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 NambuJona-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.

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

“…The residual phase problem from the Jacobian due to the curvature is mild and can be efficiently taken into account by reweighting for this theory [50]. The Lefschetz thimble integration has been examined in other models [52][53][54][55] and has been studied from other aspects [56][57][58][59][60][61][62] which involve the sign problem. This approach also shed new light on the complex Langevin sampling method [27,33,62], and vice versa [61].…”

“…In this paper, we study the path integration on the Lefschetz thimbles in the (0+1) dimensional massive Thirring model at finite chemical potential µ [63], in order to clarify the effects of the fermion determinant on the structure of the thimbles contributing to the partition function [55]. The lattice model is formulated with the staggered fermions [64,65] and a compact auxiliary vector boson (a link field).…”

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

“…This example also demonstrates that in using Lefschetz thimbles, for example, either in Euclidean semi-classics or real time semi-classics (with sign problems) [110,111] or in lattice simulations [101][102][103][104], all thimbles whose Stokes multipliers are non-zero must be summed over. Numerical evidence for the correctness of this perspective is also given in [105][106][107][108][109].…”

Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

“…(2.5) near the critical point shows that thimbles are manifolds with N real dimensions. 2 Thimbles in fermionic models have a distinct feature not present in bosonic models [10][11][12]. If one uses the standard procedure of introducing an auxiliary bosonic field through the Hubbard-Stratanovich transformation and then integrate over the fermion fields one is left with a determinant det D[φ] which, when exponentiated, adds to the auxiliary field action a term − log detD [φ].…”

“…The analysis determining which combination of thimbles is equivalent to the original integration region R N or, in other words, the determination of the integers n σ in eq. (2.8) is very involved in all but the simplest, one dimensional cases (for an example where this is possible, see [10]). …”