# :code:`e` and eignvectors :code:`v` where eigenvalue :code:`e[i]` corresponds
# to eigenvector :code:`v[:,i]`.
e, v = scipy.linalg.eigh(H)
print("{} eignvalues and {} eigvenvectors {} elements long.".format(
len(e), v.shape[1], v.shape[0]))
################################################################################
# Computing expectation values
# ------------------------------------------------------------------------------
# To interpret the results, it is informative to compute the expectations values
# related to the spin :math:`\mathbf{S}`, orbital :math:`\mathbf{L}`,
# and total :math:`\mathbf{J}`, angular momentum. We first load the relevant
# matrices for these quantities for a `p` atomic shell. We need to specifiy
# that we would like to include spin when loading the orbital operator.
orb_mom = edrixs.get_orb_momentum(l, ispin=True)
spin_mom = edrixs.get_spin_momentum(l)
tot_mom = orb_mom + spin_mom
################################################################################
# We again transform these matrices to our Fock basis to build the operators
n_fermion = 2
opL, opS, opJ = edrixs.build_opers(n_fermion, [orb_mom, spin_mom, tot_mom],
basis)
################################################################################
# Recall that quantum mechanics forbids us from knowing all three Cartesian
# components of angular momentum at once, so we want to compute the squares of
# these operators i.e.
#
# .. math::
# \mathbf{S}^2 = S^2_x + S^2_y + S^2_z\\
# The 4-rank tensor is in the complex spherical Harmonics basis
umat = edrixs.get_umat_slater('t2g', F0, F2, F4)
# Fock basis in the complex spherical Harmonics basis with the orbital ordering:
# |-1,up>, |-1,dn>, |0,up>, |0,dn>, |+1,up>, |+1,dn>
# basis: 2d list of integers with 1 or 0, the shape is (15, 6) in this case
# where, 15=6*5/2 is the total number of Fock basis and 6 is the total number of
# single-particle orbitals
basis = edrixs.get_fock_bin_by_N(norb, noccu)
# quantum number of orbital angular momentum for t2g: l=1
ll = 1
# Matrices of lx,ly,lz,sx,sy,sz,jx,jy,jz in the single-particle basis
# lx: l_orb[0], ly: l_orb[1], lz: l_orb[2]
l_orb = edrixs.get_orb_momentum(ll, True)
# sx: s_spin[0], sy: s_spin[1], sz: s_spin[2]
s_spin = edrixs.get_spin_momentum(ll)
# jx: j_so[0], jy: j_so[1], jz: j_so[2]
j_so = l_orb + s_spin
# very small Zeeman splitting along z-direction
emat_zeeman = (l_orb[2] + s_spin[2]) * 1e-10
# many-body operators of L^2, Lz
Lxyz = edrixs.build_opers(2, l_orb, basis)
L2 = np.dot(Lxyz[0], Lxyz[0]) + np.dot(Lxyz[1], Lxyz[1]) + np.dot(
Lxyz[2], Lxyz[2])
Lz = Lxyz[2]
# many-body operators of S^2, Sz
Sxyz = edrixs.build_opers(2, s_spin, basis)
S2 = np.dot(Sxyz[0], Sxyz[0]) + np.dot(Sxyz[1], Sxyz[1]) + np.dot(
Sxyz[2], Sxyz[2])
e, v = scipy.linalg.eigh(H)
return e, v
################################################################################
# The large :math:`U` limit
# ------------------------------------------------------------------------------
# Let us see what happens with :math:`U \gg t`.
e, v = diagonalize(1000, 1)
print("Energies are")
print(e)
################################################################################
# To analyze what is going on we can determine the spin expectation values
# of the cluster. Building the operators follows the same form as the
# Hamiltonian and the previous example.
spin_mom_one_site = edrixs.get_spin_momentum(ll)
spin_mom = np.zeros((3, norb, norb), dtype=np.complex128)
spin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site
opS = edrixs.build_opers(2, spin_mom, basis)
opS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1])
+ np.dot(opS[2], opS[2]))
################################################################################
# This time let us include a tiny magnetic field along the :math:`z`-axis, so
# that we have a well-defined measurement axis and print out the expectation
# values.
zeeman = np.zeros((norb, norb), dtype=np.complex128)
zeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2]
e, v = diagonalize(1000, 1, extra_emat=zeeman)