超临界区域通常被描述为均匀的，没有明确的相变。然而，其中物质表现出的不同行为（如类液态和类气态）暗示了“超临界边界”的存在。在这里，我们通过重新审视Lee-Yang（LY）理论并引入复相图（即具有复数$T$和$p$的4维相图），为这些现象提供了数学描述。传统的使用实数$T$和$p$值（物理平面）的2维相图在临界点以上缺乏LY零点，从而阻止了临界性的出现，而在这种4维场景中，平面外的零点在各种物理性质上具有临界异常。例如，当使用等压热容$C_p$（系统对$T$的响应函数）来划分超临界区域时，可以通过将其降维为具有复数$T$和实数$p$的3维相图来可视化该4维复相图。然后，我们发现由$C_p$定义的超临界边界与该3维相图中具有复数$T$的LY零点边缘在物理平面上的投影完美对应，而在传统LY理论中这些平面外的零点被忽略了。同样的关系适用于等温压缩系数$K_T$（或$κ_T$），它是系统对$p$的响应函数，此时应使用复数$p$。Widom线与LY零点边缘之间的这种相关性在三个系统中得到了证明，即van der Waals模型、2D Ising模型和水，这明确揭示了超临界物质的初始相变本质。通过LY理论的这一扩展及相关新发现，提供了一个适用于相变和超临界区域的相与相变的统一图景，该图景应适用于其他热力学状态函数的复相图。
英文摘要：
The supercritical region is often described as uniform with no definite transitions. The distinct behaviors of the matter therein (as liquid-like and gas-like), however, suggest ``supercritical boundaries". Here, we provide a mathematical description of these phenomena by revisiting the Lee-Yang (LY) theory and introducing a complex phase diagram, i.e. a 4-D one with complex $T$ and $p$. While the traditional 2-D phase diagram with real $T$ and $p$ values (the physical plane) lacks LY zeros beyond the critical point, preventing the occurrence of criticality, the off-plane zeros in this 4-D scenario possess critical anomalies in various physical properties. For example, when the isobaric heat capacity $C_p$, which is a response function of the system to $T$, is used to separate the supercritical region, this 4D complex phase diagram can be visualized by reducing to a 3D one with complex $T$ and real $p$. Then, we find that the supercritical boundary defined by $C_p$ shows perfect correspondence with the projection of the edges of the LY zeros with complex $T$ in this 3D phase diagram on the physical plane, whilst in conventional LY theory these off-plane zeros are neglected. The same relation applies to the isothermal compression coefficient $K_T$ (or $κ_T$) which is a response function of the system to $p$, where complex $p$ should be used. This correlation between the Widom line and the edges of LY zeros is demonstrated in three systems, i.e., van der Waals model, 2D Ising model and water, which unambiguously reveals the incipient phase transition nature of the supercritical matter. With this extension of the LY theory and the associated new findings, a unified picture of phase and phase transition valid for both the phase transition and supercritical regions is provided, which should apply to the complex phase diagram of other thermodynamic state functions.
Universal supercritical thermodynamics for black holes
School of Science, Hunan Institute of Technology（湖南工学院理学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Peng Huanwu Collaborative Center for Research and Education, Beihang University（北京航空航天大学彭桓武研究中心）；University of Chinese Academy of Sciences（中国科学院大学）；Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院基础物理与数学科学中心）
