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Weyl Semi-Metal Phase Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan (Dated: March 28, 2017) I. WEYL SEMIMETAL There are two important types of nodal points in 3D. One is the point degeneracy between two energy levels, the other is the point degeneracy between four energy levels, see Fig.1. To distinguish between them, from now on we call the former a Weyl point, and the latter a Dirac point. In this chapter, we only study the Weyl FIG. 2 The textures of d(k) of two Weyl points with opposite points. helicities. The figures are from somewhere on the web. A. Classification of Weyl node + The Hamiltonian near a Weyl point can be written as + H = d(k) · σ; (1.1) + where k is the momentum away from the node. If the + components of d are all linear in k, then we call it as a + - linear Weyl node. If at least one of the components of d is quadratic in k, then we call it as a quadratic Weyl node, and so on. FIG. 3 The winding number is determined by the degree of The topological charge (or Berry index) of a node is the map, which is the same irrespective of the direction one given by the first Chern number (see Sec. ??), peeps. Z 1 2 QT = d a · F; (1.2) the textures of d(k) = ±k around the nodal point are 2π S2 k shown in Fig.2. It is obvious that the winding numbers, 1 @d @d and hence the topological charges, are ±1. The sign ± is Fk = 3 d · × : (1.3) 2d @ki @kj called as the helicity (or chirality) of the Weyl point. For a general linear node, its topological charge can be The integral is over a constant-energy surface with fixed obtained simply as the sign of the Jacobian, jdj, and i; j; k are in cyclic order. We learned in Sec ?? that the integrand is just (half of) the solid angle of the @di 2 2 QT = sgn ; (1.5) image f : Sk ! Sd, thus QT is the winding number of @kj the map . For example, given which is the same for every point k. The sign simply shows that whether the map preserves or reverses the H = ±k · σ; (1.4) orientation. For a Weyl node with higher order, its topological charge can also be determined by the energy dispersion near the node. This is explained below (see the App. of Chang and Yang, 2015) : An image point dr is called a r regular point if the Jacobian j@di =@kjj 6= 0. A regu- (a) (b) lar point can have none, or several pre-image points k(`), d(k(`)) = dr; ` = 1; ··· ;N. The degree of the map f is defined as, N FIG. 1 (a) A Weyl point between two levels. (b) A Dirac X @di degf = sgn : (1.6) point between 2 double-degenerate levels. @kj (`) `=1 k=k 2 It is equal to the winding number, and thus the topolog- TABLE I Counting Weyl nodes ical charge QT = degf. Brouwer's lemma guarantees that degf is the same for every regular points (see Fig.3). time-rev symm space-inv symm min number Therefore, one can choose a convenient dr to calculate it. no no 2 For more details, one can see Sec. 10.7 of Felsager, 1998, yes no 4 and Milnor, 1965. no yes 2 For example, for the following quadratic Weyl node, yes yes unstable ! k2 k2 d(k) ' y − x ; k k ; ±k ; (1.7) 2 2 x y z 1. Multiplet of nodes due to symmetry First, in the absence of space or time symmetry, mass- the Jacobian is less lattice fermions are required to come in pairs with opposite helicities. This is the Nielsen-Ninomiya the- @di 2 2 = ∓(kx + ky): (1.8) orem (Nielsen and Ninomiya, 1981a,b), or fermion- @kj doubling theorem (see App. ??). We now consider one of the node with helicity + or −, Choosing d0 = (1=2; 0; 0), then there are two pre-images, (1) (2) k = (0; 1; 0) and k = (0; −1; 0). They contribute to H = ±vσ · (k − k0): (1.13) QT = ∓2 in total. A node with jQT j = 2 is sometimes called as a double Weyl node. Under time reversal transformation (if the pseudo-spin 2 2 If dx(k) is changed to (kx +ky)=2, then the topological behaves like a spin), charge would be 0. That is, not all quadratic Weyl nodes are double Weyl nodes. k ! −k; σ ! −σ: (1.14) So 0 B. Linear Weyl node H ! H = ±vσ · (k + k0): (1.15) In the following, we focus only on the linear Weyl Therefore, if there is TRS, then there