The obtained expressions for the observables (see Methods), namely the isobaric thermal expansion α p , the isobaric heat capacity c p and the isothermal compressibility \({\kappa }_{T}\) together with \({{\rm{\Gamma }}}_{s}\) enable us to study the behavior of the system on the verge of the critical point. Because the equations for c p , α p , \({\kappa }_{T}\) and T depend on the pressure and volume of the system, they constitute a parametric system. This characteristic of the model prevents us from obtaining an analytical expression for v. Note that Eq. (3) (Methods) clearly indicates a transcendental equation for v. We thus analyze the behavior of the various observables by varying v, which causes variations in T. We fix the critical point parameters by employing the corresponding expressions, see Methods. The parameters were adjusted12 so that \({T}_{c}\simeq 180\,{\rm{K}}\), which is in the No Man’s Land region2, but a bit lower than T c reported in ref.7 and that shown in Fig. 1. Note that the free parameters of the model reported in ref.12 could be changed in order to explore other systems of interest. Here, we focus on the analysis of \({{\rm{\Gamma }}}_{s}\) and \({{\rm{\Gamma }}}_{w}\) (see Methods) to the supercooled phase of water. Also, for the sake of completeness, we stress that we have recalculated both thermal expansion and specific heat, already reported in ref.12. Figure 2(a,b) show the p–v phase diagram for a range of temperatures and the T–v diagram. Note that when T = 0 K the resulting mapping \(p(T=0,v)\) is a straight line. This is obtained using Eq. (3). When the temperature is high, the pressure for \(v\approx {v}_{0}\) is higher than the case for low temperatures. For \(v\approx {v}_{0}+\delta v\), however, higher temperatures decrease the pressure for fixed values of v. Figure 2(b) shows that in a particular range of values of volume, for given pressure values, physical temperature values are inaccessible. Figure 2(a) shows that the point where the pressure is the same for every temperature value (blue vertical line) is the limiting value for the volume (v) for which physical values of the temperature are obtained. As discussed above, we cannot analytically obtain an expression \(v(T,p)\) because Eq. (3) is transcendental in v. Hence, we have a mapping of these physical quantities [see Eq. (4)], and we can find the corresponding v and T values for each pressure value (p). The same holds true for any other desired order of these three parameters. Figure 3(a–f) depict the behavior of the observables for the system considering 16 pressure values, varied in uniform steps from p = 1.17 kbar to 0.17 kbar. The panels a) and b) show the observables α p and c p , which were presented and discussed in ref.12 for a different range of pressure values. Here, we focus on an analysis of these observables near the critical point. Remarkably, the absolute values of α p and c p increase significantly for \(p={p}_{c}\) and \(T={T}_{c}\), a fingerprint of a phase transition and/or critical point. Figure 3(c) shows the behavior of \({{\rm{\Gamma }}}_{s}={\alpha }_{p}/{c}_{p}\), see Methods. Note the effect of pressure on \({{\rm{\Gamma }}}_{s}\) and its distinct behavior upon approaching the critical point, when comparing with c p and α p . In the immediate vicinity of the critical point, \({{\rm{\Gamma }}}_{s}\) is extremely sensitive to thermal fluctuations. Figure 3(d) shows the so-called pseudo-Grüneisen parameter \({{\rm{\Gamma }}}_{w}={w}^{2}{{\rm{\Gamma }}}_{s}\)18 (see Methods). Note that for the data set corresponding to \(p={p}_{c}\), \({{\rm{\Gamma }}}_{w}\to 0\) for \(T={T}_{c}\). The vanishing of \({{\rm{\Gamma }}}_{w}\) can be understood in terms of the behavior of the normalized speed of sound w/w c (where \({w}_{c}\approx 6.513\,{\rm{m}}\,{{\rm{s}}}^{-1}\)), shown in Fig. 3(e). In the vicinity of the critical point, \({w}_{c}\to 0\), whereas its value for temperatures far from T c increases to approximately 100w c . Physically, this finding suggests that, near the critical point, the propagation of sound waves is significantly suppressed. Interestingly, an anomalous behavior of the sound velocity was also observed close to the Mott critical end-point in strongly correlated electronic systems and associated with a diverging compressibility of the electronic degrees of freedom19,20. Figure 3(f) shows that the compressibility also presents an enhanced behavior near the critical point. Our findings are in perfect agreement with those reported in ref.21 for the various observables.

Figure 2 (a) Pressure (p) versus volume (v) phase diagram obtained from Eq. (3) for different values of temperature. The temperature was uniformly varied from 0 to 200 K, with steps of 10 K. The straight line is related to T = 0 K. Similar results were reported in ref.12. (b) Temperature (T) versus volume (v) for different values of pressure, which were also varied uniformly as in panel (a). The parameters used were the same as in12, namely \(c=6\), \(\delta \varepsilon =1000\,{\rm{J}}\,{{\rm{mol}}}^{-1}\), \({v}_{0}=(2\times {10}^{-5})\,{{\rm{m}}}^{3}\,{{\rm{mol}}}^{-1}\), \(\delta v=(0.5\times {10}^{-5})\,{{\rm{m}}}^{3}\,{{\rm{mol}}}^{-1}\) and \(\lambda =0.2\). The blue solid line indicates the lower physically valid volume in our analysis. Full size image

