Abstract
Nonsymmorphic symmetries, which involve fractional lattice translations, can generate exotic types of fermionic excitations in crystalline materials. Here we propose a topological phase arising from nonsymmorphic symmetries—the hourglass Dirac chain metal, and predict its realization in the rhenium dioxide. We show that ReO_{2} features hourglasstype dispersion in the bulk electronic structure dictated by its nonsymmorphic space group. Due to time reversal and inversion symmetries, each band has an additional twofold degeneracy, making the neck crossingpoint of the hourglass fourfold degenerate. Remarkably, close to the Fermi level, the neck crossingpoint traces out a Dirac chain—a chain of connected fourfolddegenerate Dirac loops—in the momentum space. The symmetry protection, the transformation under symmetrybreaking, and the associated topological surface states of the Dirac chain are revealed. Our results open the door to an unknown class of topological matters, and provide a platform to explore their intriguing physics.
Introduction
Topological metals or semimetals, which host robust fermionic excitations around protected bandcrossing points, have been a focus of current research. For example, Weyl and Dirac semimetals possess two and fourfold degenerate isolated bandcrossing points close to the Fermi level, around which the quasiparticles resemble the relativistic Weyl and Dirac fermions^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. Under certain symmorphic symmetry operations such as mirror or inversion, the crossing points may also form onedimensional (1D) nodal loops^{17,18,19,20,21,22,23,24,25,26,27,28,29,30} or even linked nodal loops^{31,32,33,34}, but such loops are usually vulnerable against spin–orbit coupling (SOC) and can be removed without altering the symmetry, hence they are termed as accidental nodal loops. Recently, it was realized that nonsymmorphic symmetries, which involve fractional lattice translations, could play a key role in stabilizing the bandcrossing points^{35,36,37,38,39,40,41,42,43}. They have two important effects. First, the degeneracies enabled by nonsymmorphic symmetries could be robust against SOC. Particularly, spin–orbit nodal loops with two or even fourfold degeneracy have been theoretically proposed^{44,45,46,47,48}. Second, nonsymmorphic symmetries may entangle multiple bands together, so that the resulting crossing points are unavoidable and entirely dictated by the crystalline symmetry. Such bandcrossing points are thus referred to as essential. For example, it was found that bands are entangled into groups of four and form hourglassshaped dispersion on the 2D surface of nonsymmorphic insulators KHgX (X = As, Sb, Bi)^{49,50,51}. Theoretical modeling suggested that such hourglass fermions may also exist in the bulk of 3D crystals^{52}, and interestingly, Bzdušek et al.^{53} showed that the neck point of the hourglass may trace out a Weyl chain of twofolddegenerate nodal loops, when multiple nonsymmorphic operations are present.
Although the essential bandcrossings are solely determined by the space group for which theoretical analysis has offered valuable guidelines, the search for realistic materials that exhibit them at low energy is still challenging. This is because the bands in real materials typically have complicated 3D dispersions, such that the crossing point that we are chasing may be far away from the Fermi energy. The situation could be even worse for nodal loops, since the points on the loop are not guaranteed to have the same energy, there might be large energy variation around the loop. So far, the proposed nonsymmorphic topological metals are still limited, therefore, besides exploring new topological phases, it is also urgent to discover more suitable candidate materials to expedite experimental studies of their intriguing properties.
Here we predict a remarkable topological phase that is enabled by nonsymmorphic symmetry—the hourglass Dirac chain metal, of which the existence can be argued purely from symmetry analysis. Furthermore, based on firstprinciples calculations, we demonstrate that this phase is realized in an existing material—ReO_{2}, which serves as a solid example to illustrate the essential physics. We show that fourfold degenerate lines and hourglasstype essential bandcrossings occur in the bulk band structure of ReO_{2}. Due to time reversal (\({\cal T}\)) and inversion (\({\cal P}\)) symmetries, the hourglass here is actually doubled, and the neck crossingpoint here becomes a Dirac point with fourfold degeneracy. Remarkably, close to Fermi level, the neck point traces out a Dirac chain—a chain of connected (fourfolddegenerate) Dirac loops—in the momentum space, as schematically shown in Fig. 1a, b. This chain is essential, robust against SOC, and dictated by two orthogonal glide mirror planes combined with \({\cal T}\) and \({\cal P}\) symmetries. In addition, there is another pair of isolated bulk hourglass Dirac points on a symmetry line (Fig. 1a). We clarify the protection of these exotic bandcrossings, and discuss their transformations under symmetrybreaking. At the sample surface, we find an interesting coexistence of drumheadtype surface states and surface Fermi arcs. The bulk hourglass Dirac chain as well as the topological surface states should be readily probed in experiment. Our findings provide an exciting platform to explore the intriguing topological fermions from nonsymmorphic symmetries.
