In this work, we show in pedagogical detail that the most singular contributions to the slow part of the asymptotic density-density correlation function of Luttinger liquids with fermions interacting mutually with only short-range forward scattering and also with localised scalar static impurities (where backward scattering takes place) has a compact analytical expression in terms of simple functions that have second order poles and involve only the scale-independent bare transmission and reflection coefficients. This proof uses conventional fermionic perturbation theory resummed to all orders, together with the idea that for such systems, the (connected) moments of the density operator all vanish beyond the second order - the odd ones vanish identically and the higher order even moments are less singular than the second order moment which is the only one included. This important result is the crucial input to the recently introduced "Non-Chiral Bosonization Technique" (NCBT) to study such systems. The results of NCBT cannot be easily compared with the results obtained using conventional bosonization as the former only extracts the most singular parts of the correlation functions albeit for arbitrary impurity strengths and mutual interactions. The latter, ambitiously attempts to study all the parts of the asymptotic correlation functions and is thereby unable to find simple analytical expressions and is forced to operate in the vicinity of the homogeneous system or the half line (the opposite extreme). For a fully homogeneous system or its antithesis viz. the half-line, all the higher order connected moments of the density vanish identically which means the results of chiral bosonization and NCBT ought to be the same and indeed they are.

中文翻译：

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强非均质液体的密度-密度相关函数

在这项工作中，我们在教学细节中展示了对 Luttinger 液体渐近密度-密度相关函数的缓慢部分的最奇异贡献，费米子相互作用，仅具有短程前向散射和局部标量静态杂质（其中向后散射发生）在具有二阶极点且仅涉及与尺度无关的裸透射和反射系数的简单函数方面具有紧凑的解析表达式。这个证明使用了所有阶数的传统费米微扰理论，以及对于这样的系统，密度算子的（连接的）矩都在二阶以上消失 - 奇数同样消失，高阶偶数矩比二阶矩更不奇异，二阶矩是唯一包含的。这一重要结果是最近引入的“非手性玻色化技术”（NCBT）研究此类系统的关键输入。NCBT 的结果无法与使用常规玻色化获得的结果进行比较，因为前者仅提取相关函数的最奇异部分，尽管是针对任意杂质强度和相互作用。后者，雄心勃勃地尝试研究渐近相关函数的所有部分，因此无法找到简单的解析表达式，并被迫在齐次系统或半线附近（相反的极端）进行操作。对于完全同质的系统或其对立，即。在半线上，密度的所有高阶连通矩都消失了，这意味着手性玻色化和 NCBT 的结果应该是一样的，事实上它们是一样的。