Topological phases in pyrochlore thallium niobate Tl$_2$Nb$_2$O$_{6+x}$

The discovery of new topological electronic materials brings a chance to uncover novel physics and plays a key role in observing and controlling various intriguing phenomena. Up to now, many materials have been theoretically proposed and experimentally proved to host different kinds of topological states. Unfortunately, there is little convincing experimental evidence for the existence of topological oxides. The reason is that oxidation of oxygen leads to ionic crystal in general and makes band inversion unlikely. In addition, the realization of different topological states in a single material is quite difficult, but strongly needed for exploring topological phase transitions. In this work, using first-principles calculations and symmetry analysis, we propose that the experimentally tunable continuous solid solution of oxygen in pyrochlore Tl$_2$Nb$_2$O$_{6+x}$ ($ 0\leq x \leq 1.0$) leads to various topological states. Topological insulator, Dirac semimetal and triply degenerate nodal point semimetal can be realized in it via changing the oxygen content and/or tuning the crystalline symmetries. When $x = 1$, it is a semimetal with quadratic band touching point at Fermi level. It transits into a Dirac semimetal or a topological insulator depending on the in-plane strain. When $x = 0.5$, the inversion symmetry is spontaneously broken in Tl$_2$Nb$_2$O$_{6.5}$, leading to triply degenerate nodal points. When $x = 0$, Tl$_2$Nb$_2$O$_6$ becomes a trivial insulator with a narrow band gap. These topological phase transitions driven by solid solution of oxygen are unique and physically plausible due to the variation of valence state of $Tl^{1+}$ and $Tl^{3+}$. This topological oxide will be promising for studying correlation induced topological states and potential applications.


INTRODUCTION
The remarkable discoveries of various quasiparticles in solids with or without the counterpart 3 in high-energy physics have inspired intensive studies on topological electronic materials (TEMs). [1][2][3][4][5] They are promising for future applications, owing to low-dissipation transport property and intrinsic insensitivity to environment perturbations. TEMs are characterized as having electronic structures with non-trivial topology in momentum space. Typically, TEMs can be classified into topological insulator (TI), [6][7][8] topological semimetal (TSM) [9][10][11][12][13][14] and topological superconductor. The initial impetus originates from the TI, which exhibits linear dispersive surface/edge states and can make novel quantum electronic devices compatible with current electronic technologies. Moreover, magnetically doped TIs are proved to hold quantum anomalous Hall effect. 15,16 Recently, research focus of TEMs has shifted towards TSMs, which have exotic transport properties. [17][18][19] TSMs are special metals with Fermi surfaces composed of and only of nodal points. They include four members, namely Weyl semimetal (WSM), 20,21 Dirac semimetal (DSM), 10,11 nodal line semimetal (NLSM) 22,23 and triply-or multiply-degenerate nodal point (TDNP or MDNP) semimetal. [24][25][26][27][28][29] These TSMs are distinguished from each other by the degeneracy of the nodal points and the topological protection mechanism. WSM has isolated double-degenerate nodal points at or close to the Fermi level and is topologically robust as long as the translation symmetry of lattice is preserved, while DSM has isolated four-fold degenerate nodal points and is protected by proper crystalline symmetries. NLSM contains continuous nodal points forming lines, while MDNPs host three-, six-or eight-fold degenerate nodal points. Both of them need some proper crystalline symmetries, such as rotation, mirror and/or nonsymmorphic translation. The TDNP in WC family is a crossing point formed by a nondegenerate band and a double-degenerate band. 25,27 It is identified as an intermediate state between Weyl and Dirac TSM, bringing in new interesting physics. However, there have been quite few reports on TEMs 4 discovered in oxide materials till now 12,[30][31][32][33][34] and their properties are to be extensively explored once they are available experimentally.
Tl2Nb2O6+x is in pyrochlore structure, 35 which has been known since 1960s. The ideal pyrochlore Tl2Nb2O6O'x = 1 was first discovered, and then Fourquet et al. demonstrated that there exist continuous solid solutions Tl2Nb2O6O'x (0 ≤ x ≤ 1.0) via thermogravimetric analysis (TGA), chemical analysis, and X-ray thermodiffractometry. 36 Interestingly, with the removal of (1-x) O' out of the Tl2Nb2O7, the Tl atoms could shift along [111] axis and be away from the central symmetric position, leading to spontaneous inversion symmetry breaking, which brings a very unique way to systematically tune the topological phases in it.
In this work, we propose that Tl2Nb2O6+x can have several attractive topological features as x changes. DSM, TI and TDNP semimetal states all can be realized in Tl2Nb2O6+x series via tuning the crystalline symmetry or oxidation level. When x = 1, Tl2Nb2O7 is cubic and is a zero-gap semimetal similar to HgTe with quadratic contact point (QCP) at Γ. 37 In-plane compressive and tensile strain can drive it into DSM and TI, respectively. When x = 0.5, Tl2Nb2O6.5 has no inversion symmetry and is a TDNP semimetal. When x = 0, Tl2Nb2O6 is a trivial insulator with narrow band gap. Because strain engineering greatly contributes to exploring physics [37][38][39][40] and quite a small strain is introduced here, it is feasible for experimental observation of the topological states in Tl2Nb2O6+x. Moreover, many intriguing phenomena and rich physics have been found in pyrochlore oxides, such as complex magnetic phases, superconducting and multiferroics. Thus, our studies may provide a unique platform for investigating the strongly correlated topological phases, multi-phase control and potential applications.

