Topologically Charged Nodal Surface

Transcription

1 Topologicall Charged Nodal Surface Meng Xiao * and Shanhui Fan + 1 Department of Electrical Engineering, and Ginton Laborator, Stanford Universit, Stanford, California 94305, USA Corresponding * mengiao@stanford.edu and + shanhui@stanford.edu Abstract: We report the eistence of topologicall charged nodal surface a band degenerac on a twodimensional surface in momentum space that is topologicall charged. We develop a Hamiltonian for such charged nodal surface, and show that such a Hamiltonian can be implemented in a tight-binding model as well as in an acoustic meta-material. We also identif a topological phase transition, through which the charges of the nodal surface changes b absorbing or emitting an integer number of Wel points. Our result indicates that in the band theor, topologicall charged objects are not restrict to ero dimension as in a Wel point, and thus pointing to previousl uneplored opportunities for the design of topological materials. 1

2 Classified b the dimension of band degenerac, there are three classes of three-dimensional topological semimetals: Wel and Dirac semimetals [1-17], nodal line semimetals[18-20], and nodal surface semimetals[21,22], where in the wavevector space the degenerac occurs at a point, line, and surface respectivel. However, while a Wel point is topologicall charged in that it carries a non-ero quantied total Berr flu, nodal lines and surfaces are not topologicall charged in all previous works. Here we introduce an effective Hamiltonian for a topologicall charged nodal surface, and show that such a Hamiltonian can be implemented in a tight-binding model and a realistic acoustic metamaterial. We also demonstrate a topological phase transition, where a nodal surface changes its charge b emitting or absorbing an integer number of Wel points as the sstem parameters var. Our work indicates that topologicall charged objects in a band structure is not restricted to a point, but can have a much wider set of geometries, which points to additional possibilit in topological material design. The topolog of both two-dimensional and three-dimensional band structures can be characteried b the Chern number. In a two-dimensional spinless sstem, nonero Chern number is realied b breaking time reversal smmetr. For a three-dimensional sstem, the Chern number of a closed surface can be nonero even without time reversal breaking. In this case, the topological charges, which result in nonero Chern numbers, must come from band degeneracies. Wel points [23,24] are band degenerac points which are commonl known as the topological charges in the momentum space. A simple Hamiltonian of a Wel point with charge +1 is Hˆ q q q, (1) where q, q and q are respectivel the wavevector components along the, and directions originating from the Wel point, and,, and are the Pauli matrices. The Wel Hamiltonian in Eq. (1) is isotropic along all the directions in the momentum space, and hence the Berr flu coming out from this Wel point is also isotropic. The total Berr flu coming out is 2, and hence the magnitude of Berr flu densit decas as 1/ q q q when awa from the Wel point as shown schematicall in Fig. 1(a), where the red sphere represents the Wel point and the arrows represent the direction and magnitude of the Berr flu densit. 2

3 However, Wel points are not the onl kind of geometric objects in the momentum space that carries topological charge. Here we introduce a nodal surface that also carries nonero topological charges. A Hamiltonian of a charged nodal surface with charge +1 is given b Hˆ q q q q. (2) The eigenvalues and eigenvectors of this Hamiltonian are i /,1 2 E q 1 q and v e q q, respectivel, where we have used q q cos and q q sin, and the subscripts + and correspond to the upper and lower bands, respectivel. The two eigenvalues are degenerate at q 0 and increase or decrease linearl with q when awa from the nodal surface, which indicates that the Hamiltonian in Eq. (2) represents a nodal surface. [21,22] The Berr connection is given b q 2 1 q A ˆ, (3) q 2 2q 2 1 q 2 2 which has onl one non-vanishing component along the aimuthal direction. Hence the Berr curvature also possesses onl one non-vanishing component which is along the direction. If we consider two planes sandwiching this nodal surface, then the total Berr flu passing through each plane is as lim q d ˆ A. Hence the Hamiltonian in Eq. (2) describes a nodal surface with topological charge q +1. The nodal surface (transparent red surface) together with the Berr flu densit (arrows) are shown in Fig. 1(b). The Berr flu densit distribution is quite different from that of the Wel point in Fig. 1(a), even though the both possess the same topological charge. Nodal surfaces can also possess other topological charges, as discussed in detail in Supplemental Material Sec. I. For a nodal surface with a higher topological charge, the Berr flu can be along other directions beside the direction. The Hamiltonian of Eq. (2) can be realied in a simple tight-binding model. The realiations of nodal surfaces have been discussed in Graphene networks [21] and a quasi-one-dimensional crstal famil with nonsmmorphic lattice smmetries [22]. None of these works, however, showed a nodal surface that carries a non-ero topological charge. Here we consider a lattice sstem having a two-fold screw 3

