supercell. Based on the folded band structure in a \(37\sqrt{2}\times 37\sqrt{2}\times 83\) commensurate supercell, we find the momentum difference between these two nodes to be around 0.009 Å^{−1}, which is much smaller than K_{C}. Therefore, with the inclusion of CDW, the pair of Weyl nodes in (TaSe_{4})_{2}I system is energylike and the system will become an insulator. This is consistent with experimental measurements^{23}.
Our arguments bear on understanding the origin of the CDW in Mo_{3}Al_{2}C^{34,35}, where it has been argued that the sudden change in the electronic density of states is due to Fermi surface nesting along the CDW nesting vector along the (1, 1, 1) direction^{34}. However, it is also reported that there is no sign of the semimetaltoinsulator transition in the Mo_{3}Al_{2}C^{35}. Here, we examine the band structure of Mo_{3}Al_{2}C without including SOC. We find a pair of Weyl nodes along the (1, 1, 0) (ΓM) direction in the bands that cross the Fermi energy (Fig. 2f), with the separation of the folded Weyl nodes (∽0.23 Å^{−1}) being larger than K_{C} ∽ 0.17 Å^{−1}. Therefore, the pair of Weyl nodes in Mo_{3}Al_{2}C would be momentumlike with CDW interaction. These results allow us to conclude that the CDW in Mo_{3}Al_{2}C leads to partial Fermi surface gapping but not to a metalinsulator transition.
Entangled causality between bulk and surface
We discuss the causal structure of the topological surface states in correlated Weyl semimetals by considering a path cut through the Weyl cone (away from the node) where we have a surface chiral mode connecting the gapped conduction and valence bands (Fig. 3a). The topological chiral modes exhibit opposite directions for Weyl cones of different signs of the Chern numbers. In the momentumlike case, the two surface chiral modes cannot cross (Fig. 3b). After the CDW interaction is included, there is no causal interaction between the chiral modes with opposite chirality. Thus, the surface states in the momentumlike case remain as chiral modes connecting the conduction and valance bands. In contrast, in the energylike case, the surface chiral modes with opposite chirality cross each other (Fig. 3c, top panel). As a result, in the presence of the CDW, these chiral modes can interact to open up a surface bandgap (Fig. 3c, bottom panel). Here, we show the [010] surface states under CDW interaction using the iterative green function method^{36} for momentumlike (Fig. 3d) and energylike (Fig. 3e) cases. As we can see, the causality on the surface matches that in the bulk which presents entangled causality between the bulk and surface states in topological materials. This indicates that the causal structure of two interacting Weyl quasiparticles can be determined by observing the behavior of the chiral edge states without knowing the causality in the bulk.
Causal structures with quantum horizon region
In the theory of relativity, the speed of light is a universal constant. However, in our condensed matter analog, the Fermi velocities of Weyl fermions can be different from each other and are not constrained to be the speed of light. Accordingly, we now consider the case where the two Weyl fermions have different Fermi velocities by choosing nonzero θ values in Eq. (1) (Fig. 4a). We define V_{F,H} (V_{F,L}) as the higher (lower) Fermi velocity of the two Weyl fermions. The gapping condition is found to remain unchanged when the two Weyl fermions lie inside or outside each other’s dispersion cone. Specifically, when the two Weyl nodes lie in the energylike regions [\({(\delta E/{V}_{F,H})}^{2}{(\delta {{{{{{{\bf{k}}}}}}}})}^{2} > 0\)], the whole system is gapped out by the CDW interaction. In contrast, when the two Weyl nodes lie in the momentumlike regions [\({(\delta E/{V}_{F,L})}^{2}{(\delta {{{{{{{\bf{k}}}}}}}})}^{2} < 0\)], the system remains gapless.
Because of the different Fermi velocities involved, a unique phase can emerge in our system in which the Weyl node I lies in the energylike region of the Weyl node II but the Weyl node II lies in the momentumlike region of Weyl node I (Fig. 4b). When a small nonzero CDW interaction is included, the Weyl nodes remain intact and no global band gap is seen (Fig. 4c). However, the system is near a quantum critical point and a slight increase in the strength of the interaction can drive the two Weyl fermions to fall into each other’s dispersion cone (become energylike) and open a global band gap, see Supplementary Fig. 3. Interestingly, the causal structure of the interacting Weyl system could thus be changed by tuning the strength of the interaction. These results openup opportunities for exploring causal structures beyond the framework of Einstein’s theory of relativity in which spacelike to energylike crossover is forbidden. Since we are close to a quantum critical point, we refer to this region as the “quantum horizon region” that bridges energylike and momentumlike regions.