Results
Symmetry and band structure
We demonstrate the essential physics of hourglass Dirac chain using ReO_{2} as a solid example. Singlecrystal ReO_{2} is observed with three structures denoted as α, β, and rutiletype^{54, 55}. βReO_{2} is energetically more stable, and is experimentally shown to be a stable paramagnetic metal in a wide temperature range from the ambient temperature down to the liquid helium temperature (~4.2 K)^{56, 57}. Hence we focus on βReO_{2} here. It adopts the PbO_{2}type orthorhombic crystal structure with space group no. 60 (Pbcn)^{54}. As shown in Fig. 2a, the structure is characterized by zigzag chains of Re atoms running along the caxis, and each Re atom is contained in a slightly distorted octahedron of six surrounding O atoms. As we shall see, the hourglass Dirac chain is solely dictated by the space group symmetry of the structure, which may be generated by the following symmetry operations: the inversion \({\cal P}\), and two glide mirror planes involving half lattice translations \(\tilde {\cal M}_x:\left( {x,y,z} \right) \to \left( {  x + \frac{1}{2},y + \frac{1}{2},z} \right)\) and \(\tilde {\cal M}_z:\left( {x,y,z} \right) \to \left( {x + \frac{1}{2},y + \frac{1}{2},  z + \frac{1}{2}} \right)\). Here the tilde above a symbol indicates that it is a nonsymmorphic symmetry. One also notes that combining all three operations leads to a third glide mirror \(\tilde {\cal M}_y:\left( {x,y,z} \right) \to \left( {x,  y,z + \frac{1}{2}} \right)\). The Brillouin zone of the structure is shown in Fig. 2b.
The electronic band structure of ReO_{2} is calculated by firstprinciples methods based on the density functional theory (DFT). SOC was included, and possible correlation effect of Re(5d) orbitals was tested. The details are presented in the Methods section. Following experimental results, we focus on the paramagnetic phase of βReO_{2}, the possibility of magnetic ordering at ultralow temperature (<4.2 K) will be discussed later in the Discussion section. In octahedral crystal field, Re(5d) orbitals are split into t _{2g } and e _{ g } groups, with the latter at higher energy. For Re^{4+} with 3 valence electrons, the Ret _{2g } orbitals will be halffilled, resulting in a metallic state. Figure 2c shows the calculated band structure of ReO_{2} along with the projected density of states (PDOS). Indeed, one observes a metallic phase with fairly dispersive bands around Fermi level, and the lowenergy states are dominated by the Ret _{2g } orbitals. Understanding that each band is at least twofold degenerate due to the presence of \({\cal T}\) and \({\cal P}\), two interesting type of band features can be observed from Fig. 2c: Firstly, all bands are fourfold degenerate along U–X, Z–T, and T–R (Fig. 2b); secondly, hourglassshaped dispersions appear on T–U, U–R, and T–Y (Fig. 2d–f). The neck point of the hourglass is a crossingpoint with fourfold degeneracy. In the following, we shall demonstrate each feature purely from the symmetry of the system.
