Double Dirac Nodal Line Semimetal with Torus Surface State
Xiao-Ping Li, Botao Fu, Da-Shuai Ma, Chaoxi Cui, Zhi-Ming Yu, Yugui Yao
DDouble Dirac Nodal Line Semimetal with Torus Surface State
Xiao-Ping Li, Botao Fu, Da-Shuai Ma, Chaoxi Cui, Zhi-Ming Yu, ∗ and Yugui Yao † 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 College of Physics and Electronic Engineering, Center for Computational Sciences,Sichuan Normal University, Chengdu, 610068, China
We propose a class of nodal line semimetals that host an eight-fold degenerate double Diracnodal line (DDNL) with negligible spin-orbit coupling. We find only 5 of the 230 space groups hostthe DDNL. The DDNL can be considered as a combination of two Dirac nodal lines, and has atrivial Berry phase. This leads to two possible but completely different surface states, namely, atorus surface state covering the whole surface Brillouin zone and no surface state at all. Based onfirst-principles calculations, we predict that the hydrogen storage material LiBH is an ideal DDNLsemimetal, where the line resides at Fermi level, is relatively flat in energy, and exhibits a largelinear energy range. Interestingly, both the two novel surface states of DDNL can be realized inLiBH. Further, we predict that with a magnetic field parallel to DDNL, the Landau levels of DDNLare doubly degenerate due to Kramers-like degeneracy and have a doubly degenerate zero-mode.
Introduction.
The past decade has witnessed a tremen-dous advance in the understanding of topological bandtheory, for which one of the most representative andexperimentally relevant realization may be topologicalsemimetals, where novel quasi-particles, such as Weyland Dirac fermion, appear as low-energy excitationsaround nontrivial crossings formed by conduction andvalence bands [1–4]. Various fascinating phenomena as-sociated with topological semimetals are predicted, suchas topologically protected surface states [4, 5], unusualoptical and magnetic responses [6–9], density fluctuationplasmons [10, 11], and quantized circulation of anoma-lous shift [12].In three-dimensions, the band crossing, in addition tozero-dimensional (0D) nodal points [13–19], also can be1D nodal line (NL) [20–24] or even 2D nodal surface[25–28], protected by corresponding space group (SG)symmetries. The current study of nodal line semimet-als mainly focus on the case that the line is generated byband inversion, topologically protected by π Berry phaseand characterized by drumhead-like surface state. Var-ious topological nodal line semimetals belonging to thisparadigm are predicted, and some of them have beenexperimentally confirmed [29–38]. With multiple sym-metries, the nodal line can take many different formsin Brillouin zone (BZ), such as higher-order nodal line,nodal chain, crossed nodal line, nodal box and Hopf-linkloop [39–51].In this work, we theoretically propose another possi-bility of the nodal line, namely, double Dirac nodal line(DDNL) in 3D systems with negligible spin-orbit cou-pling (SOC) effect (such as the materials with atoms ofcarbon or even lighter than carbon). This nodal line ∗ zhiming [email protected] † [email protected] is four-fold degenerate (eight-fold degenerate if includ-ing spin degree of freedom) and resides at certain high-symmetry line in BZ. For each 2D plane transverse toDDNL (setting as k x - k y plane), the band crossing on theline can be considered as a sum of two Dirac point andits effective Hamiltonian may be written as H = (cid:20) h D h (cid:48) h (cid:48)† h D (cid:21) , (1)where each entry is a 2 × h D = v x k x σ x + v y k y σ y , (2)with σ i ( i = x, y, z ) the Pauli matrix, and v x ( y ) theFermi velocity in x ( y ) direction. The two diagonal blocks( h D ) describe two 2D Dirac points with same topologicalcharge (Berry phase) of π in the k x - k y plane, and theoff-diagonal term h (cid:48) denotes the coupling between thetwo Dirac points. While Dirac nodal line has been wellstudied in previous works [21, 31], the DDNL has not yetbeen proposed.Generally, the electron bands in the systems withoutSOC effect is not degenerate. Hence, the DDNL is ratherrare and requires strict symmetries for its realization. In-deed, by an exhaustive searching over 230 SGs [52], we TABLE I. Space groups allowing for DDNL with negligibleSOC effect. The DDNL is stabilized by the nonsymmorphicoperators of systems. Here, α ∈ (0 , )SG No. BZ Location57 Γ o RT: { α, , }