Crystal structure
The ternary oxide Tl2Nb2O6+x belongs to the pyrochlore structure. The ideal structure of x = 1 is in space group Fd-3m (No. 227) (Fig. 1a), whose first Brillouin zone (BZ) is shown in Fig. 1b.
Four Tl atoms will form a tetrahedra with an O' atom at the center. Nb and O atoms are in 16c (0, 0, 0) and 48f (0.2925, 1/8, 1/8) positions, respectively, forming NbO6 octrahedra. The experimental lattice constant is a0 = 10.622 Å and is used for the calculations in the present paper. 36 Compared with the case of x = 1, the missing of O' makes the same number of Tl ions change from +3 to +1 and shift away from the centrosymmetric position (Fig. 1c). Though the distribution of O' vacancy and Tl +1 ions is somehow random in x = 0.5 case, we take away half of O' atoms in the primitive unit cell (Z = 2), and Tl atoms are shifted away from 16d to the 32e (0.507, 0.507, 0.507) positions. The lattice constant is taken as 10.6397 Å according to the experimental value in x = 0.490 case, which is the closest to 0.5. 36 The crystal structure symmetry becomes R3m (No. 160) without inversion center, being different from that of Tl2Nb2O7. When x is reduced to 0, all the Tl atoms becomes +1 and stay on the noncentral position. The lattice constant is taken as 10.6829 Å, which is the experimental value when x = 0.070. 36

Band structure of bulk Tl2Nb2O7
The 5d orbitals of Tl 3+ atom split into eg and t2g orbitals due to the crystal field formed by oxygen hexagonal bipyramid. Without considering the SOC, Tl2Nb2O7 is a QCP semimetal with a triply degeneracy at Γ point ( Fig. 1d), which is also verified by the hybrid functional HSE06 calculation (red color bands in Fig. 1d). This is the same as the results in Materiae, an online 6 database of topological materials, and other similar databases. [41][42][43] The states at Γ point mainly come from t2g orbitals composed by hybridization of Tl 5d and O 2p orbitals. When SOC is taken into consideration, SOC splitting among p orbitals is opposite to that among t2g orbitals. 44 Therefore, the final effective SOC of the Γ point is determined by the competition between Tl t2g and O p spin-orbit splitting. 44 With SOC, the QCP at Γ (Fig. 1d) splits into a double degenerate Γ7 + band and a fourfold degenerate Γ8 + states. (Fig. 1e, 2c) Γ7 + is higher than Γ8 + , which indicates that the effective SOC in these bands is negative due to the d-p hybridization as discussed in TlN. 44 The fourfold degenerate Γ8 + is half occupied and becomes another QCP similar to HgTe. 37

