Recently there has been a renewed interest in the spectra and role in dynamical properties of excited states of the spin-1/2 Heisenberg antiferromagnetic chain in longitudinal magnetic fields associated with Bethe strings. The latter are bound states of elementary magnetic excitations described by Bethe-ansatz complex non-real rapidities. Previous studies on this problem referred to finite-size systems. Here we consider the thermodynamic limit and study it for the isotropic spin-1/2 Heisenberg XXX chain in a longitudinal magnetic field. We confirm that also in that limit the most significant spectral weight contribution from Bethe strings leads to $(k,\omega )$-plane gapped continua in the spectra of the spin dynamical structure factors $S^{+-}(k,\omega )$ and $S^{xx}(k,\omega )=S^{yy}(k,\omega )$. The contribution of Bethe strings to $S^{zz}(k,\omega )$ is found to be small at low spin densities m and to become negligible upon increasing that density above $m\approx 0.317$. For $S^{-+}(k,\omega )$, that contribution is found to be negligible at finite magnetic field. We derive analytical expressions for the line shapes of $S^{+-}(k,\omega )$, $S^{xx}(k,\omega )=S^{yy}(k,\omega )$, and $S^{zz}(k,\omega )$ valid in the $(k,\omega )$-plane vicinity of singularities located at and just above the gapped lower thresholds of the Bethe-string states's spectra. As a side result and in order to provide an overall physical picture that includes the relative $(k,\omega )$-plane location of all spectra with a significant amount of spectral weight, we revisit the general problem of the line-shape of the transverse and longitudinal spin dynamical structure factors at finite magnetic field and excitation energies in the $(k,\omega )$-plane vicinity of other singularities. This includes those located at and just above the lower thresholds of the spectra that stem from excited states described by only real Bethe-ansatz rapidities.

最近，人们对自旋1/2海森堡反铁磁链在与Bethe弦有关的纵向磁场中的激发态的光谱及其在动力学性质中的作用有了新的兴趣。后者是由Bethe-ansatz复杂非真实速度描述的基本磁激发的束缚态。以前对此问题的研究涉及有限大小的系统。在这里，我们考虑热力学极限，并在纵向磁场中研究各向同性自旋1/2 Heisenberg XXX链的热力学极限。我们确认，在此限制范围内，Bethe弦谱对频谱的最大贡献也导致$（ķ，\omega ）$自旋动力学结构因子谱中的平面缺口连续 ${\mathrm{小号}}^{+-}（ķ，\omega ）$ 和 ${\mathrm{小号}}^{XX}（ķ，\omega ）={\mathrm{小号}}^{ÿÿ}（ķ，\omega ）$。Bethe琴弦对${\mathrm{小号}}^{žž}（ķ，\omega ）$被发现在低自旋密度m时很小，并且在将该密度增加到10以上时变得可忽略不计$米\approx 0.317$。对于${\mathrm{小号}}^{-+}（ķ，\omega ）$，发现在有限的磁场中该贡献可以忽略不计。我们导出线形的解析表达式${\mathrm{小号}}^{+-}（ķ，\omega ）$， ${\mathrm{小号}}^{XX}（ķ，\omega ）={\mathrm{小号}}^{ÿÿ}（ķ，\omega ）$和 ${\mathrm{小号}}^{žž}（ķ，\omega ）$ 在有效 $（ķ，\omega ）$平面位于Bethe弦状态光谱的空缺下阈值处并刚好在其上方。附带的结果是，为了提供包括相对$（ķ，\omega ）$在所有频谱具有很大频谱权重的平面位置上，我们重新研究了在有限磁场和激励能量下，横向和纵向自旋动力学结构因子的线形的一般问题。 $（ķ，\omega ）$平面附近的其他奇点。这包括那些位于和仅在实际Bethe-ansatz速度描述的激发态的光谱下阈值之上和正上方的那些。