Topological Semimetals

Chapter Topological Semimetals *Short summary for online In a lattice model for a d dimensional topological insulator, transitions between topological and trivial regimes can be induced by tuning some parameter λ. We can promote this parameter to a momentum, and thus obtain a d + dimensional lattice model. This is will be a topological semimetal, also called -gap insulator, where the bands touch at separated points. The band touching points are called Weyl nodes, they carry a topological charge, and therefore are topologically protected: continuous changes of parameters can move them, but cannot open a gap. To transition to an insulator, all Weyl nodes have to fuse with other nodes with opposite charge. Topological semimetals also have surface states, whose density depends on the orientation of the surface with respect to the crystallographic axis.. Coupling SSH Chains for a Two-dimensional Weak Topological Insulator We construct a two-dimensional chiral symmetric lattice model by connecting several SSH chains to each other through nearest-neighbor hopping. The interchain hopping amplitudes w y connect sites on opposite sublattices, so chiral symmetry is respected. As shown in Fig.., we obtain a so-called brick-wall lattice, rotated by 45 degrees. The corresponding bulk momentum-space Hamiltonian reads, ( H CD (k) = v + w x e ik x + w y e i v + w x e ik x + w y e i ). (.) Here we used v for the intracell hopping amplitudes, and w x for the hopping along the chains. Expressed in terms of the Pauli matrices, this reads, Ĥ CD (k) = (v + w x cosk x + w y cos ) ˆσ x + (w x sink x + w y sin ) ˆσ y. (.) The system has chiral symmetry, inherited from the SSH chains, because of the

CHAPTER. TOPOLOGICAL SEMIMETALS Figure.: Geometry of the model. Filled (empty) circles are sites on sublattice A (B), each hosting a single state. They are grouped into unit cells, one of which is indicated by a dotted line. The intracell hopping amplitudes v are shown in thick lines, the intercell hopping amplitudes along the x and y axes, w x and w y, are shown is thin and dashed lines. The left and right edge regions are indicated by blue and red shaded background. way we introduced with the interchain hoppings. Chiral symmetry : ˆσ z Ĥ CD (k) ˆσ z = Ĥ CD (k). (.3) It also has inversion symmetry and time-reversal symmetry, Inversion symmetry : ˆσ x Ĥ CD (k) ˆσ x = Ĥ CD ( k); (.4) Time-reversal symmetry : Ĥ CD (k) = Ĥ CD( k). (.5) For the properties we focus on now, the symmetries besides chiral symmetry are not important. They can be broken, e.g., by making the intracell hopping complex, which modifies Eq. (.) by an extra term Im(v) ˆσ y... Weak Topological Insulator We expect interesting physics if the two-dimensional model is composed of topological SSH chains (w x > v) that are weakly coupled, w y w x,v. We then have what is called a weak topological insulator: a two-dimensional insulator with a weak topological invariant, the -independent winding number along the k x axis. A weak topological insulator has protected edge states, as can be understood by treating the interchain hopping w y as a perturbation. These edge states can be seen on a numerical example shown in Fig... Consider a straight boundary at some angle α with the x axis. For w y =, we have a zero-energy eigenstate for each chain that

.. TOPOLOGICAL TRANSITION BRODENED 3 - - / / k x - Energy Energy 3 3 - - / / k x Energy 3 3 - - / / Figure.: Dispersion relation of a weak topological insulator, obtained by coupling topological SSH chains (v =, w x = 5/3), with interchain hopping w y = /3. Left: bulk bands. Middle and Right: dispersion relations at an edge along x and y. Large (red) dots indicate edge states (whose wavefunction has over 4% weight within the first 4 unit cells of the total of 5 unit cells). These edge states only show up for a cut along y. ends at the boundary, all of them on sublattice A (B). The linear density of these zero energy states is sinα. When we turn on the interchain hopping w y, the energy of these states is topologically protected in the same way as for a single SSH chain: To move away from zero energy, they would need partners on the other sublattice, but no partner is available this side of the bulk. Although the edge states are at zero energy, they are nondispersive, and hence do not take part in conduction.. Transition from Trivial to Weak Topological Insulator is Broadened: Topological Semimetal Consider how the two-dimensional chiral symmetric model can be tuned using the intercell hopping along the chains, w x, from topologically trivial to a weak topological insulator. In the case of no interchain hopping, w y =, we have a trivial insulator with w x < v, and a weak topological one with w x > v. There is a sudden transition from trivial to weak topological insulator at w x = v, when the gap closes at k x = π, which corresponds to a line spanning the two-dimensional Brillouin zone. We will see that in the presence of an interchain hopping w y, this transition is broadened. The gap closes only at isolated momenta, the so-called nodes, which traverse the Brillouin zone as the control parameter drives the system through the transition. To build on our understanding of the topology in the SSH model, we can demote to a parameter, much in the same way as we did when we connected the Qi-Wu-Zhang model to the Thouless charge pump. Looking at the bulk Hamiltonian in this way, we have a continuous set of SSH models, with complex intracell hopping parameter v SSH ( ) = v + w y cos + iw y sin, and intercell hopping w SSH ( ) = w x. The corre-