4 rotational smmetr along the direction, and time reversal smmetr. We define a compound anti-unitar smmetr operator G2 C2, which, in real space, acts as G 2 :,,, t,, h / 2, t, (4) where h represents the unit cell sie along the direction. It is eas to check that G 1 for a Bloch 2 2 wavefunction on the k / h plane with arbitrar k and k, where k, k and k are the wavevectors along the, and directions, respectivel. Therefore, k / h forms a nodal surface due to a Kramers degenerac on this surface.[22] An alternative compound smmetr G2 m also ensures the presence of a nodal surface, where m is the mirror smmetr with respect to the direction. The nodal surface as protected b either G2 or smmetr, in general, ma not have non-ero topological charge. In order to construct a charged nodal surface, one has to break either the inversion smmetr or time reversal smmetr or both as required in order to achieve a non-ero Berr flu. [25]. Therefore, the idea is to construct a charged nodal surface is to consider a sstem that has G2 or smmetr, but breaks inversion or time-reversal smmetr. The simplest model ehibits the smmetr of G2 or consists of two sublattices. In Fig. 2(a), we consider such a tight binding model, which ehibits the smmetr. Here red and blue spheres represent different sublattices, and their projections onto the plane form a heagonal lattice with a lattice constant a, with the red and blue spheres projected to the A and B sites of the honecomb lattice. The lattice constant along the direction is h, and two sublattices are on the =0 and =h/2 planes. There is a coupling between two red (blue) spheres on neighbor A (B) sites at different -planes with coupling strength t c, as well as a coupling between nearest-neighbor red and blue sphere with strength t 0. The coupling strengths are assumed to be real and hence the sstem also ehibits time reversal smmetr. Thus the sstem has G2 smmetr. On the other hand, the sstem does not have inversion smmetr. The Hamiltonian of this tight binding Hamiltonian in the reciprocal space is given b: 4

5 with Hˆ,,,,,,,, f k k k g k k k g k k k f k k k, (5) ka 3 ka f k, k, k 2tc cos ka kh 2cos khcos, 2 2 (6) kh 3 3 ka ka ka g k, k, k 2t0 cos 2cos ep i ep i (7) It is eas to show that near k / h, both f k, k, k f k, k, k and,, g k k k are proportional to k / h, which proves that such a tight-binding model possesses a nodal surface at k / h, as required b the G2 smmetr as discussed above. Figure. 2(b) shows the band structure along several directions in the reciprocal space, which shows clearl the features of a nodal surface: The bands are degenerate at k / h for arbitrar k and k, and the dispersion is linear when awa from the nodal surface. Besides the presence of a nodal surface, we also note that the band dispersions are linear along all the directions at the K point, indicating the eistence of a Wel point at the K point. Keeping to the lowest order, the Hamiltonian in Eq. (5) near H 4 / 3 a,0, / h H 4 / 3 a,0, / h can be written as 3 Hˆ 3t I c t0ahq q q 6tchq 2, (8) and where I is the 2 2 identit matri, and the upper and lower signs represent the Hamiltonian near the H and H points, respectivel. Eq. (8) and Eq. (2) share the same form, and hence the should both possess the same charge (for both H and H points). Therefore, the nodal surface at k / h of this tight-binding model is topologicall charged. This conclusion is also consistent with the presence of Wel points at K and K. The charges of these two Wel points are the same. Since the total charges inside the Brillouin one must vanish, the charges of Wel points must be compensated b the charges 5