60 Γ o RU: { , α, }
61 Γ o RT: { α, , } , RU: { , α, } , RS: { , , α }
62 Γ o RS: { , , α }
205 Γ c RM: { , , α } a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n π π2π 2π (a) (b)(c) (d) π FIG. 1. Schematic showing the surface state of nodal linesemimetal, as well as the electronic band around line. Theorange region denotes surface state. (a) Usual Dirac NL withdrumhead-like surface state. (b) A Dirac NL traversing bulkBZ should come in pair and can be considered as a variation ofusual Dirac NL due to strong anisotropy. (c-d) DDNL has twopossible but distinct surface states: (c) a torus surface statespanning the whole surface BZ, and (d) no surface state. find that only 5 SGs can exhibit the DDNL, which residesalong the high-symmetry line located at the boundary ofBZ. The five SGs and the location of DDNL in BZ aregiven in Table. I. Particularly, for all the five SGs, theDDNL is the only possible degeneracy at the correspond-ing high-symmetry line(s), indicating that the number ofthe electronic bands of materials belonging to these SGsmust be a multiple of 8 (including spin degree of free-dom). Hence, for the material belonging to these SGsand having 8 n + 4 (with n an integer) electrons, it mustbe a DDNL semimetal enforced by the filling [53].The DDNL is topologically distinct from the usualDirac nodal line (with π Berry phase) in that its Berryphase is 2 π , as it can be considered as a combination oftwo Dirac nodal lines. Since the Berry phase is definedmod of 2 π , it is trivial for DDNL. As a consequence, theDDNL features two distinct states at the boundary of sys-tem, as schematically shown in Fig. 1(c-d), which bothare completely different from the drumhead-like surfacestate in usual NL [see Fig. 1(a-b)]. One is the novel sur-face state spanning over the whole surface BZ [see Fig.1(c)]. Since the 2D surface BZ is a torus, such novel sur-face state then is termed as torus surface state [39, 54].The other case is that no surface state appears on the sur-face, even though there exists a NL (e.g. DDNL) in bulk[see Fig. 1(d)]. These two cases are topologically allowedand consistent with the trivial Berry phase of DDNL.Moreover, we predict that by applying a magnetic fieldparallel to the line, the Landau levels of DDNL are dou-bly degenerate due to Kramers-like degeneracy and havea doubly degenerate zero-mode, which shall suggest pro-nounced signature in magneto-transport.We demonstrate our ideas by the first-principles calcu-lations and model analysis of a concrete material. Wefind the hydrogen storage material LiBH is an idealDDNL semimetal candidate, where the DDNL residesat Fermi level, is relatively flat in energy, and exhibits a bc (a) (b) k y k x k z LiBH
ΓX S YZU TR ˆ Γ ˆ T ˆ R ˆ U ˜ Y ˜ Γ ˜ T ˜ Z ~ Γ ~ Z ~ M ~ X FIG. 2. (a) Crystal structure of LiBH material. (b) Bulk BZand (100), (010) and (001) surface BZ. a large linear energy range. Interestingly, both torussurface state and no surface state simultaneously appearin LiBH material. More importantly, we calculate theLL spectrum of a lattice model based on LiBH and findthe doubly degenerate zero-mode LL can be clearly ob-served. These results indicate that the novel propertiesof DDNL predicted here should be experimentally de-tected in LiBH. Thus, our work not only predicts a newsemimetal phase, but also shows an ideal material plat-form for exploring interesting fundamental physics con-nected to it.
Crystalline structure and electronic bands.
