Topological insulators Part III: tight-binding models

1 5 Topological insulators Part III: tight-binding tight-binding modelsTight-binding models are effective tools to describe the motion of electrons in solids. Here, we assume that the system is a discrete lattice andelectrons can only stay on the lattice site. The kinetic energy is included by allowing electrons to hop from one site to Example 1: a one-band modelLets consider a 1D lattice with one atom per unit cell. For each atom, we consider only one quantum state. The creation (annihilation) operatorci (ci) creates and annihilates one particle on site i. The non-interacting Hamiltonian can be written as( )H=- ij tijci cj+tjicj ci + iVici ciThe first term ijtijci cj describes hoppings from site j to i. The second term ijtjicj ci describes the hopping from i to j.

2 The last term is thepotential energy, which tells us how much energy we need to put an electron on each site Vi. Because the lattice contains only one type ofatoms, Vi=constant ( translational symmetry). Because H is Hermitian, we find that tij=tji* and Vi is a real :( )H =H( ) - ij tijci cj+tjicj ci + iVici ci =- ij tijci cj+tjicj ci + iVici ci( )- ij tij*cj ci+tjici cj + iVi*ci ci=- ij tijci cj+tjicj ci + iVici ciBy comparing the two sides, we find that tij*=tji and Vi=Vi*.Therefore, we can simplify the Hamiltonian( )H=- ij tijci cj+tij*cj ci +V ici ci=- ij tijci cj+tij*cj ci +VNIn the last term, N= ici ci is the total number of electrons in the system. Because VN is a constant, this term just shifts the total energy by aconstant, and thus has no other physical contribution (can be ignored if we are not interested in the total energy).

3 For tight-binding models , a typically approximation is to assume that electrons can only hop to its nearest-neighbor sites. In reality, long-rangehopping is allowed, but their amplitudes are small (decay exponentially as distance increases). Therefore, in many cases, we just need thenearest-neighbor hopping terms to describe the systems. Due to the translational symmetry, all nearest-neighbor hoppings shall have the samehopping strength. If we assume that this hopping strength is a real number t, the Hamiltonian is( )H=-t ij ci cj+cj ci =-t ici ci+1+ , ij implies that i and j must be neighbors. On the right hand side, means Hermitian conjugate. Notice that the second term is in factthe Hermitian conjugate of the first term, so we just use to represent series: typically, Fourier series is used to describe a periodic function in the real space, which will have a describe set of wave vectorsin the k space.

4 Here, it is the opposite. We have a continuous k-space and it is periodic (the Brillouin zone), but the real space is discrete.( )ck=a2p ici - kx( )ci=a2p BZ kck kxBecause ci,cj =dij, it is easy to check that( ) ck,ck' = 12p a ici - kxi,12p a jcj k'xj = i,j ci,cj 12p a - kxi k'xj= i,jdi,j12p a - k-k' xi= i12p a - k-k' ia=ad k-k' a =d k-k' If we know ck,ck' =d k-k' , we can also show that ci,cj =dij.( ) ci,cj = 12p a BZ kck kxi,12p a BZ k'ck' - k'xj =12p a BZ k BZ k' ck,ck' kxi - k'xj=12p a BZ k BZ k'd k-k' kxi - k'xj=12p a BZ k k xi-xj =dijHamiltonian in k-space We can transfer the Hamiltonian into the k-space. In real space, it looks like( )H=-t ici ci+1+ first term is( ) jcj cj+1= j12p a BZ kck - ka j12p a BZ k'ck' k' j+1 a= j12p a BZ k BZ k'ck ck' k'a - k'-k aj= BZ k BZ k'ck ck' k'ad k-k' = BZ kck ck kaThe second term is the Hermitian conjugate of the first term, so( ) jcj cj+1 = BZ kck ck ka = BZ kck ck - ka( )H=-t ici ci+1+ BZ kck ck ka-t BZ kck ck - ka=-2t BZ kck ckcoska= BZ k -2tcoska ck ckFor a solid, we know that the total energy of electrons (ignore interactions) is( )E= n BZ ken k nn k where n sums over all bands, en k is the dispersion relation for band n and nn k is the occupation number for the Bloch wave state in band nwith momentum k.

5 In second quantization, this formula implies that the Hamiltonian is( )H= n BZ ken k gn,k gn,kHere, gn,k is the creation operator for a Bloch wave ynk r =unk r kr. If we compare this formula with the tight-binding Hamiltonian (in k-space), we find immediately that the tight-binding model we considered here has only one energy band. And our ck operator is in fact thecreation operator for Bloch waves. And the dispersion relation for this band is ek=-2tcoska. Notice that ek is a periodic function of k withperiodicity 2p a, which is exactly what we expect for Bloch Discrete Fourier transformation and Fourier 71In the formula above, we treat k as a continuous variable. In many cases, it is useful to treat k as discrete values (where k turns into k).

