Strong-coupling perturbation theory for the extended Bose-Hubbard ...

M. ISKIN AND J. K. FREERICKS PHYSICAL REVIEW A 79, 053634 2009
ins
E CDW n A ,n B
= U n An A −1 + n B n B −1
+ zV n An B
− n A + n B
M
4
2 2
n
+
A n B +1
Un A − n B −1 + Vzn B − zn A +1 + n B n A +1
Un B − n A −1 + Vzn A − zn B +1 zt2
2 + Ot4 , 13
which is also an extensive quantity, and the odd-order terms in t also vanish. Notice that Eq. 13 reduces to Eq. 12 when
n A =n B =n 0 as expected.
The calculation of the defect state energies is more involved since it requires using degenerate perturbation theory. This is
because when exactly one extra particle or hole is added to the Mott phase, it could go to any of the M lattice sites and all of
those states share the same energy when t=0. Therefore, for both Mott-defect states with exactly one extra particle or hole, the
initial degeneracy is of order M and it is lifted at first order in t. A lengthy but straightforward calculation leads to the energy
of the Mott particle-defect state up to third order in t as
par
E Mott
n 0 = E ins
Mott n 0 + Un 0 + zVn 0 − − n 0 +1zt + n 0n 0 +1 1−z
U
n 0 z −2
U 2 + z2 −3z +3 U − V 2
+ 21−z
U −2V +
2z
U − V − n 0 +2
2U − Vzt 2 − n 0 n 0 +1
+ n 0 +1 z1−z
U 2 − 2z2 −6z +6
U − V 2 + 2z1−z
U −2V 2 + 2z2 −3z +3
+
UU − V
+ 4z2 −3z +3
+ n
U − VU −2V 0 +2 z −1
UU − V − z
4U − V 2zt3 + Ot 4 .
4z −2
UU −2V
14
This expression is valid for all d-dimensional hypercubic lattices, and it recovers the known result for the on-site BH model
when V=0 7,8. To third order in t, we obtain a similar expression for the energy of the Mott hole-defect state given by
hol
E Mott
n 0 = E ins
Mott n 0 − Un 0 −1 − zVn 0 + − n 0 zt + n 0 +1n 0 1−z
U
− n 0 n 0 +1n 0 +1 z −2
U 2 + z2 −3z +3 U − V 2
+ 21−z
U −2V +
+ 4z2 −3z +3
+ n
U − VU −2V 0 −1 z −1
UU − V − z
4U − V 2zt3 + Ot 4 ,
2z
U − V −
n 0 −1
2U − Vzt 2
+ n 0 z1−z
U 2 − 2z2 −6z +6
U − V 2 + 2z1−z
U −2V 2 + 2z2 −3z +3
+
UU − V
4z −2
UU −2V
15
which also is valid for all d-dimensional hypercubic lattices, and recovers the known result for the on-site BH model when
V=0 7,8.
On the other hand, for d1 dimensions, when an extra particle or hole is added to the CDW phase, it could go to any of
the M /2 sites in sublattice B or A, respectively. Here, we recall that n A n B is assumed in this paper. Therefore, for both
CDW-defect states with an extra particle or hole in d1 dimensions, the degeneracy is of order M /2 and it is lifted at second
order in t. This is because the states occupy one of the sublattices, and they cannot be connected by one hop, but rather require
two hops to be connected. Another lengthy but straightforward calculation leads to the energy of the CDW particle-defect state
up to third order in t as
par
E CDW
ins
n A ,n B = E CDW
−
+
n A ,n B + Un B + zVn A − +
n A +1n B +1z
Un B − n A + Vzn A − zn B − n A n B +1z
Un A − n B −1 + Vzn B − zn A +1
n B n A +1z
Un B − n A −1 + Vzn A − zn B +1 + n A n B +2
Un A − n B −2 + Vzn B − zn A +2 +
2n A n B +1z −1
Un A − n B −1 + Vzn B − zn A +2zt 2 + Ot 4 .
n B n A +1z −1
Un B − n A −1 + Vzn A − zn B
16
This expression is valid for d1-dimensional hypercubic lattices. Notice that the odd-order terms in t vanish for these
lattices. To third order in t, we obtain a similar expression for the energy of the CDW hole-defect state given by
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