Mathematical Principle of Quantum Monte-Carlo Sign problem



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.