Structure of Lefschetz thimbles in simple fermionic systems

Abstract

The Picard-Lefschetz theory offers a promising tool to solve the sign problem in QCD and other field theories with complex path-integral weight. In this paper the Lefschetz-thimble approach is examined in simple fermionic models which share some features with QCD. In zero-dimensional versions of the Gross-Neveu model and the Nambu-Jona-Lasinio model, we study the structure of Lefschetz thimbles and its variation across the chiral phase transition. We map out a phase diagram in the complex four-fermion coupling plane using a thimble decomposition of the path integral, and demonstrate an interesting link between anti-Stokes lines and Lee-Yang zeros. In the case of nonzero mass, it is shown that the approach to the chiral limit is singular because of intricate cancellation between competing thimbles, which implies the necessity to sum up multiple thimbles related by symmetry. We also consider a Chern-Simons theory with fermions in $0+1$-dimension and show how Lefschetz thimbles solve the complex phase problem caused by a topological term. These prototypical examples would aid future application of this framework to bona fide QCD.

37 pages, 17 figures. v2: minor changes, the version to appear in JHEP

References in corpus (14)

• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• QCD Critical Point and Complex Chemical Potential Singularities
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• Spontaneous CP-violation in the strong interaction at theta = pi
• A New Look At The Path Integral Of Quantum Mechanics
• RG-Whitham dynamics and complex Hamiltonian systems
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Real-Time Feynman Path Integral Realization of Instantons
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
• Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling
• Path integration and perturbation theory with complex Euclidean actions

Cited by in corpus (3)

• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• Resurgence and Dynamics of O(N) and Grassmannian Sigma Models