Band structure and topological property of strained Tl2Nb2O7
The QCP at Γ is protected by Oh point group. Breaking the Oh, this fourfold degeneracy will be lifted, and thus topological insulating states or topological semimetal states are formed. 37 In this section, we consider the topological phase transition in Tl2Nb2O7 system with strain (positive strain refers to expansion, while negative strain refers to compression). The related space group is changed from Fd-3m (No. 227) to I41/amd (No. 141). A top view of the structure without strain is shown in Fig. 2a, while its non-SOC and SOC bands are shown in Fig. 2b, 2c for comparison.
From the pictures, we can see when SOC is included, the gapless semimetal is formed owning to the fourfold degeneracy of Γ8 + , which is also similar to the case of Cu2Se. 45 In Tl2Nb2O7, Γ7 + states are higher than Γ8 + states, while in Cu2Se, Γ8 + states are higher than Γ7 + states.
A compressive strain of -1% in xy-plane is applied (Fig. 2d)  The IR of the band shown in black is B2, while that of the double degenerate bands shown in red and blue is E. 46 Thus, in the non-SOC case without the spin degree of freedom, two TDNPs related with inversion or time-reversal symmetry can be formed in the -Z to Γ and Γ to Z directions, respectively. The energy band with SOC in strain of -1% case is also calculated (Fig.   2f). SOC drives a phase transition from the QCP semimetal to Dirac semimetal, where two fourfold degenerate Dirac points are on the path -Z to Γ and Γ to Z, respectively.  (Fig. 3a). There are two branches of surface states emerging in the gap and touching at X and Z points due to Kramer's degeneracy. One branch connects to the conduction bulk bands, 8 while the other one links the valence bulk bands. Moreover, iso-surface with energy at Dirac points of the surface states is calculated (Fig. 3b). There exists a pair of surface Fermi arcs connecting two projected Dirac nodes.
An expansion strain of 1% in the xy-plane is applied to the system (Fig. 2g), and the related lattice constants are changed to a = b = 1.01 a0, and c = 0.98 a0. The non-SOC band structure is shown in Fig. 2h. There exists one band intersection along the X-Γ, which is formed by bands in red and blue. The two bands belong to IRs of the C2v point group: A2 and B2, respectively. 46 In fact, this nodal point is on a nodal line in kx-ky plane, which is protected by the coexistence of inversion and time-reversal symmetries (see Supplementary Information Fig. S2). When SOC is included, the gap is fully opened in the entire nodal line (Fig. 2i), generating a strong TI with global gap of ~13 meV. The same Wilson loop method is used here to identify the topological property of the structure with strain of 1% (see Supplementary Information Fig. S3). Topologically protected surface Dirac cone on (010) surface connecting the conduction and valence bands emerges inside the gap (Fig. 3c). These two branches of surface states also touch at X and Z points, similar to the case of -1% strain. Furthermore, iso-energy plot of surface states at energy of -6 meV in the gap is displayed (Fig. 3d).
To understand the phase transition mechanism under strain, an effective k • p model is constructed (see Supplementary Information). From Fig. S4, we can see the band structures coincide with those from first-principles calculations.

TDNPs in noncentrosymmetric Tl2Nb2O6.5
Compared with the case of x = 0, the extra O' (in the network of Nb2O6) oxidizes one of the nearest four Tl atoms to +3 and repels the other three monovalent Tl. Therefore, Tl atoms are 9 away from the centrosymmetric position. Such natural breaking of inversion symmetry provides a new material hosting intrinsic TDNPs, whose topological properties are studied by calculating the band structure with SOC (Fig 4). We can see band crossings along Γ-L ([111] axis) host massless fermions, which appear in a pair due to time-reversal symmetry. To shed light on the forming mechanism of TDNPs, wave-function analysis is performed. These bands belong to IRs of the C3v point group: the IRs of the black/green, blue and red bands are E1/2, 2E3/2, 1E3/2, respectively. 46 Thus, band crossings can form two TDNPs near Γ point, protected by C3v and time-reversal symmetries. HSE06 calculation without SOC also confirms the existence of TDNP along Γ-L direction (see Supplementary Information Fig. S5). These TDNPs are about 0.4 eV below Fermi level, which is higher than those in MoP 28 while lower than those in WC. 29 .
To understand the mechanism forming TDNPs in Tl2Nb2O6.5, we further introduce a k • p model around Γ point with SOC included. In order to make the discussions simple, we chose the  121  132  141  2  121  20  21  22  221  132  2  132  221  20  21  22  121  2  141  132  121  10  11  12 () parameters that can be obtained by fitting the first-principles results, listed as Table S1. The bands comparison between the k • p model and the first-principles calculations is shown in Fig. S6 (see Supplementary Information). Now it is easy to check that, along the kc axis, the |±1/2> band is two-fold degenerate. Meanwhile, the |±3/2> bands split into two bands which cross with |±1/2> band, forming two TDNPs (Fig. 4b).
Thanks It is noted that Tl2Ta2O6+x 35 has the same chemical and physical properties as Tl2Nb2O6+x, as well as the band topology. La2Hf2O7 is found to be a QCP semimetal in GGA calculation and it becomes a topological crystalline insulator in GGA+SOC calculation, which are the same as those in topological material database Materiae, and other similar databases. [41][42][43] The effective SOC splitting in La2Hf2O7 is found to be opposite to that of Tl2Nb2O7, while the absence of valence variation in La ions makes the oxygen content hard to be tuned in La2Hf2O7.

CONCLUSION
In summary, we propose that a pyrochlore oxide Tl2Nb2O6+x with continuous oxidation level x can host various topological phases, which is realized by a change of valence state of Tl from 1+ 11 to 3+ and the displacement of its atomic position. Tl2Nb2O7 with x = 1 is a semimetal with QCP due to cubic symmetry. When a small in-plane tensile strain is applied, a nodal line appears in the

COMPUTATIONAL METHODS
We have performed first-principles calculations within density functional theory (DFT), using the Vienna ab initio simulation package (VASP). 48,49 Exchange-correlation potential is