The wiki tells me that if you heat carbon at atmospheric pressure it eventually turns directly into a gas without being liquid first. At what pressure can you make liquid carbon? Has anyone actually ever studied liquid carbon? Surprisingly a web search did not answer these questions.
The carbon phase diagram shows that liquid carbon is only achievable at high pressure (~100 atm) and high temperature (~4500 K). This means that liquid carbon probably has very limited applications, and not many will be researching it. Funding is hard to come by when the applications of the research are not apparent.
“Surprisingly a web search did not answer these questions” — Well, a web search for liquid carbon turns a few results:
low-density liquid carbon contains predominantly twofold-coordinated chain structures (sp hybridization). As the density increased to that of solid forms, bond hybridization increased and threefold-coordinated (graphite-like) and fourfold-coordinated (diamond-like) bonds become more prevalent (indicating sp2 and sp3 hybridization, respectively). These observations are consistent with molecular dynamics calculations that rely on a tight-binding model of interatomic bonding. The fits also suggest that the bond length between carbon atoms in the liquid is significantly shorter than those in the solid, an observation also consistent with simulations.
An earlier study (1999) suggest two different liquid phases
So it's still a pretty open research question…
I have science-fiction related reasons for being very interested in liquid carbon (tidally heated carbon-rich planemo), so I've looked into it a bit. Liquid carbon exists at high pressures and very high temperatures. I think the main problem is the temperatures, since barely anything is solid under conditions where carbon is liquid, so there isn't much of anything that you can hold liquid carbon in for long periods of time to study it. This is particularly difficult for high pressures, since you would need the pressure cell to made of a material that is both strong enough to hold the high pressures in, and refractory enough to withstand the high temperatures. Usually diamond is what is used for this sort of thing (in diamond anvil cells), but that obviously is going to melt under conditions where carbon is liquid (and it should often transform into much weaker graphite before that happens). Thus, the study of liquid carbon is mostly restricted to methods that create it for only a very short amount of time — e.g., shock compression and rapid laser-heating — and theoretical models and simulations.
This paper: (THE CARBON PHASE DIAGRAM NEAR THE SOLID-LIQUID-VAPOR TRIPLE POINT, by I. I. Klimovskii and V. V. Markovets) highlights the problem with using methods like shock-compression and flash-heating that only create high-pressure high-temperature conditions for very short periods of time: Sometimes phase transitions take time — and there may be metastable states that are created in such experiments but can only last for a few seconds, so extrapolating data from such experiments to conditions where carbon might be under such conditions for long periods of time (as planetary scientists might) is not always valid. In particular, that paper presents the view that the phase diagram of carbon looks different depending on how fast you heat the carbon.
This problem doesn't just apply to experimental studies either. One very common way physicists study extreme pressure and temperature conditions (conditions that are not yet possible or feasible to study experimentally, and where simple extrapolations from lower temperatures and pressures will likely not be accurate) is to use "ab initio molecular dynamics simulations". That means that they use super computers to actually model atomic nuclei and electrons in the material using very reasonable simplifying approximation of the underlying quantum physics. (These studies are often associated with the words "density functional theory" (DFT), because that's a way of modelling electrons in complex systems like this.) Even with these simplifying approximations, it can only study extremely small amounts of matter for extremely short amounts of time, so it is actually generally much worse than things like shock compression in that way. That's not the only theoretical way people study extreme conditions, though.
That being said, some of the people who are interested in liquid carbon may actually be more interested in states that exist for short-periods of time than ones that exist for long periods of time. For example, that image that buckminst gave is from a CERN study that seems to have been done by the people making the Large Hadron Collider about what might happen to graphite target when high-energy-particle beams from the collider are dumped into it. (On Graphite Transformations at High Temperature and Pressure Induced by Absorption of the LHC Beam, by Jan M. Zazula) A lot of data about extremely high pressures and temperatures exists because of people trying to model explosions and high-velocity impacts, and one of the equations of state for carbon I found is from Sandia National Labs, which is very closely associated with the U.S. nuclear weapons program. (Multicomponent-Multiphase Equation of State for Carbon, by Gerald I. Kerley and Lalit Chhabildas) Another EOS I found for carbon specifically mentioned research into nuclear fusion as a reason for the study, though 2 of the authors were in planetary science and astronomy departments, so they were probably also thinking about carbon-rich astronomical objects (like carbon planets and maybe even white dwarfs). (A multiphase equation of state for carbon addressing high pressures and temperatures, L. X. Benedict et al.), mostly if not entirely based on DFT ab initio molecular dynamics) It's also worth noting that short simulation times and experimental times are probably more trustworthy for studying fluids than for studying solids, as far as the problem of metastable states is concerned, since fluids are already in a constant state of flux in terms of the positions of molecules to each each other, so you can't have metastable crystal structures like with solids. You can have fluids of metastable chemical compositions (you encounter them all the time), but, at very high temperatures, the molecules are likely getting created and destroyed fast enough that that isn't much of a problem either.
Speaking of nuclear things, graphite is commonly used in the cores of nuclear reactors due to its high temperature stability (at least in the absence of oxygen, with which it burns), so most papers about the properties of graphite at high temperature are related to nuclear reactors, but those rarely discuss liquid carbon from what I've seen.
