5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337

5.28 Nils Blümer
considerations alone. Often it is useful to consider a cubic representation of the angular part of
atomicdorbitals,
|dxy〉 ∝ (|2,2〉−|2,−2〉), |dyz〉 ∝ (|2,1〉+|2,−1〉), |dzx〉 ∝ (|2,1〉−|2,−1〉)
|d x 2 −y 2〉 ∝ (|2,2〉+|2,−2〉), |d 3z 2 −r 2〉 ∝ |2,0〉, (47)
expressed in terms of eigenfunctions of the angular momentum operator,
l 2 |l,m〉 = � 2 l(l +1)|l,m〉, lz|l,m〉 = �m|l,m〉. (48)
In lattices with cubic symmetry the five d orbitals are energetically split into the t2g orbitals
(|dxy〉, |dyz〉, |dzx〉) and the eg orbitals(|dx2−y2〉, |d3z2−r2〉), which give rise to one threefold degenerate
and one twofold degenerate band, respectively. Lower symmetry can lift the remaining
degeneracies; e.g., in the trigonal case the t2g orbitals are further split into one nondegenerate
a1g and one twofold degenerate eπ g band. Thus, it is possible that in some d systems only one
band crosses or touches the Fermi surface which then justifies the one-band assumption made
in (1) and used for the examples in this lecture. In general, however, the inclusion of several
orbitals per site is important. An SU(2)-invariant generalization of the Hubbard model where
the interaction is still local but the valence band is degenerate then contains additional coupling
terms21 [53, 54]
ˆHm-band = −t �
〈ij〉,νσ
�
′
+U
i;ν