Chemistry Reference
In-Depth Information
critical points have been found mainly in quantum systems, starting with Fish-
er's seminal work on the random transverse field Ising model
22,23
by means of
the Ma-Dasgupta-Hu renormalization group.
24
The above classification is based on the behavior of the
average
disorder
strength at large length scales. In recent years it has become clear, however,
that an important role is often played by strong disorder fluctuations and
the rare spatial regions that support them. These regions can show local order
even if the bulk system is in the disordered phase. Their fluctuations are very
slow because they require changing the order parameter in a large volume.
Griffiths
25
showed that the contributions of the rare regions lead to a singular
free energy not only at the phase transition point but in an entire parameter
region around it. At generic thermal (classical) transitions, the rare region con-
tributions to thermodynamic observables are very weak since the singularity in
the free energy is only an essential one.
26,27
In contrast, at many quantum
phase transitions, rare disorder fluctuations lead to strong power-law quantum
Griffiths singularities that can dominate the thermodynamic beha-
vior.
22,23,28,29
In some systems, rare region effects can become so strong that
they destroy the sharp phase transition by smearing.
30
A recent review of rare
region effects at classical, quantum, and nonequilibrium phase transitions can
be found in Ref. 31.
QUANTUM VS. CLASSICAL PHASE TRANSITIONS
In this section, we give a concise introduction into the theory of quantum
phase transitions, emphasizing similarities with and differences from classical
thermal transitions.
How Important Is Quantum Mechanics?
The question of how important quantum mechanics is for understanding
continuous phase transitions has several facets. On the one hand, one may ask
whether quantum mechanics is even needed to explain the existence and prop-
erties of the bulk phases separated by the transition. This question can be
decided only on a case-by-case basis, and very often quantum mechanics is
essential as, e.g., for the superconducting phase. On the other hand, one can
ask how important quantum mechanics is for the behavior close to the critical
point and thus for the determination of the universality class to which the tran-
sition belongs. It turns out that the latter question has a remarkably clear and
simple answer: Quantum mechanics does
not
play any role in determining the
critical behavior if the transition occurs at a finite temperature; it does play a
role, however, at zero temperature.