4 CHAPTER. TOPOLOGICAL SEMIMETALS (a) (b) (c) Figure.3: The curve on the d x d y plane, traced by the bulk Hamiltonian Ĥ CD (k x, ), at a fixed, as k x is tuned across the Brillouin zone, can be trivial (left, winding number ), or topological (right, winding number ). If, as in the example shown, none of v, w x, and w y is bigger than sum of the other two, for some value of, we will have the trivial, for others, the topological case. We then must have an intermediate value of where the gap closes: a node. sponding curves in the d x d y plane, as illustrated in Fig..3, are circles of radius w x, whose center is at (v + w y cos,w y sin ). Thus, as we tune, the center of the circle is taken on a circle of radius w y around (v,). If, as in the example shown, none of v, w x, and w y is bigger than sum of the other two, for some value of, we will have the trivial, for others, the topological case. We then must have two values of (,k x ) where the gap closes: the nodes. We have If w x < v w y : Trivial If v w y < w x < v + w y : Topological semimetal If v + w y < w x : Weak topological. We show the dispersion relations, and with edge states for x edge, y edge, in Fig..4. See Exercise.8. In the topological semimetal case we have isolated points in the Brillouin zone where the gap closes. This is all quite natural when describing the model as a set of -dimensional chiral symmetric topological insulators, but in the full D model, it can appear strange... Gap closing points In the topological semimetal phase we do not have a bulk gap: it closes at single points in the Brillouin zone, the so-called nodal momenta k. The nodes are the solutions of the equations, d x (K x,k y ) = ; d y (K x,k y ) =. (.6) These are two equations for two variables k x and, so we expect only isolated, pointlike solutions, if any. Continuously varying the parameters w x, w y, and v, each isolated pointlike node will be driven along a path in the Brillouin zone. For our two-dimensional model

.. TOPOLOGICAL TRANSITION BRODENED 5 Energy Energy..5..5..5..5. - - / /..5..5..5..5. - - / / k x - - / / - - / / - - / / - - / / - - / / k x - - / / k x - - / / k x - - / / k x Figure.4: Transition from trivial insulator to a weak topological insulator, obtained by tuning w x. Interchain coupling is w y = /3, and parameters are set so that w x + w y + v =. From left to right, v,w x = (.,.566),(.95,.76),(5/6, 5/6),(.76,.95),(.566,.). Top row: cut along y. Bottom row: cut along x. Large (red) dots indicate edge states (whose wavefunction has over 4% weight within the first 6 unit cells of the total of 3 unit cells)

6 CHAPTER. TOPOLOGICAL SEMIMETALS Γ Γ Figure.5: Paths of the nodes in the Brillouin zone. Left: v =.,w y = /3, tune w x. Right: v =.,w y = 5/3, tune w x. system, the gap closing requirements read v = w x cosk x + w y cosk y ; w x sink x = w y sink y, (.7) and can be solved as w x = (v + w y cosk y ) + w y sin K y ; (.8) K x = atan ( w y sink y, v w y cosk y ), (.9) using the atan function, which returns φ ( π,π], if supplied with arguments asinφ, acosφ for any a >. Using a parameter, e.g., w x, to tune the system across a broadened topological phase transition, we find that nodes are first born pairwise, then driven in opposite directions in the Brillouin zone, then annihilate pairwise. Examples with v =, w y = /3, and v =, w y = 5/3, are shown in Fig..5. In our two-dimensional model this follows from inversion symmetry: nodes must come in inversion symmetric pairs of (K x,k y ) ( K x, K y ), and If v > w y : v w y w x v + w y ; M X x ; (.) If v < w y : w y v w x v + w y ; X y X x. (.) However, even if inversion symmetry is broken, nodes can only be born and annihilated pairwise. As we show next, this has a deeper, topological reason..3 Topological Charge of Nodes The nodes in the topological semimetal, i.e., the isolated pointlike gap closing points, carry a topological charge, and are hence topologically protected. We show this first