We moti-vate our investigation by considering the hydrogen stor-age material LiBH [55]. This material has a orthorhombicstructure with SG Pnma (No. 62), which is one candi-date in Table I. The primitive cell of LiBH contains intotal 12 atoms with Li, B and H residing at the 4 c Wyck-off positions, as shown in Fig. 2(a). Since all the threeatoms: Li, B and H are lighter than carbon, the SOCeffect in LiBH is negligibly small, and we virtually ob-tain a spinless system. This is a precondition for realiz-ing DDNL, as the DDNL is not robust against SOC [52].Moreover, the electron number of LiBH is 20 (= 2 × a = 6 . b = 3 . c = 6 . (cid:101) C z = (cid:8) C z | (cid:9) and (cid:101) C y = (cid:8) C y | (cid:9) , and a spatialinversion P . The material also has time-reversal symme-try T with T = 1, corresponding to the negligible SOCeffect in LiBH.The calculated electronic band structure of LiBH with-out SOC is presented in Fig. 3(a). It is clearly shownthat this material is a DDNL semimetal with the lineappearing at RS path, consistent with symmetry analy-sis (see Table I). A remarkable feature of the electronicbands of LiBH is that its low-energy spectra is roughlysymmetric about Fermi level, indicating that LiBH hasan approximate chiral symmetry [57]. From Fig. 3(a),one observes that there exist two band crossings almostcutting the Fermi level. We first consider the linear cross-ing (labelled as A ) on ΓY path. Due to the presence of (cid:102) M z = (cid:101) C z P and PT symmetry, point A can not appear (b)(a) (d)(c) -4-2024 E n e r g y ( e V ) 「 「X S R T Y DDNLhourglass-NL
RS YY Γ D P P E n e r g y ( e V ) E n e r g y ( e V ) k x k y -0.2-0.1 0.0 0.1 -1.5 1.5 A D FIG. 3. (a) Electronic band structure of LiBH. (b) Schematicshowing DDNL and hourglass NL. The electronic band (c)around a generic point (D) on DDNL and (d) a generic pathpassing through hourglass NL. in isolation but reside on a NL in the k z = 0 plane [seeFig. 3(b)]. As discussed in Ref. [58], this NL is formedby the neck crossing-point of the hourglass-like disper-sion [see Fig. 3(d)] and then is termed as hourglass NL.The hourglass NL can move in the k z = 0 plane but isunremovable as its two endpoints are pinned at S point.The second and also the most striking feature of LiBHis the four-fold (eight-fold if including spin) degeneratecrossing along RS path, giving rise to a DDNL at Fermilevel. The DDNL is rather flat with an energy variationof 20 meV, due to the approximate chiral symmetry, andexhibits highly dispersive bands in the plane normal tothe line, as shown in Fig. 3(a) and 3(c). Therefore, LiBHprovides an ideal material platform to explore the novelphysics associated with DDNL and hourglass NL in ex-periments. Although the properties of hourglass NL aresimilar to usual NL [41, 46], the DDNL would featurenovel phenomena distinguished from usual NL, due to atrivial Berry phase and completely different low-energyspectrum. Hence, in the following we mainly focus onthe investigation of the interesting properties of DDNL. DDNL and effective Hamiltonian.
Since the LiBHmaterial exhibits three orthogonal two-fold screw rota-tion axes, the electronic bands at all the three bound-ary planes ( k x,y,z = π plane) are at least doubly degen-erate and the whole bulk BZ is covered by nodal sur-faces [27, 59], as shown in Fig. 3(a). The DDNL sitsat the hinge between k x = π and k y = π planes, e.g.the high-symmetry path RS, and then is formed by thecrossing of two nodal surfaces. For a generic point ( D )on RS, its symmetry operators can be generated by (cid:101) C z , (cid:102) M y = (cid:8) M y | (cid:9) and a combined operator A = (cid:101) C y T .The algebra satisfied by the three generators at D pointis equivalent to that of the three generators at S point, asboth D and S are interior points of RS line [52]. Then, forconcise we consider the algebra of the three generators at S point to explain the appearance of DDNL, which canbe written as (cid:101) C z = (cid:102) M y = 1, A = − (cid:102) M y (cid:101) C z = − (cid:101) C z (cid:102) M y , (cid:101) C z A = A (cid:101) C z , (cid:102) M y A = −A (cid:102) M y . (3)The Bloch states at S point can be chosen as the eigen-states of (cid:101) C z , denoted as | c z (cid:105) with c z = ± (cid:101) C z . Since (cid:101) C z anticommutes with (cid:102) M y ,the two states | (cid:105) and | − (cid:105) would be degenerate, as | ± (cid:105) = (cid:102) M y | ∓ (cid:105) . Also as (cid:101) C z commutes with A and A = −
