The quantum Monte Carlo (QMC) method employs stochastic but importance
Sampling over small but representative portions of the many-body Hilbert
space.
However, its application is often plagued by the sign-problem when
applied to fermion systems.
It is regarded as one of the most challenging unsolved
problems in many branches of contemporary physics.
We have made significant progress in studying the underlying mathematical
principles of the sign-problem.
It reveals a deep connection between two seemingly unrelated research
directions – QMC simulations and positivity based Kramers symmetry
and Majorana reflection in mathematical physics.
It opens up new opportunities to simulate the interplay among
interaction, topology, and electron doping in strongly correlated systems.
Majorana positivity
In collaboration with T. Xiang's group, we explored the fermion sign-problem
from a new perspective, based on the Majorana reflection positivity and
Majorana Kramers positivity [Ref. 1] .
We proved two sufficient conditions for the absence of the sign-problem,
which provide a unified framework to describe all known sign-problem-free
interacting lattice fermion models so far.
They also allow us to identify a number of new sign-problem-free
models including, but not limited to, lattice
fermion models with repulsive interactions but without particle-hole
symmetry, and interacting topological insulators with spin-flip terms.
Kramers positivity
In a previous work, we (with S. C. Zhang) proved the absence of the
QMC sign-problem based on Kramers positivity
[Ref. 2] .
This result is based on $T$-invariant decomposition to interaction terms,
where $T$ is an anti-unitary transformation satisfying $T^2=-1$, but is
not limited to the time-reversal operator.
Then the statistical weights, i.e., the determinants of
the fermion matrices, can be expressed as products of complex conjugate
pairs of the eigenvalues, thus are positive definite.
Compared with previously known sign-problem-free models, our result
does not need the fermion determinants to be factorizable, hence,
it applies to a much wider class of models.
References and talks
1. Congjun Wu, and Shou-Cheng Zhang,
"Sufficient condition for absence of the sign problem in the
fermionic quantum Monte-Carlo algorithm",
Phys. Rev. B 71, 155115 (2005),
see pdf file
.
2. Zhong-chao Wei, Congjun Wu , Yi Li, Shi-Wei Zhang, Tao Xiang,
"Majorana Positivity and the Fermion sign problem of Quantum Monte Carlo
Simulations",
Phys. Rev. Lett. 116, 250601 (2016) .
See pdf file .
Talk Fermion positivity and
the QMC sign problem
.
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Last modified: July 15, 2007.