6 of the nodal surfaces, as these points and surfaces are the onl band-degenerate features in the Brillouin one. As a direct numerical check that the nodal surface is indeed charged, Fig. 2(c) shows the Chern number as a function of k for different k planes of the upper band (red) and the lower band (blue). The change of Chern number at k 0 is due to the presence of two Wel points while the change of Chern number at k / h is due to the charged nodal surface. Hence we can conclude that this nodal surface possesses topological charge +2. As the Chern number for a fied k can be nonero, there eist onewa edge states when the sstem is finite according to the bulk-edge correspondence. Conversel, the eistence of a one-wa edge state at a fied k for a finite sstem is a direct indication that the Chern number at such a k is non-ero. [26] In Fig. 2(d), we consider a strip geometr which is periodic along the and directions and finite along the direction with a igag boundar. We fi kh /2 and show the band structure as a function of k. The can area represents the projected bulk band. The red and blue curves represent the one-wa edge states localied on the upper and lower boundaries, respectivel. The one-wa nature of these edge states can also be seen with a time-dependent simulation of the tight-binding model in a finite sstem. (Supplemental Material Sec. II) The eistence of the onewa edge states confirms the nontrivial topolog of this nodal surface. In general, in a sstem with a G2 smmetr, there is alwas a nodal surface at kh. This nodal surface ma be either topologicall trivial or non-trivial, depending on sstem parameters. Here we show that the transition between a topologicall trivial and non-trivial nodal surfaces involves the absorption and emission of Wel points b the nodal surface. As an illustration, we add an additional onsite interlaer coupling term t k h 3 3 sin 2 / 2 to the Hamiltonian in Eq. (5). This c term can be realied with onsite hopping between net nearest laers with opposite hopping phases at different sublattices. [27] Since this term vanishes at k 0 and k / h, it doesn t lift the degenerac of the nodal surface and it also preserves the eistence of Wel point at K and K. Here characteries the hopping strength and the numerical factors in the term is chosen to ensure that the topological 6

7 transition occurs at = ±1 (See Supplemental Material Sec. I) The topological charge distribution for 1 is shown in Fig. 3(a), where the red transparent plane and blue spheres represent the positivel charged nodal surface and Wel points with charge 1, respectivel. At 1, the q term in Eq. (8) vanishes at H and K. However, the topological charges in fact remains unchanged since to the net order there is still the q term. (See Supplementarl Material Sec. I) At 1, the nodal surface 3 becomes topologicall trivial with two charge 1 Wel points emerging and moving awa from it. The Wel point at K decomposes into three Wel points: two with charge 1 go awa and one with charge 1 stas. This charge distribution is shown schematicall in Fig. 3(b), where we use the gra transparent surface to represent the topologicall trivial nodal surface. A movie shows this topological transition process can be seen in Supplemental Material Movie S. 1. For the Hamiltonian shown here, the onl geometric object where degenerac can occur are the Wel points and the nodal surfaces. The process through which the nodal surface changes its charge, as illustrated above, is quite general for this class of Hamiltonian For a process through which the nodal surface is preserved, the nodal surface can onl change its charge b emitting or absorbing Wel points. We now proceed to show that topologicall nontrivial nodal surfaces can be found in acoustic metamaterials with full wave simulations using COMSOL[28]. We consider sstems consisting of acoustic resonance cavities with connection tubes controlling the hopping strength[9]. This acoustic sstem has been previousl used eperimentall to demonstrate various topological concepts.[29-31] Figure 4(a) shows the unit cell used in the simulation, where blue and ellow represent the surfaces where sound hard boundar condition and periodic boundar condition are applied, respectivel. The sstem is filled with air. It is eas to see that this unit cell possesses the G2 smmetr. Figure 4(b) shows the corresponding band structure. All bands are two-fold degenerate at the surface of k / h and ehibit linear dispersion awa near the surface. The k / h surface is therefore a nodal surface. The eistence of a Wel point at the K point is also confirmed as the band dispersion are linear along all the high smmetr direction when awa near the K point. In order to demonstrate that this nodal surface is topologicall nontrivial, we focus on the two red bands in Fig. 4(b) which possesses a relativel larger band gap when k / h or 0, and then consider a strip of unit cell as shown in Fig. 4(c), where the left panel shows the sketch of the strip and the right panel shows the details of the boundar. The strip is 7