1, the state | ± (cid:105) and its Kramers-like partner A| ± (cid:105) are linearly independent. Hence, the four states {| (cid:105) , | − (cid:105) , A| (cid:105) , A| − (cid:105)} must be degenerate at thesame energy, forming a DDNL along RS path. Basedon the quartet basis, the matrix representations of thegenerators can be expressed as (cid:101) C z = σ ⊗ σ z , (cid:102) M y = σ z ⊗ σ x , A = iσ y ⊗ σ K , (4)with K the complex conjugation. With the standard ap-proach [19, 60], the effective k · p Hamiltonian for a genericpoint on DDNL can be obtained as H DDNL = ( c + c k z ) + (cid:20) h D h (cid:48) h (cid:48)† h D (cid:21) , (5)with h D = c k x σ x + c k y σ y and h (cid:48) = αk x σ y + βk y σ x .Here, c i =1 , , , is real parameter, and α and β are com-plex parameters. Clearly, the obtained Hamiltonian (5)is consistent with Eq. (1) in Introduction, which directlydemonstrates the existence of DDNL in LiBH. Due tononvanishing off-diagonal term h (cid:48) , the two Dirac conesdescribed by h D would split at a general momentumpoint, while are sticked together along k x ( y ) = 0 axis [seeFig. 3(c)], resulting from the nodal surfaces on bound-aries. Torus surface state.
We then explore the surfacestate of LiBH. It is has been extensively shown that NLsemimetals feature drumhead-like surface state at sam-ple boundary, due to π Berry phase of the line. Thisis the case for the hourglass NL in LiBH, which has π Berry phase and leads a drumhead-like surface state at(001) surface, as shown in Fig. 4(c,f). However, both(100) and (010) surfaces, which are parallel to DDNL, donot exhibit drumhead-like surface state. More surpris-ingly, the (100) surface has a torus surface state coveringthe whole (100) surface BZ, as shown in Fig. 4(a,d), andin sharp contrast, the (010) surface does not have anysurface state [Fig. 4(b,e)].The surface state in NL semimetal generally is pro-tected by quantized π Zak phase, which is the Berryphase of a straight line normal to the surface and cross-ing the bulk BZ [61]. Here, due to the presence of P symmetry in LiBH, the Zak phase is quantized to be 0or π , corresponding two topologically distinct phases.We first study the surface state on (100) surface. Start-ing from the Zak phase Z ( k y = 0 , k z ) (with k z (cid:54) = 0) of ˜ Y ˜ Γ ˜ T ˜ Z (010) ˆ Γ ˆ T ˆ R ˆ U (001) ~ Γ ~ Z ~ M ~ X π ππ π π ππ π (100) -2-1 0 1 2 E n e r g y ( e V ) ~ Γ ~ Z ~ M ~ M ˜ Γ ˜ T ˜ T ˜ Z -2-1 0 1 2 E n e r g y ( e V ) -2-1 0 1 2 E n e r g y ( e V ) (a) (b) (c)(d) (e) (f) ˆ Γ ˆ U ˆ T ˆ T FIG. 4. (a-c) Schematic figures of the surface state for (100),(010) and (001) surfaces. The values 0 and ± π in (a-c) arethe Zak phase for lines normal to the corresponding surface.(d-f) Surface spectra on (100), (010) and (001) surfaces. In(d), the surface state spans the whole surface BZ, leading tothe torus surface state, while in (e) no surface state can beobserved. a straight line transverse to (100) surface, which is ob-tained as π in LiBH material [see Fig. 4(a)]. By movingthe straight line along k y -axis, the Zak phase shall notchange its value until the line passes through DDNL atRS path ( k y = π ). The changed value equals to the Berryphase of DDNL, namely, 2 π . This means that one alwayshas Z ( k y > π, k z ) = Z ( k y < π, k z ), as the Zak phaseis defined mod 2 π . Similarly, by moving the straightline along k z -axis, the Zak phase would change its valueby zero when the line crosses over the hourglass NL at k z = 0 plane. Therefore, the Zak phase Z ( k y , k z ) = π for any gapped state, resulting in a torus surface state in(100) surface, as shown in Fig. 4(a,d). Similar analysisapplies for (010) surface, except that Z ( k x , k z ) = 0 foreach gapped state. Hence, no surface state can be foundon this surface, as shown in Fig. 4(b,e). These unusualsurface properties also can be inferred from the geometrystructure of DDNL. Unlike the usual NL shown in Fig.1(a-b) and Fig. 4(c) , the projection of DDNL can notseparates the corresponding surface BZ as two parts [seeFig. 4(a,b)], as the BZ is periodic. Hence, for the surfaceparallel to DDNL, the whole surface BZ would share sametopological properties, leading to a torus surface state orno surface state on the boundary of system. Landau spectrum.
