We discuss a new strategy for treating the complex action problem of lattice field theories with a θ-term based on density of states (DoS) methods. The key ingredient is to use open boundary conditions where the topological charge is not quantized to integers and the density of states is sufficiently well behaved such that it can be computed precisely with recently developed DoS techniques. After a general discussion of the approach and the role of the boundary conditions, we analyze the method for 2-d U(1) lattice gauge theory with a θ-term, a model that can be solved in closed form. We show that in the continuum limit periodic and open boundary conditions describe the same physics and derive the DoS, demonstrating that only for open boundary conditions the density is sufficiently well behaved for a numerical evaluation. We conclude our proof of principle analysis with a small test simulation where we numerically compute the density and compare it with the analytical result.

我们讨论了一种基于状态密度（DoS）方法用θ项处理晶格场论的复杂作用问题的新策略。关键因素是使用开放边界条件，在该条件下拓扑电荷不会量化为整数，并且状态密度表现得足够好，因此可以使用最近开发的DoS技术精确地计算出来。在对方法和边界条件的作用进行了一般性讨论之后，我们分析了带有θ的2-d U（1）晶格规理论的方法-term，可以以封闭形式求解的模型。我们表明，在连续极限中，周期性和开放边界条件描述了相同的物理学并推导了DoS，这表明仅对于开放边界条件，密度才能很好地进行数值评估。我们通过一个小的测试模拟来结束本原理分析的证明，在该模拟中我们通过数值计算密度并将其与分析结果进行比较。