Chemistry Reference
In-Depth Information
model; and, more recently, it was combined with realistic band structure
calculations to investigate many-particle effects and strong correlations in
real materials. Reviews of these developments can be found in Refs. 139
and 140.
Let us finally point out that we have focused on bulk quantum phase
transitions. Impurity quantum phase transitions
42
require a separate discus-
sion that is beyond the scope of this chapter (Note, however, that within the
DMFT method a bulk quantum many-particle system is mapped onto a self-
consistent quantum impurity model.) Some of the methods discussed here such
as quantum Monte Carlo can be adapted to impurity problems. Moreover,
there are powerful special methods dedicated to impurity problems, most
notably Wilson's numerical renormalization group.
141-143
SUMMARY AND CONCLUSIONS
We have discussed quantum phase transitions in this chapter. These are
transitions that occur at zero temperature when a nonthermal external para-
meter such as pressure, magnetic field, or chemical composition is changed.
They are driven by quantum fluctuations, which are a consequence of Heisen-
berg's uncertainty principle. At first glance, it might appear that investigating
such special points in the phase diagram at the absolute zero of temperature is
purely of academic interest, however, it has become clear in recent years that
the presence of quantum phase transitions has profound consequences for the
experimental behavior of many condensed-matter systems. In fact, quantum
phase transitions have emerged as a new ordering principle for low-energy
phenomena that allows us to explore regions of the phase diagram where
more conventional pictures, based on small perturbations about simple
reference states, are not sufficient.
In the first part of the tutorial, we provided a concise introduction to the
theory of quantum phase transitions. We contrasted the contributions of ther-
mal and quantum fluctuations, and we explained how their interplay leads to a
very rich structure of the phase diagram in the vicinity of a quantum phase
transition. It turns out that the Landau-Ginzburg-Wilson (LGW) approach,
which formed the basis for most modern phase transition theories, can be
generalized to quantum phase transitions by including the imaginary time as
an additional coordinate of the system. This leads to the idea of the quantum-
to-classical mapping, which relates a quantum phase transition in
d
-space
dimensions to a classical one in
d
1 dimensions. We also discussed briefly
situations in which the LGW order parameter approach can break down, a
topic that has attracted considerable interest lately.
The second part of this chapter was devoted to computational
approaches to quantum phase transitions with the emphasis being on Monte
Carlo methods. If one is mainly interested in finding the universal critical
รพ
