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Register a new userHigh density QCD on a Lefschetz thimble?
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It is sometimes speculated that the sign problem that afflicts many quantum field theories might be reduced or even eliminated by choosing an alternative domain of integration within a complexified extension of the path integral (in the spirit of the stationary phase integration method). In this paper we start to explore this possibility somewhat systematically. A first inspection reveals the presence of many difficulties but - quite surprisingly - most of them have an interesting solution. In particular, it is possible to regularize the lattice theory on a Lefschetz thimble, where the imaginary part of the action is constant and disappears from all observables. This regularization can be justified in terms of symmetries and perturbation theory. Moreover, it is possible to design a Monte Carlo algorithm that samples the configurations in the thimble. This is done by simulating, effectively, a five dimensional system. We describe the algorithm in detail and analyze its expected cost and stability. Unfortunately, the measure term also produces a phase which is not constant and it is currently very expensive to compute. This residual sign problem is expected to be much milder, as the dominant part of the integral is not affected, but we have still no convincing evidence of this. However, the main goal of this paper is to introduce a new approach to the sign problem, that seems to offer much room for improvements. An appealing feature of this approach is its generality. It is illustrated first in the simple case of a scalar field theory with chemical potential, and then extended to the more challenging case of QCD at finite baryonic density.
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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.
Recently, we have proposed a novel approach (arxiv:1205.3996) to deal with the sign problem that hinders Monte Carlo simulations of many quantum field theories (QFTs). The approach consists in formulating the QFT on a Lefschetz thimble. In this paper we concentrate on the application to a scalar field theory with a sign problem. In particular, we review the formulation and the justification of the approach, and we also describe the Aurora Monte Carlo algorithm that we are currently testing.
Recently there has been remarkable progress in solving the sign problem, which occurs in investigating statistical systems with a complex weight. The two promising methods, the complex Langevin method and the Lefschetz thimble method, share the idea of complexifying the dynamical variables, but their relationship has not been clear. Here we propose a unified formulation, in which the sign problem is taken care of by both the Langevin dynamics and the holomorphic gradient flow. We apply our formulation to a simple model in three different ways and show that one of them interpolates the two methods by changing the flow time.
The complex Langevin method and the generalized Lefschetz-thimble method are two closely related approaches to the sign problem, which are both based on complexification of the original dynamical variables. The former can be viewed as a generalization of the stochastic quantization using the Langevin equation, whereas the latter is a deformation of the integration contour using the so-called holomorphic gradient flow. In order to clarify their relationship, we propose a formulation which combines the two methods by applying the former method to the real variables that parametrize the deformed integration contour in the latter method. Thr
Thimble regularisation is a possible solution to the sign problem, which is evaded by formulating quantum field theories on manifolds where the imaginary part of the action stays constant (Lefschetz thimbles). A major obstacle is due to the fact that one in general needs to collect contributions coming from more than one thimble. Here we explore the idea of performing Taylor expansions on Lefschetz thimbles. We show that in some cases we can compute expansions in regions where only the dominant thimble contributes to the result in such a way that these (different, disjoint) regions can be bridged. This can most effectively be done via Pade approximants. In this way multi-thimble simulations can be circumvented. The approach can be trusted provided we can show that the analytic continuation we are performing is a legitimate one, which thing we can indeed show. We briefly discuss two prototypal computations, for which we obtained a very good control on the analytical structure (and singularities) of the results. All in all, the main strategy that we adopt is supposed to be valuable not only in the thimble approach, which thing we finally discuss.
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