The topological feature of NL alsocan reflect in its magnetotransport [62, 63]. By applyinga magnetic field parallel to DDNL, the electronic bandsare quantized into discrete Landau levels (LLs). Gener-ally, one can investigate the LLs of DDNL in a k z -fixedplane, where the low-energy Hamiltonian is captured byEq. (5). It is well known that the Dirac Hamiltonian h D features a zero-mode LL [64]. One may wonder thatthe zero-mode LL would not appear in DDNL, due tothe presence of the coupling h (cid:48) between the two Diracequation. However, this is not the case. We find that the -0.5 0.5 0 E e n e r g y ( e V ) -0.5 0.5 0 E e n e r g y ( e V ) -1 1 0 E e n e r g y ( e V ) B (T) -π 0 π (a) (b) (c)
D-k z +k z k z FIG. 5. (a-b) Landau spectrum with (a) different k z ( B = 5T) and (b) different B (at k z = 0 plane) based on the effectivemodel (5). (c) Landau spectrum of a lattice model based onLiBH with B = 0 . φ a × b ( φ is magnetic flux quantum h/e ).The red curves denote doubly degenerate zero-mode LLs. Thecalculation details are presented in [56]. DDNL also has a zero-mode LL for each k z -fixed plane[56], as shown in Fig. 5(a,b). Moreover, these zero-modeLLs are doubly degenerate. While the B field breaks both T and (cid:101) C y symmetries, it preserves A (= (cid:101) C y T ) sym-metry, and then all the LL bands (including zero-modeLLs) of DDNL are doubly degenerate resulting from theKramers-like degeneracy produced by A = −
1. Sincethe degeneracy of the two zero-mode LLs is protected by A symmetry, it would be robust against B field and al-ways exist regardless of the filed strength. Given the factthat the zero-mode LL in graphene gives rise to manynovel phenomena [64], one can expect the DDNL mayexhibit interesting magnetoresponses distinct from theusual Dirac NLs and also 2D Dirac semimetal.Particularly, the low-energy spectrum of LiBH is soclean that the LL properties proposed here shall be ob-served in it. We further demonstrate it by calculatingthe LL spectrum of a lattice model based on LiBH mate-rial. The calculated results are given in Fig. 5(c), wherea doubly degenerate LL with almost flat dispersion oc-curring at the Fermi level, corresponding to the doublydegenerate zero-mode LL. This strongly suggests that theunusual LL spectrum of DDNL can be detected in LiBHby magnetotransport measurements. Conclusion.
In summary, we propose a new class ofsemimetal phase: DDNL in LiBH material. The DDNLcan be considered as a combination of two Dirac NLsand exhibits many distinct phenomena, such as torussurface state and unusual LL spectrum. In particular,we predict LiBH material is an ideal DDNL semimetaland demonstrate that the novel phenomena of DDNLpredicted here can be clearly observed in LiBH material.Moreover, as the torus surface in LiBH is rather flat inenergy [see Fig. 4(a)], it will be interesting to explore pos-sible unconventional superconductivity, correlation effectand magnetism in LiBH (100) surface.The authors thank J. Xun for helpful discussions.This work is supported by the National Key R&DProgram of China (Grant Nos. 2020YFA0308800,2016YFA0300600), the NSF of China (Grants Nos.12061131002, 11734003), the Strategic Priority ResearchProgram of Chinese Academy of Sciences (Grant No.XDB30000000) and Beijing Institute of Technology Re-search Fund Program for Young Scholars. [1] C.-K. 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