Other papers I've found that discuss liquid carbon include: Comparative study of melting of graphite and graphene, by Yu. D. Fomin and V. Brazhkin, which seems to spend most of the paper comparing different theoretical models to try to determine which ones are most accurate, with some comparisons to the somewhat confused experimental data; and Ultra-High Pressure Dynamic Compression of Geological Materials, by Thomas S. Duffy and Raymond F. Smith, which is clearly a planetary science paper, and briefly covers carbon on page 10, with a phase diagram, and also goes into a lot of detail about experimental methods in the first half of the paper.
It might also be mentioned in some other papers, and, though it isn't about liquid carbon in the strictest since, this paper does make a somewhat dubious attempt to model convective cooling (implying liquid like behavior over geological time frames) in a graphite planetary crust subjected to high heat fluxes from the planetary interior: Thermal evolution of rocky exoplanets with a graphite outer shell, by K. Hakim, et al.
I believe all these links are to free access papers, so you can skim through all of them to see if there's something in them that interests you.
So what are some things we know or suspect about liquid carbon? If you look at the phase diagram, you can see that, in the lower-pressure part of the liquid range, you can see that increasing the pressure increases the melting point. This is because the solid is denser than the liquid, and increasing the pressure tends to make things denser, so it freezes the liquid at higher and higher temperatures. At high enough pressures, though, the melting curve bends around the other way, and increasing the pressure actually LOWERS the melting point. Just like with water, this indicates that, at high enough pressures, liquid carbon is actually DENSER, than graphite. This melting point maximum (which that phase diagram puts around 0.2~0.3 GPa, although experimental sources cited in the Sandia National Labs paper, put it much higher, at around 5~6 GPa, and I trust those more) is the first indication of some kind of change in the nature of the liquid as pressure increases (a "liquid-liquid phase transition"). The "comparitive study of melting" paper indicates that this transition is gradual, not sudden, and I think the Sandia National Labs paper partially explains it in terms of the molecular structure of the liquid, with polyatomic molecules, like C2, C3, C4, etc, and, apparently, especially C3, existing at lower pressures (and temperatures), but mostly monatomic carbon atoms, not bonded into molecules, at higher pressures (and temperatures). Like all substances made of atoms, the liquid becomes progressively more and more metallic as the pressure increases (and as the temperature increases, as long as it stays at liquid/solid like densities because a "metallic gas" is really a plasma) because this makes the electron shells of the atoms tend to overlap more and more.
After the stable solid allotrope becomes diamond, the melting point increases with temperature, because diamond, which is much denser than graphite, is denser than the liquid at those pressure. However, if you look at the "L. X. Benedict et al." and "Ultra-High Pressure Dynamic..." papers, you'll see phase diagrams that go to much higher pressures, and see that there is also, at least according to predictions, a point where the melting point of diamond also drops with pressure, again almost certainly indicating a liquid state denser than the solid state, until even higher-pressure solid polymorphs appear, and according to both papers, though the don't completely agree with each other, the cycle repeats at least one more time.
One minor not about these extremely high pressure and high temperature phase diagrams is that some of the "liquid" might technically not be liquid, but really supercritical fluid, and arguably supercritical plasma. (The abstract of a 1973 paper behind a pay wall here: (Thermodynamic properties of carbon up to the critical point, H. R. Leider et al.) puts the critical point at 6810K and 2200 atm, which I notice is in the range covered by the CERN phase diagram buckminst posted, but is not marked on it.) That's mostly a technicality at temperatures anywhere near the melting point, but it does mean that, at any pressure above 0.22 GPa (or whatever the actual critical pressure is if they're wrong), the "liquid" has no boiling point, but will just become less and less dense as the temperature increases.
The other thing you'll notice on that phase diagram is the "metastable liquid phase", near the graphite-vapor-liquid triple point. I don't know exactly what that's about, but it is probably what the first paper I linked to (about the phase diagram near the triple point) was about. That paper mentioned a phase transition from graphite to "carbyne", taking about 2-3 seconds, and that "carbyne" (I think solid "carbyne", though maybe its a liquid phase) exists between 2600 K and 3800 K. It's extremely unclear to me what they mean by "carbyne", and it doesn't look like they ever properly define it. (You can look at some the different definitions Wikipedia lists for this word yourself, though it would probably be more informative to go through the sources that paper cites.) My best guess is that it refers to the solid state of those polyatomic carbon molecules like C3 and C7 that the Sandia paper and the paper behind the paywall both mention, or maybe it's liquid C3. Alternatively, maybe it's "linear acetylinic carbon" (...-C≡C−C≡C−C≡C−...), though it seems odd to me that that would be stable at high temperatures, or even just a bunch of carbon atoms not bonded to each other (which sounds really odd and unlikely to me, although I do know that there is such a thing as glassy carbon, and some other weird metastable states that I've seen mentioned here: Heat-treated glassy carbon under pressure exhibiting superior hardness, strength and elasticity, M. Hu, et al., but that paper only covers temperatures far below the melting point of graphite).