Chemistry Reference
In-Depth Information
QUANTUM PHASE TRANSITIONS:
COMPUTATIONAL CHALLENGES
Computational studies of quantum phase transitions generally require a
large numerical effort because they combine several formidable computa-
tional challenges. These include (1) the problem of many interacting degrees
of freedom, (2) the fact that phase transitions arise only in the thermodynamic
limit of infinite system size, (3) critical slowing down and supercritical
slowing down at continuous and first-order transitions, respectively, and
(4) anisotropic space-time scaling at quantum critical points. In disordered
systems, there is the additional complication (5) of having to simulate large
numbers of disorder configurations to obtain averages and probability distri-
butions of observables. In the following, we discuss these points in detail.
1.
The Quantum Many-Particle Problem
Computational studies of
quantum phase transitions require simulating interacting quantum many-
particle systems. The Hilbert space dimension of such systems increases
exponentially with the number of degrees of freedom. Thus, ''brute-force''
methods such as exact diagonalization are limited to very small systems
that are usually insufficient for investigating properties of phase transitions.
In many research areas of many-particle physics and chemistry, sophisticated
approximation methods have been developed to overcome this problem, but
many of those approximations are problematic in the context of quantum
phase transitions. Self-consistent field (SCF) or single-particle-type approx-
imations such as Hartree-Fock or density functional theory (see, e.g.,
Refs. 43-45), by construction, neglect fluctuations because they express the
many-particle interactions in terms of an effective field or potential. Because
fluctuations have proven to be crucial for understanding continuous phase
transitions (as discussed in the section above on Phase Transitions and Cri-
tical Behavior), these methods must fail at least in describing the critical
behavior close to the transition. They may be useful for approximately locat-
ing the transition in parameter space, though. Other approximation meth-
ods, such as the coupled cluster-method,
46
go beyond the self-consistent
field level by including one or several classes of fluctuations. However, since
the set of fluctuations included is limited and has to be selected by hand,
these methods are not bias free. Quantum critical states are generally very
far from any simple reference state so they are particularly challenging for
these techniques. One important class of methods that are potentially
numerically exact and bias free are quantum Monte Carlo methods.
47-49
They will be discussed in more detail later in this chapter. However, quan-
tum Monte Carlo methods for fermions suffer from the notorious sign pro-
blem that originates in the antisymmetry of the many-fermion wave function
and hampers the simulation severely. Techniques developed for dealing with
the sign problem often reintroduce biases into the method, via, for instance,