Figure 3 (a) Isobaric thermal expansivity α p , (b) isobaric heat capacity c p , (c) Grüneisen parameter \({{\rm{\Gamma }}}_{S}={\alpha }_{p}/{c}_{p}\), (d) Grüneisen parameter \({{\rm{\Gamma }}}_{w}={w}^{2}{\alpha }_{p}/{c}_{p}\), (e) speed of sound w normalized by its value on the critical point, namely \({w}_{c}\approx 6.513\,{\rm{m}}\,{{\rm{s}}}^{-1}\), (f) isothermal compressibility \({\kappa }_{T}\) for different values of pressure. The employed parameters were the same as presented in the caption of Fig. 2. The critical temperature is indicated by the vertical blue solid lines. Further details are discussed in the main text. Full size image

The Maxwell-relation \({(\frac{\partial v}{\partial T})}_{p}=-\,{(\frac{\partial S}{\partial p})}_{T}\) and the negative thermal expansivity shown in Fig. 3 indicate that the entropy (S) of the system is enhanced when approaching the liquid-liquid critical point, i.e., by applying pressure, the high- and low-density phases mix and the entropy increases. It is noteworthy to point out that we also find this in the finite-T critical end-point reported for molecular conductors8,9 and the quantum critical points in heavy-fermion compounds22,23. The high- and low-density phases produce two different energy scales. Because the degree of H-bonding depends on temperature and pressure, a scaling cannot be applied successfully24,25. Reference6 indicates that water molecule interactions create an open H-bond structure that has a lower density than other configurations. We can capture the energy scales associated with the H-bond configurations that correspond to the low- and high-density phases using a compressible Ising-like model and two accessible system volumes. In particular, the capture of the energy scales associated with H-bonds is, in our analysis, represented by the vanishing of one of the possible volumes associated with the sites. Using the Landau theory26, we find that, by decreasing the order parameter fluctuations, a divergence in both the correlation length7 and relaxation time27 are expected. Reference28 reports a connection between the entropy-dependent relaxation time and \({{\rm{\Gamma }}}_{s}\). We here suggest that this is also true for supercooled water.

In what follows, we use the compressible cell Ising-like model to study the Ising-nematic phase recently detected in the low-doping regime of Fe-based superconductors29. An electronic nematic phase is essentially a melted stripe phase30. Figure 4 shows that as the pressure is increased for \(v={v}_{0}+\delta v\), the temperature decreases. The limiting volume value for such a behavior is \(v={v}_{0}+0.17\delta v\) for \(\lambda =0.2\).

Figure 4 Temperature (T) versus pressure (p) phase diagram for three different values of volume (v), as indicated in the label. A linear relation is observed between T and p for all values of v. For values close to the upper limit of the volume, the pressure reaches negative values and the angular coefficient of the mathematical relation between T and p changes sign. For high values of v (more precisely, for \(v > {v}_{0}+0.17\delta v\) in the case where \(\lambda =0.2\)), the angular coefficient is negative, indicating a decrease in temperature as pressure increases. The values of v 0 and δv employed here are the same as in Fig. 2. Full size image

In the case of the proposed nematic phase in Fe-based superconductors, the pressure variation is caused by the chemical pressure introduced in the system by the doping effect on the crystal lattice. As the pressure (doping) is varied, the critical point signature vanishes (see Fig. 3). We obtain the same behavior shown in Fig. 4 (red curve) experimentally for the 122 doped Fe-based superconductors31. In particular, the thermal expansion signatures are suppressed upon doping31. Comparing the pressure versus temperature phase diagram reported in ref.31 for the 122 doped Fe-based superconductor with our results, we see that the regime that better illustrates the nematic phase is the one where \(v={v}_{0}+0.7\delta v\), since the critical point signature is shifted for lower values of T as p increases (see Fig. 4). Because there is a substantial number of free parameters that compose the current Ising-like model, we leave the fitting of the experimental results reported in ref.31 to future research. Here we used the compressible cell Ising-like model to simulate the doping effect in single crystals by assuming there are only two different volumes in the melted electronic nematic phase30. When the system is doped, the electronic nematic phase associated with two coexisting volumes (see the figure in ref.30) is suppressed, and the reported superconductivity appears, e.g., for Ba(Fe 1−x Co x ) 2 As 2 single crystals31,32. Yet, it is worth mentioning that for the limit case where \(v={v}_{0}+0.17\delta v\), we have no pressure variations for a wide range of temperature values. Since pressure and volume are conjugated variables, this behavior can be associated to the Invar effect, which has been widely investigated in iron-nickel alloys, see e.g.33.

Finally, we highlight our main findings. We have used an energy-volume coupled Ising-like model to calculate the Grüneisen parameter for the liquid-liquid transition in supercooled water12. We find that the behavior of the Grüneisen parameter is enhanced near pressure and temperature values that display anomalous behavior and thus supports the presence of a liquid-liquid critical point governed by two distinct energy scales. Yet, such proposal is corroborated by the singular behavior of the isothermal compressibility, sound velocity and pseudo-Grüneisen parameter in the vicinity of the liquid-liquid critical point. Since the first submission of this manuscript, the compressible cell Ising-like model employed here has been used to describe the two-critical-point scenario34. In addition to exploring the critical behavior of water and its other phases, our model can also be applied to other systems by adjusting its parameters. The application of the model to describe the nematic phase in the low-doping regime of Fe-based superconductors revealed that the low-doping regime is well-described by choosing values near the upper boundary values of the volume of each cell, namely, \(v\approx {v}_{0}+\delta v\). The latter corresponds to a lower-density configuration, in agreement with the theoretical description of the nematic phase for Fe-based superconductors32. Our analysis of the Grüneisen parameter \({{\rm{\Gamma }}}_{s}\) and pseudo-Grüneisen parameter \({{\rm{\Gamma }}}_{w}\) can be applied to investigate the critical behavior in any two-state system. One needs only to adjust properly the critical parameters according with the system of interest.