This work extracts, by means of an exact analysis, the singular behavior of the dynamical response functions—the Fourier transforms of dynamical two-point functions—in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes that describe the local behavior of the response function near a threshold. The singular behavior is derived starting from first principles considerations: the method of analysis does not rely, at any stage, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behavior and of the associated edge exponents that arise in the vicinity of the dispersion relation of one massive excitation (hole, particle, or bound state). In addition, the present analysis shows that due to the lack of strict convexity of the particle dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These originate from multi-particle/hole/bound state excitations maximizing the energy at fixed momentum.

中文翻译：

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关于 XXZ 自旋 1/2 链的无质量区域动态响应函数的奇异性

这项工作通过精确分析提取了动力学响应函数的奇异行为——动力学两点函数的傅里叶变换——在 XXZ spin-1/2 的无质量区域的各种激发阈值附近链。该分析产生描述阈值附近响应函数的局部行为的边缘指数和相关幅度。奇异行为是从第一原则考虑出发的：分析方法不依赖于，在任何阶段，关于与场论或其他现象学方法的一些假设对应。该分析基于作者最近获得的 XXZ 链响应函数的无质量形状因子扩展。它证实了基于非线性 Luttinger 的关于幂律行为和相关边缘指数的预测，这些边缘指数出现在一个大规模激发（空穴、粒子或束缚态）的色散关系附近。此外，目前的分析表明，由于粒子色散关系缺乏严格的凸性，并且由于束缚态的慢速分支的存在，存在具有不同边缘指数结构的激发阈值。这些源于多粒子/空穴/束缚态激发，在固定动量下最大化能量。