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Phase transitions are common and important phenomena in nature[1]. For example, phase transitions between gas, liquid, and solid states can be well described by statistical physics, thermodynamics, and condensed matter physics. However, such theories do not apply to complex substances with liquid polymorphisms. Specifically, complex substances[2-4], such as water, silicon, sulfur, phosphorus, gallium, and hydrogen, exhibit anomalous properties in their liquid states. Despite having distinct electronic and molecular structures, these substances display similar anomalous properties in their liquid states, including structural, dynamic, and thermodynamic anomalies[5-7]. According to the liquid-liquid critical point scenario, single-component substances may exhibit a low-density liquid (LDL), a high-density liquid (HDL), and a transition between them (LLPT). The liquid-liquid critical point (LLCP) and the resulting critical fluctuation may be the source of the anomalous properties of complex liquids. The phase transitions in complex substances, such as LLPT, are intriguing and important phenomena. However, verification of the LLCP scenario is challenging because LLCPs are usually buried in the deeply supercooled region where crystallization occurs rapidly and is not easily accessible experimentally. How to trace LLPT and LLCP through easily detectable phase intervals has still been at the forefront of this field.

References：

1. X. Yu, R. Huang, H. Song, L. Xu, et al. "Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains", Phys. Rev. Lett. 129, 210601 (2021)

2. R. Li, G. Sun, L. Xu, et al. "Anomalous properties and the liquid-liquid phase transition in gallium", The Journal of Chemical Physics, 145, 054506 (2016)

3. R. Z. Li, J. Chen, X. Z. Li, E. G. Wang, L. Xu*. “Rationalizing the liquid-liquid phase transition in high pressure hydrogen using the concept of the Widom line”. New Journal of Physics 17, 063023 (2015).

4. Z. Sun, D. Pan, L. Xu*, and E. G. Wang*,  “The role of proton ordering in adsorption preference of polar molecule on ice surface”, Proc. Natl. Acad. Sci. USA 109, 13177-13181 (2012)

5. L. Xu*, P. Kumar, S. V. Buldyrev, S.-H. Chen, P. H. Poole, F. Sciortino and H. E. Stanley. “Relation between the widom line and the strong-fragile dynamic crossover in systems with a liquid-liquid phase transition.”, Proc. Natl. Acad. Sci. USA 102, 16558 (2005)

6. L. Xu*, S. V. Buldyrev, F. W. Starr, F. Mallamace, and H. E. Stanley. “Appearance of a fractional Stokes-Einstein relation in water and a structural interpretation of its onset.”, Nature Physics 5, 565-569 (2009).

7. J. Luo, L. Xu*, E. Lascaris, H. E. Stanley, and S. V. Buldyrev, "Behavior of the Widom Line in Critical Phenomena", Phys. Rev. Lett. 112, 135701 (2014).