.3. TOPOLOGICAL CHARGE OF NODES 7 for the two-band model, and then generalize..3. Two-band Model: Charge is Winding of d Around Node Since our two-dimensional semimetal is a two-band model, we can gain more insight into the nature of the nodes by taking a look at the d field, defined by Ĥ(k x, ) = d(k) ˆσ. An example for w x = w y = v = is shown in Fig..6. Here, the two nodes are at we have K x = K y = ±π/3, and you can see that the d field winds around these points. We can put the calculation into a convenient form using d(k) = d(k)cosφ(k) x + d(k)sinφ(k)ỹ, (.) where x and ỹ are unit vectors along the two axes. Put a discrete lattice on the Brillouin zone. We assign numbers to the vertices, edges, and plaquettes of the lattice, as vertex k a : φ(k a ) = atan (d x (k a ),d y (k a )); ( ) edge k a k b : φ ab = arg e iφ(k b) iφ(k a ) ; plaquette p : Q p = π (φ ab + φ bc + φ cd + φ da ), (.3a) (.3b) (.3c) where the plaquette p has boundary a b c d a. The quantity Q p Z counts the vorticity of the vector field d in the plaquette p. A discrete Stokes theorem, for a disk S composed of plaquettes, reads φ ab = π Q p. (.4) (ab) S p S It can be proven by just rearranging the sums on the right-hand-side. The discrete Stokes theorem has two important consequences. Jump in weak topoogical invariant happens when = K y. Total topological charge has to be. This is because the quantities we defined in the Brillouin zone are all periodic. Continuum Limit We can put this in a more convenient form by introducing a vector field A(k x, ), which is in some sense a gradient of φ(k x, ). This is easiest to define using square plaquettes (can be done for plaquettes of arbitrary shape), with corners k a = (k x δ k /, δ k /); k b = (k x + δ k /, δ k /); The definition reads, k c = (k x + δ k /, + δ k /); k d = (k x δ k /, + δ k /). A(k x, ) = φ ab + φ dc δ k x + φ ad + φ bc δ k ỹ. (.5)

8 CHAPTER. TOPOLOGICAL SEMIMETALS Energy ky / - / - - - / - - / / kx kx / - ky Figure.6: Brillouin zone, with d vectors. v = wx = wy =. It is easy to show that Q p corresponds to the flux of the curl of the vector field A on the plaquette p. The Stokes theorem above reads, for a disk S with boundary S, I S Z Ads = S Ad S (.6) Since A is a gradient, its curl vanishes, except at points where it is not well defined. Linearizing Around the Nodes: Charge is Determinant Two-band case, linearized around a Weyl node K, we have, using the small distance p = k K, p py vxx vxy σ x H CD x ; (.7) vyx vyy σ y {z } v vax = wa sin Ka ; vay = wa cos Ka for a {x, y}. (.8) For the Hamiltonian to be Hermitian, the matrix v of coefficients must be real. We can map how the vector field d(p) winds around the node by solving the linearized equations for dx = and dy = : dx = : py = vxx px ; vyx dy = : py = vxy px. vyy (.9)

.4. EDGE STATES IN PART OF THE BRILLOUIN ZONE 9 The winding number is then found to be.3. General, multiband case ν = signdetv. (.) The formulas obtained through linearization also apply to the multiband case. In the vicinity of each node, only two energy levels are interesting anyway, we can use the restriction of the Hamiltonian to those levels. To build more general formulas, use chiral symmetry of the Hamiltonian, ( H(k x, ) = h(k x, ) h (k x, ) ). (.) We can apply the same formalism as before, but use the phase of the complex number deth(k) instead of the phase of the vector d. vertex k a : det[h(k a )]; (.a) ( ) deth(kb ) edge k a k b : φ ab = arg ; (.b) deth(k a ) plaquette p : Q p = π (φ ab + φ bc + φ cd + φ da ), (.c).4 Edge States in Part of the Brillouin Zone Along a translation invariant edge, we can always see the model as coupled chiral symmetric chains. Zero-energy edge states appear when there is a local imbalance of sublattices. This imbalance remains after the coupling of the chains. Thus we expect that there are zero-energy bound states at the edge. At the edge, the natural boundary condition might be not the one with all unit cells maintained. This complicates things in a real physical system..5 Coupling QWZ Layers for Weak Topological Insulator Same logic, applied to Qi-Wu-Zhang model. Ĥ Weyl (k) = sink x ˆσ x + sin ˆσ y + (u + wcosk z + cosk x + cos ) ˆσ z. (.3) This is the model (with w = ) cited in Vishwanath. We do not have chiral symmetry here, neither time-reversal symmetry, but we do have inversion symmetry and mirror symmetry along z: Inversion symmetry : ˆσ z Ĥ Weyl (k) ˆσ z = Ĥ Weyl ( k); (.4) Mirror symmetry : Ĥ Weyl (k x,,k z ) = Ĥ Weyl (k x,, k z ). (.5)

