Chemistry Reference
In-Depth Information
behavior (i.e., the overall scaling scenario, the critical exponents, and the
critical amplitude ratios), using a purely classical simulation scheme based
on quantum-to-classical mapping is often most efficient. We illustrated this
approach for the transverse-field Ising model with and without dissipation,
for the bilayer Heisenberg antiferromagnet, and for dirty bosons in two
dimensions. If one is interested in nonuniversal questions such as quantitative
results for critical coupling constants or observables, a true quantum
algorithm must be used. We have reviewed several quantum Monte Carlo
approaches to quantum spin and boson Hamiltonians and discussed their
results for the quantum phase transitions in these systems. We have also
considered fermionic systems and the extra complications brought about by
the generic appearance of the notorious sign problem.
It is probably fair to say that Monte Carlo simulations of model systems
that are free of the sign problem (bosons, spin systems without frustration, and
some special fermionic systems) have become so powerful that the properties
of their quantum phase transitions can be determined quantitatively with high
precision (see, e.g., the accuracy of some of the exponent values quoted in the
preceding sections). For many frustrated spin systems, in contrast, the results
are limited to a qualitative level, and for quantum phase transitions in generic
fermionic systems (with sign problem), direct computational attacks are still of
limited utility.
ACKNOWLEDGMENTS
The author has benefited greatly from discussions with many friends and colleagues, in
particular, D. Belitz, A. Castro-Neto, A. Chubukov, P. Coleman, K. Damle, V. Dobrosavljevic,
P. Goldbart, M. Greven, S. Haas, J. A. Hoyos, F. Igl ยด i, T. R. Kirkpatrick, A. Millis, D. Morr,
M. Norman, P. Phillips, H. Rieger, S. Sachdev, A. Sandvik, J. Schmalian, Q. Si, R. Sknepnek,
G. Steward, J. Toner, M. Vojta, and A. P. Young.
Some of the work described here has been supported by the National Science Foundation
under grant nos. DMR-0339147 and PHY99-07949, by Research Corporation and by the Univer-
sity of Missouri Research Board. Parts of this work have been performed at the Aspen Center for
Physics and the Kavli Institute for Theoretical Physics, Santa Barbara.
REFERENCES
1. T. Andrews,
Phil. Trans. R. Soc. A
,
159
, 575 (1869). The Bakerian Lecture: On the Continuity
of the Gaseous and Liquid States of Matter.
2. J. D. van der Waals, PhD Thesis, University of Leiden, 1873. On the Continuity of the Gas and
Liquid State.
3. K. G. Wilson,
Phys. Rev. B
,
4
, 3174 (1971). Renormalization Group and Critical Phenomena.
I. Renormalization Group and the Kadanoff Scaling Picture.
4. K. G. Wilson and J. Kogut,
Phys. Rep.
,
12
, 75 (1974). The Renormalization Group and the
e
Expansion.
