Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density

TL;DR

We study the Lefschetz thimble decomposition of the complexified path integral for the (0+1)-D massive Thirring model at finite density. By identifying all critical points and determinant zeros, we map how thimbles contribute as the chemical potential $\mu$ is varied, revealing Stokes transitions and a crossover where multiple thimbles matter. In the uniform-field subspace, the single thimble dominates at small and large $\mu$, while multi-thimble contributions are essential to reproduce the crossover and to approach the continuum and low-temperature limits. The results provide quantitative guidance for Monte Carlo simulations on Lefschetz thimbles and highlight the interplay between thimble weights and phase cancellations.

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.

Figures (14)

• Figure 1: (a) Fermion number density and (b) scalar condensate with $L=8$, $ma=1$ for $\beta =1$ (solid), 3 (dashed), and 6 (dotted).
• Figure 2: Number density $\left < n \right >$ on the $T$-$\mu$ plane in the continuum limit ($g^2/m=1/2$).
• Figure 3: Critical points (green dots) and determinant zeros (red dots) of the Thirring model with $L=4$ and $ma=1$ at $\hat{\mu}=0$ within the subspaces of $n_-=0$ (left) and 1 (right). We set $\beta = 3 - 0.1 {\rm i}$. Gradient flows are drawn with arrows. We assign numbers to the critical points as $\sigma_{i, \bar{i}}$ here. The downward ${\cal J}_{\sigma}$ and upward ${\cal K}_{\sigma}$ cycles of a critical point $\sigma$ are shown with solid and dashed lines, respectively. The brighter background indicates the larger value of ${{\rm Re} S}$ (in arbitrary unit).
• Figure 4: Downward flow, critical points of free theory for $\beta = 3 - 0.1 {\rm i}$ in the complex $z$ plane. The horizontal (vertical) axis is for the real (imaginary) part.
• Figure 5: Downward gradient flows, critical points (green) and zero points (red) in complex $z$ plane of the Thirring model with $ma=1$, $\beta=3$ and $L=4$ for (a) $\mu=0.6$, (b) 1.2, and (c) 1.8. The downward (upward) cycles, ${\cal J}_{\sigma}$ (${\cal K}_{\sigma}$), are depicted with solid (dashed) lines.