must be another node. Its gauge structure is similar to that of a magnetic nodal point at −k0 with the same helicity. monopole. For example, for the nodal point in Eq. (1.4), Under space inversion transformation, the Berry curvature is k ! −k; σ ! σ: (1.16) 1 k^ F = ∓ ; (1.9) So 2 k2 0 H ! H = ∓vσ · (k + k0): (1.17) which is the same as the magnetic field of a monopole. (Cf: Berry curvatures of various 2D systems in Sec. ??.) Therefore, if there is SIS, then there must be another Once a Weyl node exists, it is stable against perturba- nodal point at −k0 with opposite helicity. tions. Consider When both TR and SI symmetries exist, each node would have two monopoles with opposite charges. The H = ±vk · σ + H0; (1.10) net topological charge of a nodal point (with 4 levels) is zero, and the Dirac node is not stable against perturba- H0 is an arbitrary perturbation that can be expanded by tions (see TableI). Pauli matrices, Note that when there is only SIS (but no TRS), then the minimum number of Weyl points in a solid is 2. If H0 = a(k) + b(k) · σ (1.11) there is only TRS (but no SIS), then the minimum num- ber is 4, since TRS-doublet would have a partner dou- blet with opposite helicity, as required by the Nielsen- X @b 2 = a(k) + b(0) · σ + σ · kj + O(k ): (1.12) @k Ninomiya theorem. j j 0 It's obvious that the second term shifts the position of C. The Burkov-Balent multilayer model the node, the third renormalizes the velocity of the Weyl electron, but no gap is opened. That is, the Weyl point Without diving into the subject of point group sym- remains intact under an arbitrary perturbation. It could metry, here we introduce a simplified model proposed by disappear only by merging with another node with op- Burkov and Balent (Burkov and Balents, 2011). Consider posite topological charge. a structure with alternating layers of normal insulators 3 NI For multiple-layers, we have TI t d ^ X y s H = [vτz(σ × k?) · z^ + tsτx + mσz] cl cl td l X y y + td(τ+cl cl+1 + τ−cl cl−1); (1.19) l T (t > t ) = NI (t < t ) = TI where τ± = (τx ±iτy)=2, cl = (clu; cld) is a 2-component s d s d y y operator such that τ+cl cl+1 = clucl+1d ··· etc. FIG. 4 A multi-layer structure with alternating layers of nor- Assume there are N (NI+TI)-layers with period d, and mal insulator (NI) and topological insulator (TI). impose the periodic BC along z-axis. Using the Fourier transformation, t t 1 X d d cy = p eildkz cy : (1.20) TI l N kz TI kz Weyl Weyl one has, NI m NI X h y H^ = vτ (σ × k ) · zc^ c QAH × z ? kz kz kz 0 ts 0 m ts y + mσzc ck (a) m=0 (b) m ≠ 0 kz z + t τ cy c s x kz kz FIG. 5 (a) With both TRS and SIS, the critical line at ts = td i + t (e−ikz dτ cy c + eikz dτ cy c ) (1.21) are composed of unstable Dirac points. (b) After breaking the d + kz kz − kz kz TRS with magnetization m, a Dirac point in (a) separates to ! −ikz d two Weyl points and one has a finite region of Weyl semimetal X h0 + mσz ts + tde y = c ckz ; ikz d kz phase. The gapped phase on the lower left is a quantum ts + tde −h0 + mσz anomalous Hall phase. As m gets larger, the point × is first kz X engulfed by the Weyl phase, then the QAH phase. ≡ H cy c ; (1.22) kz kz kz kz (NI) and topological insulators (TI) stacked along the z- where axis (see Fig.4). Coupling of the top and down surface Hk = τzh + mσz states (SS) of a TI layer is written as ts; coupling of SS z 0 −ikz d ikz d between nearest-neighbor TI layers is written as td. + tsτx + td(e τ+ + e τ−); (1.23) When the intra-layer coupling is larger than the inter- and h = v(σ × k ) · z^. Each k -subspace is independent layer coupling (ts > td), the whole structure is similar to 0 ? z of each other. a NI. On the other hand, when td > ts, the whole struc- ture is similar to a TI. By tuning the relative strength Under the unitary transformation (Burkov et al., 2011), between ts and td, one can induce a topological phase transition at certain critical value. At that value the bulk ! gap is expected to close, probably producing a point de- 1 0 U = (1.24) generacy.
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