Parameters
Average sign ⟨s⟩:
0.70
Computing...
Sign Problem Physics
The sign problem: Fermionic partition function Z = Σ_C s(C)·|W(C)| involves configurations with positive and negative weights s(C)=±1. Observable ⟨O⟩ = ⟨Os⟩₊/⟨s⟩₊.
Average sign: ⟨s⟩ = Z_F/Z_B = e^{-β(F_F-F_B)} → exponentially small at low T or large N. Error ~ 1/(⟨s⟩√M) diverges as sign → 0.
Sign-free cases:
• Half-filling, bipartite lattice: particle-hole symmetry → s>0 always
• Repulsive Hubbard at half-fill: exact sign cancellation
• 1D systems: no sign problem for many models
DQMC (Blankenbecler-Scalapino-Sugar): Write e^{-βH} as product of matrices via Suzuki-Trotter; integrate out fermions → det[M_↑]·det[M_↓].
Average sign: ⟨s⟩ = Z_F/Z_B = e^{-β(F_F-F_B)} → exponentially small at low T or large N. Error ~ 1/(⟨s⟩√M) diverges as sign → 0.
Sign-free cases:
• Half-filling, bipartite lattice: particle-hole symmetry → s>0 always
• Repulsive Hubbard at half-fill: exact sign cancellation
• 1D systems: no sign problem for many models
DQMC (Blankenbecler-Scalapino-Sugar): Write e^{-βH} as product of matrices via Suzuki-Trotter; integrate out fermions → det[M_↑]·det[M_↓].