Tight binding models from band representations

Tight binding models from band representations Jennifer Cano Stony Brook University and Flatiron Institute for Computational Quantum Physics

Part 0: Review of band representations

BANDREP http://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl Input: orbital, Wyckoff position, space group Output: irreps at high-symmetry points compatibility relations band connectivity

Example: p6mm

Compatibility relations constrain connectivity

Compatibility relations constrain connectivity 2 possible connectivities for the band rep Connected Disconnected decomposable connectivity indicated in BANDREP Main point: real space symmetry highly constrains band connectivity

Goals of this lecture Given Wyckoff position and orbital, write band rep matrices Build a tight-binding model with symmetries

Part 1: Bloch s theorem https://en.wikipedia.org/wiki/bloch_wave#proof_of_bloch's_theorem

Proof of Bloch s theorem 1. define simultaneous eigenstates of H, T H(r + R) =H(r) R = n 1 e 1 + n 2 e 2 + n 3 e 3,n i 2 Z Hamiltonian commutes with translations: [H, T ei ]=0 Simultaneous eigenstates: T ei n,k = e 2 ik i n,k H n,k = E n,k n,k k k 1 g 1 + k 2 g 2 + k 3 g 3 e i g j =2 ij ) n,k (r R) =T n 1 e 1 T n 2 e 2 T n 3 e 3 n,k (r) =e ik R n,k

Proof of Bloch s theorem 2. construct real space periodic function u k (r) =e ik r k(r) u k (r + R) =e ik (r+r) k(r + R) = e ik (r+r) e ik R k(r) = e ik r k(r) = u k (r) u has periodicity of lattice, not periodicity of BZ ψ has periodicity of BZ, not periodicity of lattice

Should we use u or ψ? u is the natural way to define the Berry phase r k u k is well-defined and periodic r k k = r(e ik r u k )=e ik r (r k u k + iru k ) stay tuned for Barry s lectures blows up away from origin

Part 2: tight-binding model formalism Following notes by Yusufaly, Vanderbilt and Coh, available at: http://www.physics.rutgers.edu/pythtb/_downloads/pythtb-formalism.pdf

Tight-binding degrees of freedom Atoms located at positions: r Each atom has orbitals labelled by j = 1,, n R,,j(r) =',j (r R r ) unit cell atom orbital assume orthonormality: h R,,j R 0,,ii = R,R 0, ij position operator: h R,,j r R 0,,ii =(R + r ) R,R 0, ij

Tight-binding Hamiltonian H i, j (R) h R 0,,i H R 0 +R,,ji = h 0,,i H R,,ji Translation invariance only specify atoms/orbitals and R

Two natural choices of Fourier transform Choice 1: k,ji = X k e ik (R+r ) R,,ji H k i, j h k i H k j i = X R e ik (R r +r ) H i, j (R) Choice 2: k,ji = X k e ik R R,,ji H k i, j h k i H kji = X R e ik R H i, j (R) Note: only different if more than one site per cell!

Pros/cons of Fourier transform choices Choice 2: H k i, j h k i H kji = X R e ik R H i, j (R) Hamiltonian is periodic in Brillouin zone Choice 1: H k i, j h k i H k j i = X R e ik (R r +r ) H i, j (R) Hamiltonian is not periodic in Brillouin zone Pro: Symmetries without translations are k-independent (will prove) Since we are mostly concerned with symmetries, we use Choice 1 in these lectures!!!

Choice 2: Periodicity of Hamiltonian Define diagonal matrix: V (k) i, j = ij e ik r H k+g i, j = eig (r r ) H k i, j = V (G) 1 H k V (G) i, j Notes: H is not periodic in BZ, but H(k) and H(k+G) have same eigenvalues (of course)

Fourier conventions: analogy to u and ψ Hamiltonians generate eigenvalue equations: H k i, jc nk j H k i, j C nk j = E nk C nk i = E nk Cnk i Eigenstates: X i X i C nk i k ii C nk i k ii C n,k+g i = C n,k i BZ periodic, like ψ C n,k+g i = e ig r C n,k i not BZ periodic, like u Using Choice 1 instead of Choice 2 is similar to using u instead of ψ (which we argued earlier was more natural for topological applications)

Example 1: atoms in 1d 1 orbital/site trivial subscripts: r = r 1 = 0 i =1 H(R = ±ˆx) = t H(R 6= ±ˆx) =0 H k = X R e ik R H(R) = t(e ik ˆx + e ik ˆx )= 2t cos k k=-π k=0 k=π Γ X

Example 2: square lattice H(R) = ( t if R = ±ˆx, ±ŷ 0 else H k = 2t(cos k x + cos k y ) How to plot 2d spectrum? identify high-symmetry path M Γ X Γ X M Γ

Example 3: interpenetrating square lattices r A =(0, 0) B t t t t t t t t r B =(1/2, 1/2) only one orbital/site i = 1 H AA (0) =µ A A H BB (0) =µ B H AA (±ˆx) =H AA (±ŷ) =H BB (±ˆx) =H BB (±ŷ) = t H AB (0) =H AB ( ˆx) =H AB ( ˆx ŷ) =H AB ( ŷ) = t 0 label by R, not rab

