Abstract
In this article, we apply the path optimization method to handle the complexified parameters in the 1+1 dimensional pure $U(1)$ gauge theory on the lattice. Complexified parameters make it possible to explore the Lee-Yang zeros which helps us to understand the phase structure and thus we consider the complex coupling constant with the path optimization method in the theory. We clarify the gauge fixing issue in the path optimization method; the gauge fixing helps to optimize the integration path effectively. With the gauge fixing, the path optimization method can treat the complex parameter and control the sign problem. It is the first step to directly tackle the Lee-Yang zero analysis of the gauge theory by using the path optimization method.
Highlights
Exploring the phase structure of theories and models with finite external parameters such as the temperature (T), the chemical potential (μ) and the external magnetic field (B) is an important subject to understand our Universe
We show the numerical results of the 1 þ 1 dimensional Uð1Þ gauge theory on the lattice with the complex coupling by using the path optimization method
We show the results of the 1 þ 1 dimensional Uð1Þ gauge theory only with the single plaquette; i.e., the simplest setting of the theory
Summary
Exploring the phase structure of theories and models with finite external parameters such as the temperature (T), the chemical potential (μ) and the external magnetic field (B) is an important subject to understand our Universe. The reason why we use the imaginary chemical potential to perform the Lee-Yang zero analysis in QCD is that there is the sign problem at complexified external parameters and the Monte-Carlo calculation sometimes fails; see Ref. If we can directly perform the Monte-Carlo calculation with finite complexified parameters, there is the possibility that we can better understand properties of the phase structure via the Lee-Yang zero analysis. Such complexification of chemical potential may be related to the investigation of the confinement-deconfinement transition at finite density [13,14].
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