Fourfold degenerate highsymmetry nodal lines
Let us first investigate the first feature regarding the fourfold degeneracy along the three highsymmetry lines. Consider the U–X line at k _{ x } = π and k _{ y } = 0 (in unit of the inverse of respective lattice parameter). It is an invariant subspace of \(\tilde {\cal M}_x\), so each Bloch state \(\left u \right\rangle\) there can be chosen as an eigenstate of \(\tilde {\cal M}_x\). Since
the \(\tilde {\cal M}_x\) eigenvalue g _{ x } must be ±i on U–X. Here T _{010} denotes the translation along y by one unit cell, and \(\overline E\) is the 2π spin rotation. The commutation relation between \(\tilde {\cal M}_x\) and \({\cal P}\) given by
means that \(\left\{ {\tilde {\cal M}_x,{\cal P}} \right\} = 0\) on U–X. Consequently, each state \(\left u \right\rangle\) and its Kramersdegenerate partner \({\cal P}{\cal T}\left u \right\rangle\) must share the same \(\tilde {\cal M}_x\) eigenvalue. For example, assume \(\left u \right\rangle\) has g _{ x } = +i (denoted as \(\left { + i} \right\rangle\)), then
where in the second step we used the fact that \({\cal T}\) is an antiunitary operator. Same result holds for a state with g _{ x } = −i. On the other hand, U–X is invariant under another antiunitary symmetry \(\tilde {\cal M}_z{\cal T}\), which also generates a Kramerslike degeneracy since \(\left( {\tilde {\cal M}_z{\cal T}} \right)^2 =  1\) on U–X. Note that
where the minus sign is due to the anticommutativity between two spin rotations, i.e., {σ _{ x }, σ _{ z }} = 0, so that \(\left[ {\tilde {\cal M}_x,\tilde {\cal M}_z} \right] = 0\) on U–X. Following similar derivation in Eq. (3), one finds that \(\left u \right\rangle\) and \(\tilde {\cal M}_z{\cal T}\left u \right\rangle\) have opposite g _{ x }. Thus, the four states, \(\left\{ {\left u \right\rangle ,{\cal P}{\cal T}\left u \right\rangle ,\tilde {\cal M}_z{\cal T}\left u \right\rangle ,{\cal P}\tilde {\cal M}_z\left u \right\rangle } \right\}\) at the same kpoint on U–X must be linearly independent and degenerate with the same energy. The fourfold degeneracy along Z–T and T–R can also be derived in a similar way (Supplementary Note 1).
Hourglass dispersion and Dirac chain
Next, we turn to the second feature regarding the hourglass dispersion. Consider the line U–R. It is invariant under both \(\tilde {\cal M}_x\) and \(\tilde {\cal M}_z\). From Eq. (4), \(\left[ {\tilde {\cal M}_x,\tilde {\cal M}_z} \right] = 0\) on U–R, so each state \(\left u \right\rangle\) there can be chosen as simultaneous eigenstate of both operators, with eigenvalues \(\left( {g_x,g_z} \right) = ( \pm i, \pm 1)e^{  ik_y/2}\). Using the commutation relation in (2) and \(\tilde {\cal M}_z{\cal P} = T_{111}{\cal P}\tilde {\cal M}_z\), one finds that
so the Kramers pair \(\left u \right\rangle\) and \({\cal P}{\cal T}\left u \right\rangle\) at any kpoint on U–R share the same (g _{ x }, g _{ z }) eigenvalues. In addition, points R and U are timereversal invariant momenta. At R = (π, π, π), (g _{ x }, g _{ z }) = (±1, ±i), hence if \(\left u \right\rangle\) has eigenvalues (g _{ x }, g _{ z }), its Kramers partner \({\cal T}\left u \right\rangle\) must have (g _{ x }, −g _{ z }). Similarly, at U = (π, 0, π), since (g _{ x }, g _{ z }) = (±i, ±1), \({\cal T}\left u \right\rangle\) must have eigenvalues (−g _{ x }, g _{ z }) if \(\left u \right\rangle\) has (g _{ x }, g _{ z }).
Focusing on the eigenvalue g _{ x }, the analysis shows that the four states in the degenerate quartet (may be chosen as \(\left\{ {\left u \right\rangle ,{\cal T}\left u \right\rangle ,{\cal P}\left u \right\rangle ,{\cal P}{\cal T}\left u \right\rangle } \right\}\)) at R all have the same g _{ x } (+1 or −1); whereas at point U, they consist of two states with g _{ x } = +i and two other states with g _{ x } = −i. Hence there has to be a switch of partners between two quartets along U–R, during which the eight bands must be entangled to form the hourglasstype dispersion. The situation is schematically shown in Fig. 3a. It is important to note that the fourfolddegenerate neck crossingpoint (denoted as D on U–R) is protected because the two crossing doubly degenerate bands have opposite g _{ x } (with each degenerate pair sharing the same g _{ x }, as shown in Eq. (5) and illustrated in Fig. 3a).