We summarize the full causal structure of the interacting Weyl system in Fig. 4d where the quantum horizon region is shown as a ‘thick horizon’ bounded by V_{F,H} and V_{F,L} via the equations:
Based on this causal structure, we comment on the reason why the CDW interaction is always attractive in the sense that interaction moves the two Weyl quasiparticles toward each other. Recall that the CDW interaction tends to open a global band gap and drives the system into an insulator phase. However, in a Weyl system, a global band gap between the bands forming the Weyl nodes can only be opened by annihilating the two interacting Weyl quasiparticles. Since causal interaction is only possible between energylike Weyl quasiparticles, the CDW interaction can be expected to move the two Weyl quasiparticles closer in the energymomentum space (Supplementary Figs. 2, 3). Therefore, it is only possible to have a crossover from the quantum horizon region to the energylike region but not to the momentumlike region with increasing interaction strength.
When considering the highenergy analogs of Weyl fermions in condensed matter systems, Weyl cone is often identified as a welldefined relativistic quasiparticle by using a proper linear approximation for its dispersion cone around the node. However, although the nonlinear dispersion terms violate the Lorentz invariance in highenergy physics, it is natural for condensed matter systems that the dispersion cone of the Weyl node becomes nonlinear due to quadratic and higherorder corrections. Due to the nonlinear dispersion, it’s also natural that the two Weyl nodes may not both lie inside (outside) of each other’s dispersion cone and thus impact the emergence of the quantum horizon region in the causal structure, see Supplementary Fig. 4. The nonlinear dispersion cones indicate that the Eq. (3) can be further generalized to the case simply based on whether the two interacting Weyl quasiparticles are inside or outside of each other’s dispersion cone as follows:
Based on our generalized conclusions for the Lorentz violating cases, we conjecture that the casual structure of interacting typeII Weyl quasiparticles^{37}, where the Lorentz invariance is violated and the Fermi velocities for the two branches of the dispersion cones possess the same sign, criteria similar to those in Eq. (4) would be applicable. Note that there is a connection between the behavior of the energy spectrum behind the event horizon of a blackhole and the typeII Weyl fermions^{30}, suggesting that the causal structure of the interacting typeII Weyl fermions may hide yet more profound new physics that would be interesting to explore. Finally, we note that the causal structure shown in Eq. (3) and Eq. (4) can also be derived from the lowenergy effective k⋅p description, so that our gapopening condition is universal and independent of the choice of the model and the form of the interaction, see the Supplementary Fig. 5–6.
Discussion
In analogy with the spacetime light cone and the related causalitydriven event horizon in relativistic physics, we have unveiled the causal structure of the energymomentum space in the condensed matter context and show that it consists of energylike, momentumlike, and quantum horizon regions. Our analysis reveals that a correlated Weyl system can realize a topological metalinsulator transition only when a pair of interacting Weyl nodes with opposite topological charges are energylike, otherwise they are forbidden to interact to produce a band gap. We also demonstrate that only when the interacting Weyl fermions are energylike that the two opposite chiral surface modes can have a causal connection. In this sense, the quantum information (causality) stored in a volume element is thus also encoded on its surface, much like in the case of quantum black holes, where the quantum state outside a black hole horizon carries information about the internal state of the black hole. This result points to an interesting connection between the interacting Weyl systems and quantum black holes. Finally, our study indicates the presence of a quantum horizon region as a thick horizon in the causal structure of the interacting Weyl system.
Although we have focused on WeylCDW systems, our results are applicable more generally to interacting Weyl fermions in topological systems. For example, Weyl physics can be simulated in a 3D optical lattice^{38}, where the CDW effect could be produced by introducing a period2 superlattice (dimerization) using two additional orthogonal optical waves at double the inplane wavelengths^{39}. In our case, the CDW is generated via the spontaneous breaking of the translational symmetry, which leads to a pairing interaction between the Weyl fermions. In view of the universality of Weyl physics, however, our formalism will apply to more general pairing interactions that break the symmetries of the system. Our study, for the first time, shows how the key concepts of causality and the associated event horizon in spacetime can be carried over into the field of correlated Weyl materials, and thus unveils fundamental connections between condensed matter and highenergy physics.
Causality was long considered as the timeordered relationship between causes and effects until the advent of Einstein’s theory of relativity, in which causality is defined by the light cone in spacetime. We have introduced causal structure in the energymomentum space of the condensed matter systems. We can expect a far richer tapestry of possibilities driven by causal physics in the context of condensed matter systems, just as many more exotic fermionic excitations are supported by the vacuum of crystalline materials compared to that of free space.
