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Sign problems in Fermionic QMC

2010年11月23日 留下评论 Go to comments

Well-known cases free of sign problem:

  • Negative-U Hubbard model.  Determinantal QMC has no sign problem if one decouples the on-site interaction in the right way(charge channel).
  • Hubbard model(positive U) at half-filling. Particle-hole symmetry prevents negative probability from happening. Another way to see this is that it can be mapped onto negative-U Hubbard model.  Not restricted to square lattice, any biparticle lattice is OK. For example honeycomb lattice.

Congjun Wu proves a general theorem which gives a sufficient condition under which QMC has no sign problem. Basically, there must be some anti-unitary symmetry(like time-reversal symmetry for spin 1/2). The proof was particularly for determinantal QMC. It requires that for arbitrary Hubbard-Stratonovich field configuration, the resulting determinant  must have certain anti-unitary symmetry, which means that eigenvalues come in complex conjugate pairs.  So the determinant is always positive. Notice that the determinant is generally a time-ordered product, it’s not necessarily Hermitian.

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