6 Here, we briefly discuss about the discrete Fourier transform. Consider a discrete function fi, where i=1, 2, 3 ..N marks different lattice addition, we assume the periodic boundary condition fN+i=fi. We can define( )f k=1N jfj - ka jwhere a is the lattice constant and j marks different lattice it is easy to prove that( )fj=1N kf k ka jSame as j, the wave vector k here is also a discrete variable. This is because fN+i=fN (our system has a finite size)( )fj+N=1N kf k k aj+aN =1N kf k ka j kaN( )fj=1N kf k ka jIf we compare the two equations, we find that kaN=1, which implies( )k=2pmNa=2pmLHere L=Na is the size of the system. For infinite systems, L , the discrete sum k turns into an integral as j, k also have a periodicity, and the periodicity is also N.

7 This periodicity is just the Brillouin km=2pm L and by definition,( )f km=1N jfj - kmaj=1N jfj - 2pNam aj=1N jfj - 2pNmjFor km+N, by definition( )f km+N=1N jfj - km+Naj=1N jfj - 2pNa m+N aj=1N jfj - 2pNmj - 2pjBecause j is an integer, the factor - 2pj=1. Therefore( )f km+N=1N jfj - 2pNmj=f kmAs a result, we can limit the value of m to be -N 2 m<N 2. If m is not in this range, the value of f km can be obtained using the periodiccondition f km+N=f km( )m=-N2,-N2+1, ..N2-1,For the wave vector k, this means that ( )f k+2p a=f kSo, we can confine the value of k into the range of -p a k<p a, which is the first Brillouin zone. For k outside the first Brillouin zone, wecan find the corresponding f k using the periodicity f k+2p a=f k( )k=-pa,-pa+1 2pL,-pa+2 2pL.

8 Pa-2pL,Two useful Identities:72 ( )1N k ka j - ka j'=dj,j'( )1N j ka j - k'aj=dk,k' Example 2: a two-band model in 1 DNow, let us consider a slightly more complicated situation, a 1d chain formed by two different types of atoms a and b).( )H=-t i ai bi+bi ai+1+ +Va iai ai+Vb ibi biHere, the position of a sites in the jth unit cells is( )r=a j+raand the position of b sites in the jth unit cells is( )r=a j+rbwhere, a is the lattice constant (the size of a unit cell)Discrete Fourier transform (as discussed in the previous section)( )ak=1N iai kx( )ai=1N kak - kx( )bk=1N iai kx( )bi=1N kak - kxTherefore, we know that( ) jaj bj= j1N kak - k aj+ra 1N k'bk' k' aj+rb =1N k k'ak bk' j - k aj+ra k' aj+rb = k k'ak bk'1N j - k-k' aj - kra k'rb= k k'ak bk'dk,k' - kra k'rb= kak bk k rb-ra This conclusion is generic.

9 In k-space, hoppings from b to a result in a term in Hamiltonian kak bk k rb ra . The phase factor here isdetermined by how far the electron hop, rb-ra. If we apply the same conclusion to the hopping from a to b, we find immediately that ( ) jbj aj+1= kbk ak k ra-rb If rb-ra=a2,( ) jaj bj= kak bk ka 2( ) jbj aj+1= kbk ak ka 2 For the potential term, we also know that( ) iai ai= kak 73( ) ibi bi= kbk bkTherefore, in the k-space, the Hamiltonian looks like( )H=-t k ak bk ka 2+bk ak ka 2+bk ak - ka 2+ak bk - ka 2 +Va kak ak+Vb kbk bk=-2t k ak bkcoska2+bk akcoska2+Vaak ak+Vbbk bk We can write it in a matrix form:( )H= k ak ,bk Va-2tcos ka2 -2tcos ka2 Vb akbk For any tight-binding models with two quantum states per unit cell, the Hamiltonian can be written in terms of a two-by-two matrix inthe k space:( )H= k ak ,bk H k akbk where H is a 2 2 Hermitian matrix as a function of k.

10 It is called the kernel of the Hamiltonian. Because H k contains all the information ofthe Hamiltonian H, it is often called the Hamiltonian in literature. However, it is important to keep in mind that H k is only part of theHamiltonian. The Hamiltonian H must be gauge invariant, but H k is not. For example, if we change a a f, the Hamiltonian H isinvariant, but the kernel H k is the model considered here,( )H k =Va-2tcos ka2 -2tcos ka2 VbFor more generic cases, if one have m quantum states per unit cell, H(k) will be a m m Hermitian we have a matrix, we know what to do. We need to find the eigenvalue and eigenfunctions of the Hamiltonian, diagonalizing thematrix. The next two sections discuss the physical meanings of the eigenvalues and eigenvectors of H k.