Chemistry Reference
In-Depth Information
If the boson density is an integer (per site), and, in the absence of disor-
der, charge (amplitude) fluctuations are small. If we set
^
e
i
y
i
and inte-
grate out the amplitude fluctuations, we obtain a phase-only model that can be
written as an O(2) quantum rotor model:
i
¼j
^
j
i
2
X
i
t
X
h
2
U
q
H
QR
¼
i
cos
ð
y
y
Þ
½
33
i
j
2
qy
;
i
i
j
This Hamiltonian describes, among other systems, an array of coupled Joseph-
son junctions.
In the spirit of this section, we now discuss the quantum-to-classical
mapping for the dirty boson problem. We first consider the case of integer
boson density and no disorder, i.e., the Hamiltonian in Eq. [33]. In this
case, the quantum-to-classical mapping can be performed analogously to the
transverse-field Ising model: The partition function is factorized using the
Trotter product formula leading to a path integral representation. By reinter-
preting the imaginary time direction as an extra dimension and rescaling space
and time appropriately (which does not change universal properties), we
finally arrive at an isotropic three-dimensional classical
XY
model with the
Hamiltonian
K
X
h
H
cl
k
B
T
¼
cos
ð
y
i
y
j
Þ
½
34
i
;
j
i
where
. This is again a well-known
model of classical statistical mechanics that can be simulated efficiently using
Monte Carlo cluster algorithms and series expansions (see, e.g., Ref. 82). The
resulting critical exponents are
y
i
is a classical angle in the interval
½
0
;
2
p
a
0
:
015,
b
0
:
348,
g
1
:
318,
d
4
:
780,
n
0
:
672, and
Z
0
:
038. Since space and time enter symmetrically, the dyna-
mical exponent is
z
1.
The general case of noninteger boson density and/or the presence of the
random potential is more realistic. However, it leads to broken time-reversal
symmetry for the quantum rotors because the particle number is represented
by the quantity canonically conjugate to the phase variable, that is, by angular
momentum. The quantum-to-classical mapping procedure sketched above,
therefore, leads to complex weights in the partition function, and the system
cannot be interpreted in terms of a classical
XY
model. Wallin et al.
81
found
an alternative quantum-to-classical mapping that avoids the complex weight
problem. They expressed the partition function in terms of the integer-valued
angular momentum variables of the rotors. The resulting link current (Villain
83
)
representation is a classical
¼
ð
2
þ
1
Þ
-dimensional Hamiltonian:
K
X
i
H
cl
k
B
T
¼
1
1
2
J
i
;
t
ð
m
þ ~
J
i
;
t
Þ
½
35
v
i
;
t
