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

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.

9 pages, 7 figures; (v2) references added, explanations in Sec.II becomes more in detail;(v3) typos corrected

References in corpus (7)

• Structure of Lefschetz thimbles in simple fermionic systems
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• A New Look At The Path Integral Of Quantum Mechanics
• Real-Time Feynman Path Integral Realization of Instantons
• Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

Cited by in corpus (11)

• Structure of Lefschetz thimbles in simple fermionic systems
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Monte Carlo calculations of the finite density Thirring model
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• One-dimensional QCD in thimble regularization
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Resurgence and Dynamics of O(N) and Grassmannian Sigma Models
• Performance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density
• A quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field
• Lefschetz-thimble approach to the Silver Blaze problem of one-site fermion model