Furthermore, since the whole k _{ x } = π plane is invariant under \(\tilde {\cal M}_x\), g _{ x } is well defined for any state on this plane. Hence the above argument applies to any path lying on the k _{ x } = π plane and connecting points U and R, which should feature an hourglass spectrum with fourfolddegenerate crossingpoint in between. The crossingpoint must trace out a closed Dirac loop L _{1} on this plane, as indicated in Fig. 3b. One also notes that not only U, actually any point P on U–X has fourfold degeneracy with two g _{ x } = +i and two g _{ x } = −i, as we analyzed before. Thus hourglass pattern is guaranteed to appear on any path connecting R to an arbitrary point on U–X [Fig. 3b].
Similar analysis as in the last two paragraphs applies to the k _{ z } = π plane, with the role played by \(\tilde {\cal M}_x\) replaced by \(\tilde {\cal M}_z\). It shows that hourglass pattern appears on any path connecting U to an arbitrary point on Z–T or T–R, and the neck point of the hourglass traces out a second Dirac loop L _{2}, as illustrated in Fig. 3c. Interestingly, L _{1} and L _{2} are orthogonal to each other, and they touch at the point D on the U–R line. Thus they constitute a Dirac chain in the momentum space, as shown in Fig. 1a.
Figure 3d, e shows the locations of the Dirac loops obtained from DFT calculations, which are consistent with our symmetry analysis. The chain is close to the Fermi level and has small energy variation (<0.2 eV). We stress that the presence of such bandcrossing pattern is solely determined by the space group (plus \({\cal T}\)). However, whether such crossings could manifest around Fermi level and have relatively small energy variation will depend on the specific material.
Up to now, one may wonder whether there exists a third loop on the k _{ y } = π plane, given that \(\tilde {\cal M}_y\) is also a symmetry. It turns out not to be the case. Consider any state \(\left {g_y} \right\rangle\) on k _{ y } = π plane with \(\tilde {\cal M}_y\) eigenvalue g _{ y }, one can show that
Thus each Kramers pair \(\left u \right\rangle\) and \({\cal P}{\cal T}\left u \right\rangle\) have opposite g _{ y }, which is in contrast with Eq. (5) for the other two planes. As a result, \(\tilde {\cal M}_y\) can no longer protect the neck crossingpoint, since each doubly degenerate band have both \(\tilde {\cal M}_y\) parities and two such bands would generally hybridize to open a gap. Thus a third Dirac loop on the k _{ y } = π plane does not appear. This is indeed confirmed by our DFT result. Nevertheless, symmetry does dictates hourglass dispersion with an isolated Dirac point on T–Y (Fig. 2f), due to the presence of additional \(\tilde {\cal M}_x\) symmetry on this line (Fig. 1a) (see Supplementary Note 2 for the analysis).
Surface states
Nodal loops could feature topological drumheadlike surface states^{17}. We find similar phenomena for the Dirac chain here. For example, on the (001) surface, the projected loop L _{2} is centered around \({\bar{\mathrm X}}\) point, around which one indeed observes a pair of drumhead surface bands emanating from the projected bulk bandcrossing point (Fig. 4a, b). Note that at the surface, due to the broken inversion symmetry, the spindegeneracy of the surface bands are lifted by the strong SOC. From a slab calculation, we verify that on each surface (top or bottom), the two drumhead surface bands are indeed spinsplit and nondegenerate (Supplementary Fig. 3). Similar observation is made on the (100) surface as well.
