Calculate the band dispersion and the density of states for the Kagome and pyrochlore lattices. We
assume that there is one orbital on each lattice site and the site energy is 0. The hopping
amplitude between nearest neighbors is t=1  and 0 between any further neighbors, i.e. hopping take
place only of the marked bonds in the lattice.
Add a site potential. On the red sublattice ad a site potential E and plot the bandstructures for
values E = 0, 0.2, 0.4, 0.6, and 8.
Hint: Identify the periodic lattice and its unit cell, perform the Fourier transformation, solve
the eigenvalue problem for each k-vector. To calculate DOS for a uniform k-mesh in the primitive
cell of reciprocal lattice and compute a histogram of the eigenenergies arising on this mesh. Use a
finer mesh for smoother DOS.
Kagome and pyrochlore lattice
kagome lattice (2D)
pyrochlore lattice (3D)
1

Calculate the band dispersion and density of states for the lattices in the picture. There are two
distinct lattice sites with energies Ea (black) and Eb (red) and two hopping amplitudes t (full)
and t’ (dotted). Perform the calculation
for
Ea = Eb  and t=t’
Ea = Eb  and t=2t’=1
Ea = 2t+Eb  and t=2t’=1
Hint: Identify the periodic lattice and its unit cell, perform the Fourier transformation, solve
the eigenvalue problem for each k-vector. To calculate DOS for a uniform k-mesh in the primitive
cell of reciprocal lattice and compute a histogram of the eigenenergies arising on this mesh. Use a
finer mesh for smoother DOS.
Intercalated Kagome lattice
Image
2

Calculate the band dispersion and density of states for a triangular lattices with 120 def spin
order. Consider non-interacting electrons on triangular lattice (calculate the band dispersion and
density of states). Add a local exchange field which has a direction as indicated in the picture.
Hint: Use the enlarged unit cell indicated in the figure. Note that the local term depend on the
lattice site (sublattice) and mixes the up and down spin directions (i.e. spin is not a good
quantum number). Use t=1 and several different values of b (starting from 0).
120 deg order on triangular lattice
3
H_0=t_sum_langle.pdf H_i(b)=_&_textco.pdf

Fermions vs hard-core bosons
Calculate and compare the spectra of non-interacting spinless fermions and had-core bosons on a
hexagon
and 3x3 square lattice with periodic boundary conditions. For hexagon consider the nearest-neighbor
(nn) hopping t=1 and next-neighbor hopping t’=0 and t’=t. For the 3x3 lattice consider only nn
hopping. Study all possible charge sectors, i.e., N=0,1,….,6 (or up to 9). Hard-core boson obey
bosonic anti-commutation relations:
Hint: The Hilbert space of hard-core bosons is isomorphic to the Hilbert space of fermions (site
double occupancy is forbidden). The fermionic and ‘bosonic’ Hamiltonian in occupation-number basis
has the same zeros, but possibly different signs of the non-zero terms.
Image
t
t’
4

Heisenberg  ring
Study the S=1/2 and S=1 Heisenberg model on the 5-ring and 6-ring using exact diagonalization.
Calculate the spin-spin correlation function as a function of temperature T.
Hint: There are two and three states per site for S=1/2 and S=1, respectively. The scalar product
S.S can be represented using raising and lowering operators S+, S- and Sz. Make sure to use the
correct values of the matrix elements. Note that one can set J=1, since only the ratio T/J matters.
5
tilde_H_=J_sum_l.pdf
Show that the correlations are isotropic, i.e.,
langle_S^z_iS^z_.pdf

Hubbard ring with nn interaction
Study 6-site Hubbard model with nn interaction. Calculate the spin-spin and charge-charge
correlation functions (1st, 2nd and 3rd neighbors) as a function of temperature (assume canonical
ensemble). Perform the calculation for 3 and 4 electrons. Use parameters t=1, U=1 and V=0, 1, and
2.
Hint: You can use the notebooks provided with the course.
6
H=t_sum_i_sigma_.pdf n_i_sigma_equiv_.pdf n_i_equiv_n_i_up.pdf

7
1
2
3
1
4
5
6
4
7
8
7
9
1
2
3
1
Heisenberg  model on triangular cluster
Study the ground state S=1/2 Heisenberg model on the 9-site cluster below. Calculate the spin-spin
correlation function for various pairs of sites. Determine the total spin moment S of the ground
state.
Hint: The exercise is analogy of the Heisenberg ring (the difference is only connectivity of the
lattice). Check the solution for FM coupling (J<0), what is the expected ground state energy and
its degeneracy (use this as a benchmark for the correctness of your set-up). The actually
interesting case is the AFM coupling (J>0).
tilde_H_=J_sum_l.pdf
Show that the correlations are isotropic, i.e.,
langle_S^z_iS^z_.pdf

2-orbital Hubbard model
Study the Hubbard 4-ring built from 2-orbital atoms. Calculate the spectrum the eigenstates of the
problem in the 5  and 6 electron sector. Calculate the spin-spin correlation function S in the
ground state (states in case of degeneracy)
for J from J=0 to J=1. Extend the calculation to a finite temperature and calculate S(J=0.2) for
temperatures from T=0.01 to T=1. (t=1, t'=0.1, U=4, U’=U-2J, J’=J, Δ=2)
Image
Hint: Diagonalize the different Sz sector separately. Note that total numbers of a and b electrons
are conserved. You can use the routines H1gen and H2gen of modules.jl Represent the results as
graphs Ei vs J, S vs J, S vs T
Image
Atomic Hamiltonian:
8
S^gamma_1_=_sum_.pdf
t
a
b
t'
H_2=&U(n_ia_upar.pdf