[PDF] Topological condensed matter physics Problem set 3 - TU Dresden

PDF document for free

1. PDF document for free

39567_7ub3_sheet.pdf

Technische Universitat Dresden Dr. S. Rachel

Institut fur Theoretische Physik Dr. T. MengTopological condensed matter physics

Problem set 3Summer term 2016

1. Ribbon spectra and edge states9 Points

a)1 Point

Let us consider the tight-binding chain we discussed in the very rst lecture, governed by the Hamiltonian

H 1=tX icy ici+1+ h:c: ; for which we computed the energy spectrum (i.e., the bandstructure) using a Fourier transformation. The energy spectrum readsE1(k) =2tcos(k) where we set the lattice spacing toa1. Now suppose the chain consists ofN= 7 sites and open boundary conditions (OBC) are imposed (i.e., the rst and the last sites arenotcoupled by a hoppingt). Write down the real-space Hamiltonian for this system and determine the discrete eigenenergies by diagonalizing the real-space hopping matrix. While this can be done analytically you should use a computer program such asmathematica,maple, or apython implementation. Repeat the calculation forN= 8 and compare both spectra withE1. Eventually add to the Hamiltonian the termt(cy

1cN+cy

Nc1) in order to restore periodic boundary conditions (PBC). Now consider again the discrete energies forN= 7 andN= 8 and compare toE1. What do you observe? b)2 Points We generalizeH1to the two-dimensional square lattice case, H 2=tX hijic y icj+ h:c: ;

andhijidenotes all combinations wherejis a neighboring site ofi(but each pair appears only once). For

the case of PBC, perform a Fourier transformation ofH2and calculate the energy{momentum relation E

2(kx;ky). In the following, we consider a so-called nanoribbon (a cylinder) which exhibits OBC in the

x{direction and PBC in they-direction. We denote the position of a lattice site by (m;n) corresponding

to ~Rmn=m^~ex+n^~ey. Along they{direction we can perform a Fourier-transformation while keeping the real-space representation along thex{direction: c ym;n=1N yX k ye ikyncy m;k y:

Determine the Bloch matrixh2(ky) in this hybrid representation for a ribbon with a width of eight sites.

It is again possible to impose PBC. Compute the spectrum ~E2(ky) of the ribbon. Check whether you

recover the same energies you obtained for the case where you performed a full Fourier transformation.

Do you observe a dierence in the ribbon spectrum for OBC compared to PBC? 1 c)2 Points We add to the previously considered square lattice ribbon a uniform magnetic eld of strength= 1=3

piercing through the lattice (i.e., the magnetic eld islocallyperpendicular to the ribbon). Using the

knowledge you gained from the lecture about the Hofstadter butter y determine the ribbon-Bloch matrix

structure. Choose the Landau gauge such that the magnetic unit cell is increased inx-direction, the ribbon

length must hence be a multiple of three. Do you observe a dierence in the ribbon spectrum for OBC

compared to PBC?Hint:a ribbon length of six magnetic unit cells is sucient. It might be wise to think

about an implementation where the ribbon width enters as a parameter. d)4 Points Eventually we consider the HamiltonianH2(without magnetic eld) on a honeycomb-lattice nanoribbon

(aka \carbon nanotube") with zigzag edges. Now we have two atoms per unit cell, for a ribbon consisting

ofNunit cells along thexdirection we have to deal with a 2N2Nmatrix. Try to nd the correct

ribbon-Bloch matrix. Note that the \path" ofky{independent hopping is not a straight line perpendicular

to the edge (as it is the case for the square lattice), but a zigzag line instead. Diagonalize it as a function

ofkyfor dierent ribbon lengths for both PBC and OBC. Eventually add the Semeno term, H S=MX i(1)cy ici(1) where= 1 on sublatticeAand= 0 on sublatticeB. The Semeno mass is nothing than a staggered sublattice potential term. How doesHSaect the OBC and PBC spectra?

Extra problem:2 Points

Usually a honeycomb ribbon hasA{sites on one zigzag edge andB{sites on the other edge. What happens if you considered a ribbon which hasB{sites on one edge andA{sites on the other edge? What happens, if both zigzag edges consist ofAsites? Repeat these considerations with a nite Semeno mass. In case of questions or lack of clarity please write us an email for clarication or further advice. 2

Physics Documents PDF, PPT , Doc

[PDF] bitesize physics

1. Science

2. Physics

3. Physics

[PDF] books regarding physics

[PDF] bsc physics how many years

[PDF] butler physics

[PDF] butter physics

[PDF] butterfly physics

[PDF] butterfly physics problem

[PDF] can i take physics over the summer

[PDF] careers concerning physics

[PDF] careers with physics

12345 Next 200000 acticles