Chemistry Reference
In-Depth Information
Figure 5
Scaling analysis of the Binder cumulant
B
of the classical Hamiltonian,
Eq. [24], at criticality (
a
¼
0
:
6,
e
J
¼
0
:
00111,
K
¼
1
:
153) with a dynamical exponent
z
¼
2. (Taken with permission from Ref. 64.)
Werner et al.
64
used these techniques to investigate the phase diagram of
the Hamiltonian and the quantum phase transition between the ferromagnetic
and paramagnetic phases. They found the critical behavior to be universal (i.e.,
independent of the dissipation strength
a
for all
a
6¼
0). The exponent values
are
985, which agree well with the results of
perturbative renormalization group calculations.
67,68
The other exponents can
be found from the scaling and hyperscaling relations of Eqs. [6] and [7].
n
0
:
638
;
Z
0
:
015, and
z
1
:
Diluted Bilayer Quantum Antiferromagnet
We have seen that dissipation can lead to an effective long-range inter-
action in time and thus break the symmetry between space and time directions
in the last example. Another mechanism to break this symmetry is quenched
disorder (i.e., impurities and defects), because this disorder is random in space
but perfectly correlated in the time direction.
Consider the bilayer Heisenberg quantum antiferromaget, Eq. [20]. Ran-
dom disorder can be introduced by randomly removing spins, for example,
from the system (in an experiment, one would randomly replace magnetic
atoms with nonmagnetic atoms). If the substitutions in the two layers are
made independent of one another, the resulting unpaired spins lead to com-
plex weights in the partition function that cannot be expressed in terms of a
classical Heisenberg model. Here, we therefore consider dimer dilution, that
is, the case where the corresponding spins in the two layers are removed
together. The Hamiltonian of the dimer-diluted bilayer Heisenberg quantum
