THE STRONG COUPLING EXPANSION
35
In the leading approximation the ground state is described by the
wave function:
Tìm = - m
!>=n - 1
a\ defC i\ Î )j, H - i
4- Ò")
2x1/2
(3.5)
(The Hilbert space in our problem is formed by the direct product of the
two dimensional spaces at each site y. The operators Xy act as Pauli
matrices on the space labelled by y and as a unit operator on all other
spaces.) The ground state energy corresponding to (3.5) is given by:
(3.6)
The first excited state in this leading approximation is obtained as:
n
y^yo
= l>’0>
(3.7)
These states (labelled by jo) have energy:
El — Eq = 2u
(3.8)
We see that in the leading approximation we have a nondegenerate
ground state and a highly degenerate first excited level, separated by the
gap 2m. We shall see now that in the next approximation this degeneracy is removed.
If we denote the second term in (3.4) by V we have, first of all to
consider matrix elements (j'\ V\y}. Due to the obvious relations
we find that:
T,"lT> = |0>
Oi ^yi)
\y>
(3.9)
0 ^'|Kb> = t;Xôy..±6
5
= 2uó,..
(3.10)
We see that while Ho described individual and independent spins, the
term V describes hopping of their excitations from one site to another.