Example 3: interpenetrating square lattices, cont. H k AA = H k BB = 2t(cos k x + cos k y )+µ A 2t(cos k x + cos k y )+µ B H k AB = t0 (e ik (0+r B) + e ik ( ˆx+r B) + e ik ( ˆx ŷ+r B) + e ik ( ŷ+r B) ) = t 0 (e ik ( 1 2, 1 2 ) + e ik ( 1 2, 1 2 ) + e ik ( 1 2, 1 2 ) + e ik ( 1 2, 1 2 ) ) = 4t 0 cos k x 2 cos k y 2 Γ X M Γ

Part 3: tight-binding symmetries

How do crystal symmetries act on tight-binding states? g = {P v} r! P r + v point group element translation Action on TB states: g R, ii =[U g ] j, i R 0, ji R, β fixed by spatial rotation: P (R + r )+v = R 0 + r U describes orbital rotation U is independent of R (will prove)

How do crystal symmetries act on Fouriertransformed TB basis? g k ii = X R e ik (R+r ) g R, ii = X R e ik (R+r ) R 0, ji[u g ] j, i = X R e i(p k) (P (R+r )) R 0, ji[u g ] j, i = X R e i(p k) (R0 +r v) R 0, ji[u g ] j, i = P k j i[u g ] j, i e i(p k) v Proof (promised earlier) that if v=0, then g is k-independent More generally, g splits into k-independent matrix and k-dependent phase

How does the tight-binding Hamiltonian transform under symmetries? Real space: ghg -1 = H Momentum space: H k i, j h k i H k j i = h k i ghg 1 k j i Expand RHS: = h k i g k 0 l ih k0 l H k 00 mih k00 m g 1 [U g ] i, l P k 0,ke ik v k 0,k 00 H k0 l, m [Ug 1 ] m, j P 1 k,k 00e ik00 ( P 1 v) k j i Conclude: H k i, j = h U g H P 1k i Ug 1 i, j The k-independent matrix U determines symmetry of Hamiltonian (even in non-symmorphic group!!)

How to find little group irreps Little group: G k = {g gk = k + G} In our gauge, Ug does not commute with H k : U g H k U 1 g = H k+g = V (G) 1 H k V (G) Recall: V (k) i, j = ij e ik r Instead: [V (G)U g,h k ]=0 Can simultaneously diagonalize H k and V(G)Ug (not Ug) Little group characters from eigenvalues of V(G)Uge -i(pk).v (This must be the case, otherwise little group irreps would be k-independent!!) Note: G is different for different choices of g

Example 1: 1d chain with inversion, s and p orbitals What is U? inversion does not mix orbitals diagonal H k = µs 0 0 µ p onsite U = z ) H k = z H k z + ts 0 0 t p nearest neighbor, same orbital cos k + 0 itsp sin k +... it sp 0 nearest neighbor, different orbital orbital How to implement in real space? H ss(pp) (ˆx) =H ss(pp) ( ˆx) H sp (ˆx) = H sp ( ˆx)

Example 1: 1d chain with inversion, s and p orbitals Compute inversion eigenvalues H = 1 2 (µ s µ p t s + t p ) z +( )I +... H X = 1 2 (µ s µ p + t s t p ) z +( )I +... U g = z V (k) =I Inversion eigenvalues determined by sign of: (µ s µ p ) ± ( t s + t p ) Four possibilities for inversion eigenvalue of lower band: ++, +-, -+, -- Upper band always opposite of lower: --, -+, +-, ++ Can understand from band representations Four possibilities are four different EBRs

-π Γ π Example 2: pg (Non-symmorphic) group generated by {my ½ 0} r 1 =0, r 2 =1/2 Frieze group What is U? glide exchanges orbitals U g = x xh k x = H k H k = µi +(t cos k 2 + t0 sin k 2 ) x +... both sites have same chemical potential! two types of nearest neighbor hopping Spectrum: t = t = 1

Example 2: pg Compute eigenvalues -i +1 -i H k = µi +(t cos k 2 + t0 sin k 2 ) x +i -1 +i -π Γ Band crossing is required!! π Recall: Little group characters from eigenvalues of V(G)Uge -i(pk).v, where Pk = k+g U g = x G =0) V (0) = I e i(pk)v = e ik/2 At k, matrix form of glide: xe ik/2

Exercises 1. Inversion symmetry in 1d with atoms at the general position. Consider a 1d chain of atoms with two sites per cell at x = ±x0, invariant under inversion symmetry. a) What is the matrix form for inversion symmetry? b) Construct a tight-binding model. c) Show that the tight-binding Hamiltonian is identical to that for s and p orbitals at x=0 after a basis transformation. d) What happens when x0 = 1/4? 2. Glide with time-reversal. Consider pg, generated by the glide, {my 1/2, 0}. a) Why is the term cos(k)σx forbidden? Hint: what is V(G)? b) How does time reversal change the band structure? (Time-reversal is implemented by complex conjugation and k -k.) c) Bonus: what if SOC is included? Hint: see Hourglass fermions, by Wang, Alexandradinata, Cava, Bernevig, Arxiv:1602.05585, Nature 532,189-194 (2016) 3. Rectangular lattice. Consider a rectangular lattice of s orbitals, without C4 symmetry. a) What extra terms can be added to the Hamiltonian with C4 symmetry? b) Along which path should the spectrum be plotted to see all high-symmetry lines? 4. k-independence. Prove that when there is only one atom (perhaps with many orbitals) in the unit cell, then there exists a choice of origin such that the little group irreps are k- independent.