CHAPTER. TOPOLOGICAL SEMIMETALS Neither inversion symmetry nor mirror symmetry play any important role here, they can be broken. For example, we can break inversion symmetry with w sin(k z ) ˆσ x + w cos(k z ) ˆσ y. Weak Chern Insulator. Bulk invariant. Edge states..6 Transition from Trivial to Weak Topological Broadened: Topological Semimetals Uncoupled layers: w =. As u is tuned, we transition from trivial (u < ) to weak Chern insulator with Q = + ( < u < ), to weak Chern insulator with Q = ( < u < ), to topological (u > ). At the transition points, the gap closes along lines spanning the whole Brillouin zone. The reason the transition broadens as a result of the coupling can be understood along the same lines as the lower dimensional case. We can treat the momentm k z as a parameter. The three-dimensional model is then described by a torus that is moved periodically in d-space. INCLUDE A PICTURE. In the presence of coupling, each gap closing along the lines at a single u value is turned into the trajectory of the node, at a finite range of u. u < w : w < u < + w : + w < u < w : w < u < w : w < u < w : Trivial; Semimetal, (,, ±π); Weak Chern; Semimetal, (,π, ±π) and (π,, ±π); Weak Chern; w < u < + w : Semimetal, (π,π, ±π); + w < u : Trivial. Weyl nodes are confined to four columns in the Brillouin zone, k x =,π and =,π, every Weyl node at k comes with its inversion symmetric partner at k. Breaking inversion symmetry turns the columns into helical columns..7 Topologically Protected Nodes: Gap Closing Points.7. Two-band Model: Nodes are Sources or Sinks of d. We can gain more insight into the nature of the nodes by taking a look at the d field. We can put the calculation into a convenient form using Ĥ Weyl (k) = d (k)σ + d(k) ˆσ = E (k) u (k) u (k) + E + (k) u + (k) u + (k), (.6)

.7. TOPOLOGICALLY PROTECTED NODES: GAP CLOSING POINTS d-vectors around the Weyl nodes..5. kz.5...5..5 k x..5..5.. Figure.7: Brillouin zone, with d vectors. u =, v =.5. with E (k) <, E + (k) >, and u + (k) u (k) = for every momentum k. In our model, d (k) =, and hence, E (k) = E + (k), which is important because it puts the nodes at the Fermi surface for half filling. We will use a simplified notation: d(k) = d(k)/ d(k) ; (.7) k = k a : d a = d(k a ); a = u (k a ). (.8) The main idea of the calculation is best appreciated by starting in the continuum. In a volume V with surface S = V, we can find the total number of sources/sinks of the vector field d(k), by calculating the solid angle the vector field d sweeps over as we scan over the surface. For the discretization we will need a geometrical formula for the solid angle. We could do this using ( ) ) ( d Ω abc = tan a d b d c, (.9) + d a d b + d a d c + d b d c but this is a bit cumbersome, and it would restrict us to triangular plaquettes. We can do something more efficient, by using the knowledge obtained previously that the solid angle corresponds to the discrete Berry phase, Ω abc = F abc = arg( a b b c c a ), (.3) which is valid for plaquettes with any number of nodes on the boundary.

CHAPTER. TOPOLOGICAL SEMIMETALS The discretization is obtained as node a : a ; (.3) edge a b : a b ; (.3) plaquette p : F p = arg (ab) p a b ; (.33) volume element v : Q v = π F p. p v (.34) We have to pay attention to the proper orientations of the boundaries: for plaquettes these are counterclockwise, for volume elements, the plaquettes should all be defined with normal oriented out from the middle of the volume element. We can plug in the arguments from the section on the Chern number, and obtain that the quantities Q p are integers, and they detect the nodes. A discrete Gauss theorem, for a volume V composed of volume elements, F ab = Q v. (.35) (p) V v V This has two important consequences. Jump in weak topoogical invariant happens when = K y. Total topological charge has to be. This is because the quantities we defined in the Brillouin zone are all periodic. Continuum Limit We can put this in a more convenient form by introducing a vector field B(k x, ), the Berry curvature. This is easiest to define using cubic volume elements (possible for volume elements of arbitrary shape), with corners ( k a = k x δ k, δ k,k z δ ) ( k ; k b = k x + δ k, δ k,k z δ ) k ; ( k c = k x + δ k, + δ k,k z δ ) ( k ; k d = k x δ k, + δ k,k z δ ) k ; ( k e = k x δ k, δ k,k z + δ ) ( k ; k f = k x + δ k, δ k,k z + δ ) k ; k g = ( k x + δ k, + δ k,k z + δ k ) ; k h = ( k x δ k, + δ k,k z + δ k The components of the field B are defined using the Chern numbers of the appropriate plaquettes, calculated so the normals are always along the positive direction, i.e., ). B(k x,,k z ) = F adhe + F bcg f δ k x + F ae f b + F dhgc δ k ỹ + F abcd + F e f gh δ k z. It is easy to show that Q v corresponds to the integral of the divergence of the vector field B on the volume element v.