Interestingly, we find that the pair of isolated Dirac points on T–Y also generates surface Fermi arcs. As shown in Fig. 4c for the (010) surface, the arcs connect the surfaceprojections of the bulk Dirac points around the protection of T point in the surface Brillouin zone, similar to the Dirac semimetals Na_{3}Bi and Cd_{3}As_{2} ^{6, 7}. In Weyl semimetals, the surface Fermi arcs are dictated by the nontrivial topological charges (Chern number of ±1) associated with the Weyl points. However, a Dirac point carriers zero topological charge (because it consists of two Weyl points with opposite charges), hence the protection from topological charge is not guaranteed. In the current case, to reveal the possible mechanism that protects the Fermi arcs, we notice that the plane in Brillouin zone containing the points Z, T, and S is invariant under \({\cal T}\) and without bandcrossing. Hence a 2D \({\Bbb Z}_2\) invariant can be defined for this plane and is found to be nontrivial (Supplementary Fig. 5), which dictates a Kramers pair of surface states on the \({\tilde {\mathrm T}}\)\({\tilde {\mathrm S}}\) line of the (010) surface. Thus, the Fermi arcs on this surface cannot be eliminated and is protected by the nontrivial bulk \({\Bbb Z}_2\) invariant.
Discussion
Our work not only reveals a hitherto unknown topological phase, it also finds an existing material for its realization. The hourglass Dirac chain revealed in ReO_{2} represents an essential bandcrossing: it is robust against SOC and dictated by the crystalline symmetry. We also studied a few other materials with the same space group symmetry, and indeed the same qualitative features can be observed in their band structures (Supplementary Note 8).
As the bandcrossings discussed here are all dictated by symmetry, they must be kept as long as the space group symmetry is maintained. In Fig. 5a, b, we demonstrate that when we distort the crystal lattice while maintaining the symmetry, the shape and the size of the chain can change, but it cannot be destroyed. In contrast, if we break the symmetry, e.g. by varying the angle between a and b axis away from 90° (corresponding to some shear strain) to change the lattice from orthorhombic to monoclinic, the chain will lose (part of) its protection. In this case, the distortion breaks \(\tilde {\cal M}_x\) but still preserves \(\tilde {\cal M}_z\) and \({\cal P}\), thus the Dirac loop on the k _{ z } = π plane is still protected (Fig. 5c), whereas the loop on the k _{ x } = π plane and the Dirac point on T–Y are removed. These are confirmed by the DFT calculation.
Compared with the Weyl chain proposed in ref. ^{53}, the Dirac chain here is fundamentally different due to the doubled degeneracy, similar to the distinction between Dirac and Weyl points. The added degeneracy comes from the preserved inversion symmetry, which is explicitly broken for the Weyl chain case. We stress that this doubling in degeneracy actually poses a more stringent condition regarding the symmetry protection of the bandcrossings: Superposing two copies of Weyl chain does not necessarily lead to a Dirac chain—they may hybridize and open a gap; the Dirac chain requires additional symmetry protection than the Weyl chain. Indeed, if there were no such degeneracy in the current case, the (missing) loop on the k _{ y } = π plane would be well protected. In terms of surface states, the Dirac chain metal here possesses a pair of spinsplit drumhead surface bands for a single surface, although there is no spinsplitting in the bulk bands; whereas for the Weyl chain metals, both surface and bulk bands are without spindegeneracy, and a surface typically has only one drumhead surface band. Furthermore, like the Dirac semimetal, the Dirac chain metal may also be considered as a parent phase for other topological phases (including Weyl chain metal) under symmetry breaking. For example, by suitably breaking \({\cal T}\) or \({\cal P}\), we can transform a Dirac chain metal into a Weyl chain metal or a nodalloop metal (Supplementary Note 6).
Several recent works proposed the nodallink semimetal phase^{31,32,33,34}, which also contains multiple nodal loops. However, a link structure (in which the loops do not touch each other) is topologically different from a chain. The nodallink models in those works do not require any nonsymmorphic symmetry; the linked loops are generally vulnerable against SOC; and the link is not essential in the sense that it can be removed without breaking the symmetry of the system. These features are distinct from those of the Dirac chain studied here.
In experiment, the paramagnetic metal phase of βReO_{2} was shown to persist from room temperature down to the liquid helium temperature (4.2 K), and no magnetic ordering has been found^{56, 57}. Hence the exotic band features reported here should be readily accessible for experimental measurements. In DFT + U calculations, we find that a relatively large U value could drive the system towards an antiferromagnetic (AFM) insulator phase (Supplementary Note 7), indicating the possibility of magnetic ordering at very low temperatures (at least < 4.2 K according to experiment). Interestingly, we find that even in the AFM state (if it indeed exists), the Dirac chain or one of the Dirac loops may still be preserved if the magnetic moment is aligned along certain highsymmetry directions (Supplementary Fig. 10).
