Matter-antimatter coexistence method for finite density QCD toward a solution of the sign problem
Toward the lattice QCD calculation at finite density, we propose "matter-antimatter coexistence method", where matter and anti-matter systems are prepared on two parallel ${\bf R}^4$-sheets in five-dimensional Euclidean space-time. We put a matter system $M$ with a chemical potential $μ\in {\bf C}$ on a ${\bf R}^4$-sheet, and also put an anti-matter system $\bar M$ with $-μ^*$ on the other ${\bf R}^4$-sheet shifted in the fifth direction. Between the gauge variables $U_ν\equiv e^{iagA_ν}$ in $M$ and $\tilde U_ν\equiv e^{iag \tilde A_ν}$ in $\bar M$, we introduce a correlation term with a real parameter $λ$. In one limit of $λ\rightarrow \infty$, a strong constraint $\tilde U_ν(x)=U_ν(x)$ is realized, and therefore the total fermionic determinant becomes real and non-negative, due to the cancellation of the phase factors in $M$ and $\bar M$, although this system resembles QCD with an isospin chemical potential. In another limit of $λ\rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $μ$ and $-μ^*$. For a given finite-volume lattice, if one takes an enough large value of $λ$, $\tilde U_ν(x) \simeq U_ν(x)$ is realized and phase cancellation approximately occurs between two fermionic determinants in $M$ and $\bar M$, which suppresses the sign problem and is expected to make the lattice calculation possible. For the obtained gauge configurations of the coexistence system, matter-side quantities are evaluated through their measurement only for the matter part $M$. The physical quantities in finite density QCD are expected to be estimated by the calculations with gradually decreasing $λ$ and the extrapolation to $λ=0$. We also consider more sophisticated improvement of this method using an irrelevant-type correlation.