Methods
CDW tightbinding model in real space
Consider a cubic system with one atom per unit cell and lattice parameter a. We first construct a twoband model with four Weyl points with broken inversion symmetry,
By taking Fourier transform of the lattice tightbinding model in momentum space, we get the hopping parameters as following,
where r = r_{i} − r_{j} and r_{i}, r_{j} are the displacements of lattice sites, and σ is the Pauli matrix which describes the orbital degree of freedom on the atom. Let a = 1 and the twoband Hamiltonian without interaction term is
where c_{i} = (c_{i,1}, c_{i,2}) and c_{i,1}, c_{i,2} are the electron annihilation operators with the orbital (pseudospin) index 1, 2 on the atom at the site r_{i}. Then we consider the semi1D CDW as Peierls dimerization in a double supercell along the x direction. We build a doublecell supercell along the \(\hat{x}\) direction and we denote the electron annihilation operators of the two atoms within the supercell at \({{{{{{{{\bf{r}}}}}}}}}_{i}^{{\prime} }\) as c_{i} and d_{i}, where c_{i} = (c_{i,1}, c_{i,2}), d_{i} = (d_{i,1}, d_{i,2}) and \({{{{{{{{\bf{r}}}}}}}}}_{i}^{{\prime} }\) as the displacements of super lattice sites. We can write the Hamiltonian on the supercell basis as
where τ is the Pauli matrix describing the degree of freedom between two atoms. Then, the Peierls dimerization can be realized by modifying the real space hopping strength between the two nearest neighbor atoms along the CDW direction in the supercell with a strength δ, and the interaction terms can be expressed in the supercell basis as
Then the full Hamiltonian becomes
It’s worth noting that the δτ_{x} terms are in the offdiagonal blocks of the four orbitals basis, which reflects the nature of Weyl nodes interacting with each other through CDW.
Firstprinciples calculations
Firstprinciples calculations for (TaSe_{4})_{2}I were performed using OpenMX code, where the generalizedgradient approximation, normconserving pseudopotentials, and optimized pseudoatomic basis functions were adopted^{40,41,42,43}. Three, two, two, and one optimized radial functions were allocated for the s, p, d, and f orbitals, respectively, for each Ta atom with a cutoff radius of 7 Bohr, denoted as Ta7.0s3p2d2f1. For the Se and I atoms, Se7.0s3p3d2f1 and I7.0s3p3d2f1 were adopted, respectively. A cutoff energy of 300 Ry was used for numerical integrations and for the solution of the Poisson equation. The firstprinciples calculations on Mo_{3}Al_{2}C were carried out using the Vienna Ab initio Simulation Package with the projector augmented wave potentials^{44}. The exchangecorrelation function was treated within the Perdew–Burke–Ernzerhof generalized gradient approximations^{41}.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request.
Code availability
All code used to generate the plotted band structures is available from the corresponding author upon request.
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Acknowledgements
We thank Justin Ripley for a helpful discussion. G.C. acknowledges the support of the National Research Foundation, Singapore under its Fellowship Award (NRFNRFF1320210010) and the Nanyang Assistant Professorship grant from Nanyang Technological University. The work at Northeastern University was supported by the Air Force Office of Scientific Research under award number FA95502010322, and it benefited from the computational resources of Northeastern University’s Advanced Scientific Computation Center (ASCC) and the Discovery Cluster. M.Z.H. was supported by the US DOE under the Basic Energy Sciences program (grant number DOE/BES DEFG0205ER46200). S.M.H. is supported by the NSTCAFOSR Taiwan program on Topological and Nanostructured Materials, Grant No. 1102124M110002MY3. C.C.L. acknowledges the National Science and Technology Council (NSTC) of Taiwan for financial support under Contract No. 1102112M032016MY2. F.C.C. acknowledges the support by the National Center for Theoretical Sciences and the Ministry of Science and Technology of Taiwan under grant no. MOST1102112M110013MY3. S.Y.X. acknowledges the support of the Center for the Advancement of Topological Semimetals (CATS), an Energy Frontier Research Center (EFRC) funded by the US Department of Energy (DOE) Office of Science, through the Ames Laboratory under contract DEAC0207CH11358 (fabrication and measurements), the STC Center for Integrated Quantum Materials (CIQM), National Science Foundation (NSF) award no. ECCS2025158 (data analysis), and the Corning Fund for Faculty Development. H.L. acknowledges the support by the National Science and Technology Council (NSTC) in Taiwan under grant number MOST 1112112M001057MY3. T.R.C. was supported by the Young Scholar Fellowship Program from the Ministry of Science and Technology (MOST) in Taiwan, under a MOST grant for the Columbus Program MOST1102636M006016, the National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences, Taiwan. Work at NCKU was supported by MOST, Taiwan, under grant MOST1072627E006001 and Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at NCKU.
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All the authors contributed to the intellectual content of this work. W.C.C. and G.C. initiated the project. W.C.C. proposed the conceptual idea of causal structure in energymomentum space and performed numerical simulations and analytical calculations with assistance from G.C., R.M., and S.M.H.. G.M. and G.C. performed firstprinciples calculations and analysis with assistance from C.C.L., T.R.C., F.C.C, H.L., and A. B. The materials search was done by G.C. with help from G.M., I.B., J.X.Y., Z.J.C., C.C.L. S.Y.X., and M.Z.H. W.C.C. wrote the original draft, and G.C., R.M., I.B., and A.B. revised the draft. G.C. and A.B. were responsible for the overall research direction, planning, and integration among different research units.
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Chiu, WC., Chang, G., Macam, G. et al. Causal structure of interacting Weyl fermions in condensed matter systems. Nat Commun 14, 2228 (2023). https://doi.org/10.1038/s4146702337931w
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DOI: https://doi.org/10.1038/s4146702337931w
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