Abstract
We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multi thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multiple thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.
Highlights
First principles, many attempts have been made to circumvent the sign problem in lattice QCD simulations, the complete resolution is still not available [1]
We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential
We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multiple thimbles must contribute to the path integration
Summary
The (0+1)-dimensional lattice Thirring model we consider in this paper is defined by the following action [21, 22, 63], L. It is not real-positive for μ = 0 in general, but instead it has the property (det D[A]|+μ)∗ = det D[−A]|+μ = det D[A]|−μ This fact can cause the sign problem in Monte Carlo simulations. The path-integration over the field An can be done explicitly and the exact expression of the partition function is obtained (Nf = 1) as e−βL Z = 2L−1. The μ-dependence of these quantities are shown in figure 1 for L = 8, ma = 1, and β = 1, 3, and 6 It shows a crossover behavior in the chemical potential μ (in the lattice unit) around μmˆ + ln(I0(β)/I1(β)).
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