Chemistry Reference
In-Depth Information
at about 1.5K. This transition is a thermal continuous phase transition analo-
gous to the Curie point of iron discussed above. Applying a magnetic field per-
pendicular to the ordering direction of the ferromagnet induces quantum
fluctuations between the spin-up and spin-down states and thus reduces the
transition temperature. At a field strength of
H
c
50 kOe (corresponding to
a field of
B
c
5 T), the transition temperature drops to zero. Thus, at 50 kOe
LiHoF
4
undergoes a quantum phase transition from a ferromagnet to a para-
magnet. At first glance, quantum phase transitions would seem to be a purely
academic problem since they occur at isolated values in parameter space and at
zero temperature, which is not accessible in a real experiment. However, it is
now clear that the opposite is true; quantum phase transitions do have impor-
tant, experimentally relevant consequences, and they are believed to provide
keys to many new and exciting properties of condensed matter, including
the quantum Hall effects, exotic superconductivity, and non-Fermi liquid
behavior in metals.
The purpose of this chapter is twofold: In the following two sections,
Phase Transitions and Critical Behavior and Quantum vs. Classical Phase Tran-
sitions, we give a concise introduction into the theory of quantum phase transi-
tions, emphasizing similarities with and differences from classical thermal
transitions. After that, we point out the computational challenges posed by
quantum phase transitions, and we discuss a number of successful computa-
tional approaches together with prototypical examples. However, this chapter
is not meant to be comprehensive in scope. We rather want to help scientists
who are taking their first steps in this field to get off on the right foot. Moreover,
we want to provide experimentalists and traditional theorists with an idea of
what simulations can achieve in this area (and what they cannot do,
yet).
Those readers who want to learn more details about quantum phase transitions
and their applications should consult one of the recent review articles
6-9
or the
excellent textbook on quantum phase transitions by Sachdev.
10
...
PHASE TRANSITIONS AND CRITICAL BEHAVIOR
In this section, we briefly collect the basic concepts of the modern theory
of phase transitions and critical phenomena to the extent necessary for the
purpose of this chapter. A detailed exposition can be found in, e.g., the text-
book by Goldenfeld.
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Landau Theory
Most modern theories of phase transitions are based on Landau theory.
12
Landau introduced the concept of an
order parameter
, a thermodynamic quan-
tity that vanishes in one phase (the disordered phase) and is nonzero and gener-
ally nonunique in the other phase (the ordered phase). For the ferromagnetic
