Chemistry Reference
In-Depth Information
System (f) again displays the fine structure associate with the appearance of an
insulating domain in the second shell. These quantitative results can be used to
identify the superfluid-insulator transition in experiments.
Fermions
So far, we have discussed quantum Monte Carlo approaches to quantum
phase transitions in boson and spin systems. In these systems, the statistical
weight in the Monte Carlo procedure is generally positive definite, so there
is no sign problem. Note that for spin systems, this is only true if there is no
frustration. Frustrated spin systems in general do have a sign problem.
Unfortunately, the sign problem generally exists for fermions because it
is rooted in the antisymmetry of the many-fermion wave function. The
problem can be understood by recognizing that boson and spin operators on
different lattice sites commute. The signs of the matrix elements appearing in a
quantum Monte Carlo scheme are thus determined locally. Contrarily, fer-
mion operators on different lattice sites anticommute, thus leading to extra
nonlocal minus signs. In fact, it was shown that a generic solution to the
sign problem is almost certainly impossible to obtain by proving that the
sign problem belongs to the NP (nondeterministic polynomial) hard computa-
tional complexity class.
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This means that a generic solution of the sign
problem would also solve all other NP hard problems in polynomial time.
One way of circumventing (if not solving) the sign problem is to force the
nodes of the many-fermion wave function to coincide with that of a trial wave
function. The resulting fixed-node quantum Monte Carlo method
88,113
has
been successful in determining with high precision the ground-state properties
of real materials. It is clear that the accuracy of the method depends crucially
on the quality of the trial wave function. This implies that the fixed-node
Monte Carlo method will work well if the ground-state properties are under-
stood at least qualitatively. However, quantum critical states are, in general,
very far from any simple reference state, which means that simple trial wave
functions cannot be constructed easily. Fixed-node methods are therefore not
well suited for studying the properties of fermionic systems close to quantum
phase transitions (although they may be useful for locating the transition in
parameter space).
While a general solution of the fermionic sign problem is likely
impossible to find, several nontrivial fermionic systems exist for which the
sign problem can be avoided. Hirsch et al.
90
developed a world-line Monte
Carlo simulation scheme for fermions that, in strictly one dimension, avoids
the sign problem. (Generalizations to higher dimensions still suffer from the
sign problem.) A more general quantum Monte Carlo approach to fermions
is the so-called determinantal Monte Carlo method.
114
Its basic idea is to
decouple the fermion-fermion interactions bymeans of aHubbard-Stratonovich
transformation,
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leading to a system of noninteracting fermions coupled to a