.7. TOPOLOGICALLY PROTECTED NODES: GAP CLOSING POINTS 3 The Gauss theorem above reads, for a volume element V with boundary V, B(k)d k = div B(k)d 3 k (.36) V Because of the connection between the Chern number and the solid angles, V B = curl d. (.37) Since B is a curl, its divergence vanishes, except at points where it is not well defined. Linearizing Around the Nodes: Charge is Determinant Two-band case, linearized around a Weyl node K, we have, using the small distance p = k K, ( ) px p Ĥ y p z v xx v xy v xz ˆσ x v yx v yy v yz ˆσ y ; (.38) v zx v zy v zz ˆσ z In general, the matrix v of coefficients is real. We can map how the vector field d(p) winds around the node,... The charge of the node is then found to be.7. General, multiband case ν = signdetv. (.39) The formulas obtained through linearization also apply to the multiband case. In the vicinity of each node, only two energy levels are interesting anyway, we can use the restriction of the Hamiltonian to those levels. For more general formulas, we can plug in the formulas for Chern number of multiband systems. Ĥ(k a ) = N F m= E m (k a ) a m a m + N tot m=n F + E m (k a ) a m a m ; (.4) We are interested in the nodes between the N F th and the (N F +)th energy states. This is crucial if for all k, we have m N F E m (k) and m > N F E m (k). The derivations for the two-band case remain valid, with the replacement a b det M (ab). Put a discrete three-dimensional lattice on the Brillouin zone. We assign objects to the vertices, edges, plaquettes, and cubes of the lattice, vertex a : { a, a,..., a NF }; edge a b : detm (ab), with M mn (ab) = a m b n ; plaquette p : F p = arg (ab) p detm (ab) volume element v : Q v = π F p. p v

4 CHAPTER. TOPOLOGICAL SEMIMETALS Weak invariant trivial, no edge state bulk Fermi surface / /4.. Right edge Fermi surface Weak invariant topological Weak invariant trivial, no edge state Left edge Fermi surface bulk Fermi surface kz - /4 - / - / - /4 /4 /.8.6.4.. Figure.8: Edge states of a Weyl semimetal. Color: inverse of the localization length of edge state. Fermi arcs are equipotential lines, shown in thick lines, continuous for energy E = and above, slashed for negative energy. Spacing between equienergy lines is..8 Fermi Arcs on Surface The edge states exist only in parts of the Brillouin zone: those parts where the weak Chern number is nonzero. These are the Fermi arcs. Problems Nodes protected by extra symmetries POSSIBLE EXERCISE: Even if chiral symmetry is broken, these other symmetries can stabilize the Weyl nodes. This is trivial in two-level systems, how does it work out in general? Transition from weak topological Discuss what happens to the two-dimensional chiral symmetric lattice model, as w x is tuned from w x = to w x =, if v = and w y =. Answer: If we repeat the same process, but with topological chains, v < w y, we have If w x < w y v : Weak topological; (.4) If w y v < w x < v + w y : Semimetal; (.4) If v + w y < w x : Weak topological. (.43) Paths of nodes when v is tuned If v is tuned, show that, assuming w y < w x without loss of generality, w x w y v w x + w y ; X y M (.44)

.8. FERMI ARCS ON SURFACE 5 Energy Energy Energy Energy 4 4 4 4 4 4 4 4 - - / / Figure.9: Edge states of a Weyl semimetal. Color: inverse of the localization length of edge state. Fermi arcs are equipotential lines, shown in thick lines, continuous for energy E = and above, slashed for negative energy. Spacing between equienergy lines is.

6 CHAPTER. TOPOLOGICAL SEMIMETALS