5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337
5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337
5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337
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5.28 Nils Blümer<br />
considerations alone. Often it is useful to consider a cubic representation of the angular part of<br />
atomicdorbitals,<br />
|dxy〉 ∝ (|2,2〉−|2,−2〉), |dyz〉 ∝ (|2,1〉+|2,−1〉), |dzx〉 ∝ (|2,1〉−|2,−1〉)<br />
|d x 2 −y 2〉 ∝ (|2,2〉+|2,−2〉), |d 3z 2 −r 2〉 ∝ |2,0〉, (47)<br />
expressed in terms of eigenfunctions of the angular momentum operator,<br />
l 2 |l,m〉 = � 2 l(l +1)|l,m〉, lz|l,m〉 = �m|l,m〉. (48)<br />
In lattices with cubic symmetry the five d orbitals are energetically split into the t2g orbitals<br />
(|dxy〉, |dyz〉, |dzx〉) and the eg orbitals(|dx2−y2〉, |d3z2−r2〉), which give rise to one threefold degenerate<br />
and one twofold degenerate band, respectively. Lower symmetry can lift the remaining<br />
degeneracies; e.g., in the trigonal case the t2g orbitals are further split into one nondegenerate<br />
a1g and one twofold degenerate eπ g band. Thus, it is possible that in some d systems only one<br />
band crosses or touches the Fermi surface which then justifies the one-band assumption made<br />
in (1) and used <strong>for</strong> the examples in this lecture. In general, however, the inclusion of several<br />
orbitals per site is important. An SU(2)-invariant generalization of the Hubbard model where<br />
the interaction is still local but the valence band is degenerate then contains additional coupling<br />
terms21 [53, 54]<br />
ˆHm-band = −t �<br />
〈ij〉,νσ<br />
�<br />
′<br />
+U<br />
i;ν