The Dirac chain, the hourglass dispersion, and the surface states are close to the Fermi level. They could be directly imaged in ARPES measurement and compared with our calculation results. Besides ARPES, we also suggest several interesting effects derived from the nontrivial bulk and surface states that could be useful for characterizing Dirac chain metals.
As for the bulk states, it was predicted that under an external magnetic field parallel to the Dirac loop plane, there will appear an almost flat Landau band at the loop energy^{58}. This will lead to a pronounced peak in the density of states which can be detected by the scanning tunneling spectroscopy. For a Dirac chain as in Fig. 1a, one expects that the peak will be most pronounced when the B field is along the ydirection (parallel to both loops), and it will be relatively small when the field is not parallel to either loop. In addition, it has been shown that the nodalloop dispersion leads to distinct scaling in optical absorption that Imε(ω) scales as 1/ω, where ε is the dielectric function and ω is the light frequency^{59}.
The drumheadtype surface states may also lead to several interesting effects. It has been argued that they could produce a huge surface density of states, which may offer a route toward hightemperature superconductivity^{60}. The recent work by Li et al.^{26} attributed the unusually high surface density of states on the Be (0001) surface to the drumhead surface states, which combined with the strong electron–phonon coupling found on that surface^{61} may lead to a surface superconductivity (yet to be confirmed by experiment). Interestingly, the giant enhancement of the Friedel oscillation on the Be (0001) surface was also found to be due to these nontrivial surface states^{26}. In addition, with electron–electron interaction, the drumhead surface states may lead to a surface ferromagnetism, as discussed by Liu and Balents^{62}. These effects are also expected for the Dirac chain metals, and they can be detected by surfacesensitive probes such as scanning tunneling microscopy/spectroscopy (for Friedel oscillation and superconductivity) and surface magnetooptic Kerr effect (for surface magnetism). More interestingly, the orthogonal loops dictate the presence of drumhead surface states on multiple surfaces. For the case in Fig. 1a, the drumhead surface states would appear on (100) and (001) surfaces but not on the (010) surface. Thus, the different surfaces of a Dirac chain material could exhibit very different behaviors, e.g. in terms of the Friedel oscillation strength and the possible surface superconducitivity/ferromagnetism, as determined by the surface orientation relative to the chain.
Methods
Firstprinciples calculation
The firstprinciples DFT calculations are performed by using the Vienna Abinitio Simulation Package^{63, 64}. The projector augmented wave (PAW) method^{65} was employed to model the ionic potentials, and the generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE)^{66} realization was adopted for the exchangecorrelation functional. The energy cutoff was set as 400 eV. Energy and force convergence criteria are set to be 10^{−6} eV and 0.01 eV Å^{−1}, respectively. Γcentered kmesh with size 11 × 11 × 11 was used for the Brillouin zone sampling. The surface states are studied using the method with maximally localized Wannier functions^{67,68,69}. As Re(5d) orbitals may have correlation effects, we also validate our result by using the GGA + U method^{70}. Several onsite Hubbard U values from 0 to 1.5 eV were tested, which yield no appreciable difference. Hence in the main text, we focus on the GGA results. The experimental values of the lattice parameters (a = 4.809 Å, b = 5.643 Å, c = 4.601 Å)^{54} were used in the calculation.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank D.L. Deng for valuable discussion. This work is supported by Singapore Ministry of Education Academic Research Fund Tier 2 (MOE2015T22144) and Tier 1 (SUTDT12015004). We acknowledge computational support from the Texas Advanced Computing Center and from the National Supercomputing Centre Singapore.
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S.S.W. initiated the work. S.S.W. and X.L.S. performed the firstprinciples calculations. Y.L., S.S.W., Z.M.Y. and S.A.Y. conducted the symmetry analysis. All authors contributed to the analysis of the results. X.L.S. and S.A.Y. supervised the project.
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Wang, SS., Liu, Y., Yu, ZM. et al. Hourglass Dirac chain metal in rhenium dioxide. Nat Commun 8, 1844 (2017). https://doi.org/10.1038/s41467017019863
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