Chemistry Reference
In-Depth Information
Figure 9
Binder cumulant of the classical Hamiltonian in Eq. [31] at the critical point.
(
Main panel
) Power-law scaling plot. (
Inset
) Scaling plot according to activated scaling.
(Taken with permission from Ref. 79.)
phase transitions in disordered systems according to the effective dimensional-
ity of the defects.
31
Dirty Bosons in Two Dimensions
The examples discussed so far are all
magnetic
quantum phase transi-
tions. Our last example in this section on quantum-to-classical mapping is a
quite different transition, viz. the superconductor-insulator transition in
two-dimensional dirty boson systems. Experimentally, this transition can be
realized in helium absorbed in a porous medium or in granular superconduct-
ing films as an example.
The minimal model for describing the superconductor-insulator transi-
tion in the general case of both charge and phase fluctuations being relevant
is the boson Hubbard model with a random local chemical potential.
80,81
The
Hamiltonian (defined on a square lattice) takes the form
2
X
i
X
t
X
h
U
H
BH
N
i
Þ
N
i
ð
^
i
^
þ
^
j
^
¼
ð
m
þ
v
i
zt
Þ
½
32
j
i
i
i
;
j
i
Here,
U
is the onsite repulsion,
is the chemical potential,
z
is the number of
nearest neighbors, and
v
i
represents the random onsite potential. The hopping
strength is given by
t
, and
^
m
i
,
^
i
are the boson creation and destruction opera-
tors at site
i.
The number operator is given by
N
i
¼
^
i
^
i
.
