Chemistry Reference
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bosonic field. The fermions can now be integrated out in closed form, and the
partition function is given as the sum over configurations of the bosonic field
with the weight being a fermionic determinant. This sum can be performed
by Monte Carlo sampling. In general, the fermionic determinant will have a
fluctuating sign, again reflecting the fermionic sign problem. In some special
cases, however, the determinant can be shown to be positive definite. For
instance, the determinant is positive definite for the two-dimensional repulsive
Hubbard model on bipartite lattices at exactly half filling (because of particle-
hole symmetry).
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For the attractive Hubbard model, sign-problem free algo-
rithms can even be constructed for all filling factors, and such algorithms have
been used to study the superconducting transition in two and three spatial
dimensions. In two dimensions, the transition is classified as being of the
Kosterlitz-Thouless type.
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In three dimensions, the model displays a con-
ventional second-order transition, and an interesting crossover takes place
between the Bardeen-Cooper-Schrieffer (BCS) and the Bose-Einstein condensa-
tion (BEC) scenarios.
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Another attack on the sign problem is by the so-
called meron-cluster algorithm that can be applied to certain fermionic
Hamiltonians.
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It has been used, for example, to study the effects of disorder
superconductivity in fermion models with attractive interactions.
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Despite this progress, the utility of quantumMonte Carlo simulations for
studying quantum phase transitions in fermionic systems is still rather limited.
Many of the most interesting problems, such as the ferromagnetic and antifer-
romagnetic quantum phase transitions
9,36,123
in transition-metal compounds
and heavy-fermion materials, are still too complex to be attacked directly by
microscopic quantum Monte Carlo methods.
OTHER METHODS AND TECHNIQUES
In this section we discuss briefly—without any pretense of completeness—
further computational approaches to quantum phase transitions. The
conceptually simplest method for solving a quantum many-particle problem is
(numerically) exact diagonalization. However, as already discussed in the
section on Quantum Phase Transitions: Computational Challenges, the
exponential increase of the Hilbert space dimension with the number of degrees
of freedom severely limits the possible system sizes. One can rarely simulate
more than a few dozen particles even for simple lattice systems. Systems of
this size are too small to study quantum phase transitions (which are a property
of the thermodynamic limit of infinite system size) with the exception of,
perhaps, certain simple one-dimensional systems. Even in one dimension,
however, more powerful methods have largely superceded exact diagonaliza-
tion.
One of these techniques is the density matrix renormalization group
(DMRG) proposed by White in 1992.
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In this method, one builds the