8 periodic along the and directions, and confined with sound hard boundaries along the direction. For a fied k = p / 2h, the band structure as a function of k is shown in Fig. 4(d). Can area shows the projection of the bulk band, and red and blue curves represent the dispersion of the surface states localied on the upper and lower edges, respectivel. As there are no other band degenerate points ecept for the Wel points and the nodal surface, the eistence of one-wa edge states indicates that the nodal surface is topologicall nontrivial. Our work here indicates the eistence of a charged nodal surface, a new tpe of geometric object in momentum space that carries topological charge. Sstem possessing such a charged nodal surface represents a new class of topological semimetal. For a nodal surface with ero charge, lifting the G2 smmetr results in the creation of a band gap. In contrast, for the sstem considered here that has a topologicall charged nodal surface, when the smmetr G2 is broken, charged nodal surface will become Wel points and the semimetal propert is still topologicall protected. We have provided a phsical implementation of an acoustic metamaterial that possesses such a topologicall charged nodal surface. We believe that such a topologicall charged nodal surface can be realied in electronic and electromagnetic sstems as well. This work is supported b the U. S. Air Force of Scientific Research (Grant No. FA ), and the U. S. National Science Foundation (Grant No. CBET ). 8

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10 Figures Fig. 1 (color online) Berr flu densit distributions of a Wel point (a) and a charged nodal surface (b). The red sphere and transparent red plane represent the Wel point and charged nodal surface, respectivel. The arrows represent the direction and amplitude of the Berr flu densit. 10

11 Fig. 2 (color online) (a). A sketch of the tight binding model. Each unit cell consists of two lattice sites, represented b red and blue spheres respectivel. Their projections onto the plane form a heagonal lattice with a lattice constant a. The lattice constant along the direction is h, and the red and blue spheres are located on =nh and =(n+1/2)h plane, respectivel, where n is an integer. Bonds represent hopping between different sites. Hopping strengths of the can and black bonds are given b t c and t 0, respectivel. This model ehibits the smmetr. (b). The band structure of the tight binding model in (a), where tc 0.1 and t The reciprocal space is shown in the inset with positions of relevant high smmetr points marked. Here the red transparent surface represents the nodal surface on which two bands are alwas degenerate. (c). The Chern numbers as a function of k. Red and blue curves represent the Chern numbers for the upper and lower bands, respectivel. (d). Projected band (gre) for a finite strip of the periodic sstem shown in (a). The strip is periodic along the and directions, and is terminated with a igag boundar on both ends along the direction. We fi kh /2. Blue and red curves represent the surface states localied at the lower and upper igag boundaries, respectivel. 11

12 Fig. 3 (color online) Topological charge distributions before the topological transition (a) and after the topological transition (b). Here red and blue spheres represent Wel points with positive and negative charges, respectivel. Red and gra surfaces represent the nodal surfaces with positive and ero charges, respectivel. 12

13 Fig. 4 (color online) (a). A unit cell of the acoustic sstem under consideration, where blue and ellow represent the surfaces where hard boundar and periodic boundar conditions are applied, respectivel. The sstem is filled with air (densit kg / m and speed of sound 343m/s ) (b). The band structure along various directions in the first Brillouin one, where can corresponds to the projected bulk bands studied in (d). (c). The unit cell used to calculate the projected band in (d). The left panel shows the schematic setup and the right panel shows the details of the boundar. The strip is periodic along the and directions, and confined with the sound hard boundar condition along the direction. The number of unit cell along the direction is chosen to be large enough such that the dispersions of the surface states no longer change as the number of unit cell further increases. (d). The can area shows the projection of the bulk bands as highlighted in (b), and red and black curves represent the dispersion of the surface states localied on the upper and lower edges, respectivel. The parameters used are a 40cm, h 24cm, r 5cm, w0 2.4cm, w 1.6cm, and u 12cm. We fi kh /2 in d. c 13