Chemistry Reference
In-Depth Information
then be used in variational or diffusion Monte Carlo simulations to provide
accurate results for the correlation energy and for other quantities. In contrast,
different problems arise for materials whose behavior is not even qualitatively
understood, such as when dealing with many strongly correlated electron sys-
tems. These systems are often studied by using simple models that capture the
new properties of a whole class of materials without adding too many (realis-
tic) details. However, the absence of even a qualitative understanding of such
systems severely hampers the construction of trial wave functions with the
right properties (symmetries, etc.). Ideally, this class of problems should be
studied by (bias-free) methods that do not rely on trial wave functions at all.
Studies involving the simulation of quantum phase transitions belong to
the second class of problems. While variational or diffusion Monte Carlo cal-
culations can be very useful in locating approximately the quantum phase
transition of a particular system in parameter space, they are much less suita-
ble for studying the quantum critical state itself (because it is generally far
away from any simple reference state). Significant progress in simulating
quantum phase transitions of boson and spin systems has been achieved by
path-integral (world-line) Monte Carlo
89,90
and the related stochastic series
expansion (SSE) method
91,92
in recent years. Fermion systems pose a much
harder problem to solve because the antisymmetry of the many-fermion
wave function generically leads to the notorious sign problem, an issue that
we shall return to at the end of the section. In the following we introduce
briefly the world-line and SSE methods and then discuss a few representative
examples of quantum phase transitions in boson and spin systems.
World-Line Monte Carlo
The world-line Monte Carlo algorithm is a finite-temperature method
that samples the canonical density matrix of a quantum many-particle system.
It may appear counterintuitive at first glance to use a finite-temperature method
to study quantum phase transitions that occur at zero temperature, but a finite-
temperature method is suitable for the following two reasons: (1) One of
the (experimentally) most interesting regions of the phase diagram close to a
quantum critical point is the quantum critical region located at the critical
coupling strength but at comparatively high temperatures (see the section on
Quantum vs. Classical Phase Transitions). Finite-temperature methods are
thus required to explore it. (2) Assessing the dependence of observables on
temperature is an efficient tool for determining the dynamical scaling beha-
vior of the quantum critical point (analogous to finite-size scaling, but in the
imaginary time direction).
The general idea
89,90
of the world-line Monte Carlo algorithm is similar
to that of the quantum-to-classical mapping discussed in the last section. The
Hamiltonian is split into two or more terms
ΒΌ
P
i
H
i
such that the matrix
H
