Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models

TL;DR

This work develops a Picard–Lefschetz framework for zero-dimensional $O(n)$ sigma models with near‑exact continuous symmetry. By introducing small symmetry-breaking perturbations, it shows that downward flows slow along pseudocritical orbits before branching to form middle-dimensional Lefschetz thimbles, and that only specific combinations of thimbles survive as symmetry is restored due to non‑compact directions of the complexified group. The authors provide a practical algorithm: first solve the slow flows on the pseudocritical orbit, then extend to full tenable cycles, a method verified in both bosonic $O(n)$ and fermionic models (e.g., $O(2)$ with fermions). The results illuminate how symmetry and its breaking shape thimble structure and offer a path toward handling sign problems and spontaneous symmetry breaking within Lefschetz-thimble path integrals.

Abstract

Zero-dimensional $O(n)$-symmetric sigma models are studied by using Picard--Lefschetz integration method in the presence of small symmetry-breaking perturbations. Due to approximate symmetry, downward flows turn out to show significant structures: They slowly travel along the set of pseudo classical points, and branch into other directions so as to span middle-dimensional integration cycles. We propose an efficient way to find such slow motions for computing Lefschetz thimbles. In the limit of symmetry restoration, we figure out that only special combinations of Lefschetz thimbles can survive as convergent integration cycles: Other integrations become divergent due to non-compactness of the complexified group of symmetry. We also compute downward flows of $O(2)$-symmetric fermionic systems, and confirm that all of these properties are true also with fermions.

Figures (7)

• Figure 1: Downward flows of the reduced Hamilton system ($p_{\theta}=0$) in the $rp_r$ plane with $\alpha=0.1$. Two points with $(r,p_r)=(0,0)$ and $(1,0)$ are critical points of this system. Red solid lines represent Lefschetz thimbles $\mathcal{J}$, i.e., downward flows emanating from critical points. Green dashed lines are their homological duals $\mathcal{K}$, which are characterized by downward flows getting sucked into critical points. Blue arrows show Hamiltonian vector fields.
• Figure 2: Flows projected on the critical orbit $T^*S^1$ induced by the perturbation $\varepsilon\Delta H$ at $\alpha=0.1$. Black circles at $\theta=0,\pi$ correspond to critical points in the perturbed system. Red solid and green dashed curves are downward flows projected onto $T^*S^1$, which emanate from and get sucked into critical points, respectively. Blue arrows show Hamiltonian vector fields.
• Figure 3: Behaviors of downward flows (\ref{['Eq:coupled']}) in the $\sigma_1\sigma_2$-plane when $p_r=p_{\theta}=0$ and $\alpha=0$. Black blobs are critical points of the symmetry-broken system. Red arrowed curves are typical solutions starting from the global minimum $(r,\theta)=(1+\varepsilon/2,\pi)$ with $0<\varepsilon\ll 1$ ($\varepsilon=0.1$ in this figure). Before branching into the $r$ direction, every flow from the global minimum moves slowly along the critical orbit $r=1$.
• Figure 4: Behaviors of solutions to the differential equation (\ref{['Eq:coupled']}) in the $\sigma_1\eta_2$-plane with $\theta=p_r=0$ when $\alpha=0$. Red arrowed curves show typical solutions starting from $(r,\theta)=(1-\varepsilon/2,0)$ with $\varepsilon=0.1$.
• Figure 5: Schematic figure for global structure of Lefschetz thimbles $\mathcal{J}_{(1+\varepsilon/2,\pi)}$ and $\mathcal{J}_{(1-\varepsilon/2,\pi)}$ at $\alpha=0^+$, which are shown with green and blue surfaces, respectively. In this limit, flows into $p_r$ direction of these thimbles are quite small unless $r\simeq 0$, and we totally neglect that direction. Three black blobs represent critical points of this system, and arrowed lines show slow motion with time scale $1/\varepsilon$ along the critical orbit $T^*S^1$.