Prediction of thermally driven quasi-1D superionic states in carbon hydride under giant planetary conditions

Domain: biology
Published: 2026-03-16
Editorial outcome: major_revision

Abstract
Planetary interiors experience high-pressure–high temperature conditions that give rise to unconventional states of matter, reshaping our understanding of planetary dynamics and the generation of magnetic fields. Here, using first-principles computational simulations in combination with machine-learning interatomic potentials, we predict a distinct atomic state, termed a quasi-1D superionic phase, that emerges in a stable carbohydride (CH) compound under giant planetary interior conditions. This phase originates from temperature-induced transformations and features a chiral carbon framework intertwined with dynamic hydrogen helices. At 0 K, electronic redistribution along the hydrogen sublattice induces metallization. In contrast, upon heating, carbon atoms form a rigid lattice, and hydrogen exhibits rotational motion in the xy-plane and diffusion along the z-axis, resulting in anisotropic mobility. A high-pressure–temperature phase diagram reveals sequential transitions from solid to quasi-1D superionic, 3D superionic, and fluid states. The quasi-1D superionic CH phase exhibits pronounced anisotropy in electronic, thermal, and ionic conductivity, with electronic transport predominating and the ionic contribution remaining negligible. This anisotropic behavior provides a microscopic mechanism for directional energy and charge transport under high-pressure and high-temperature conditions, offering insight into how structural anisotropy can govern transport properties in materials subjected to ultra-high pressures. This anisotropic behavior provides a microscopic mechanism for directional energy and charge transport under extreme conditions, offering new insights into the behavior of high-pressure materials and magnetic phenomena in giant and sub-Neptune exoplanets.

Introduction
Density profiles of Uranus and Neptune suggest that the ice giants contain a massive intermediate layer of hot ices, commonly approximated by the idealized “Synthetic Uranus” composition derived from solar atomic abundances [1–3]. In this idealized model, the layer is composed primarily of water, methane, and ammonia, and is sandwiched between a rocky core and a gaseous hydrogen–helium envelope under high pressures and high temperatures [4–6]. These icy constituents do not exist as conventional ices but instead as dense exotic phases in their interiors, reflecting the complex interplay of thermodynamics and physics under planetary conditions [7–22].

In many compressed molecular systems containing two or more elements, the atoms can exhibit two or more of the following dynamical behaviors: solid, plastic, superionic, and fully fluid [23–29]. These regimes are not universally realized across all systems (e.g. chain-melted phase [30, 31]), and their occurrence depends sensitively on bonding characteristics, atomic mass contrast, and thermodynamic conditions. Each regime displays a characteristic diffusion pattern of its constituent species.

To describe these regimes consistently, we introduce a simplified measure of diffusional dimensionality (DF), representing the number of spatial directions along which atoms exhibit long-range diffusion, rather than formal thermodynamic degrees of freedom. In solid phases, atoms are tightly bound and move only within vibrational modes, corresponding to DF = 0. Even in solids, limited diffusion can occur through vacancy-mediated atomic jumps between neighboring sites; however, over long timescales, atoms remain on average close to their lattice positions, thereby preserving long-range order. Plastic phases allow rotational or orientational motion while retaining translational order, which can be described on a spherical surface where the radial distance r is fixed and the angular coordinates (θ,ϕ) are free, giving DF = 2. In contrast, superionic phases are represented in Cartesian coordinates, where one sublattice becomes diffusive along all three translational directions (x,y,z) while the other remains crystalline (DF = 3 for a binary system). Fully fluid phases exhibit unrestricted motion of all atoms in three dimensions, corresponding to DF = 6. Upon heating, the binary system may sequentially transition from a solid to a plastic, superionic, and finally a fluid state (DF = 0 → 2 → 3 → 6, as shown in Fig. 1). Similar temperature-driven transitions occur in confined or two-dimensional systems, where diffusion evolves with dimensional constraints [32]. The DF scheme serves as a conceptual framework for visualizing atomic mobility rather than a strict count of microscopic degrees of freedom. Yet it neglects anisotropic motion and the coupling between translational and orientational dynamics, which strongly affect transport under high-pressure and high-temperature conditions. Recent studies show that in confined or channel-like structures, such anisotropy can lead to one-dimensional superionic conduction [33], emphasizing the role of directional transport in materials relevant to thermoelectrics, energy harvesting, and anisotropic quantum devices [34–36]. Moreover, under the high pressure–high temperature conditions typical of ice giant interiors, anisotropic conduction may play a critical role in shaping planetary magnetic fields [1, 10, 37].

Recent studies [38–41] demonstrate that helium migration within molecular frameworks (e.g., H2O, NH3) is finely modulated by the interplay between weak van der Waals interactions and strong covalent bonding, which raises critical questions about analogous hydrogen diffusion mechanisms under high-pressure and high-temperature conditions akin to planetary interiors. Of particular interest is methane (CH4), which destabilizes above 95 GPa, dissociating into hydrogen-rich phases and carbon allotropes such as diamond [42, 43]. Understanding the stability and diffusive behavior of C–H compounds in the high-pressure environments of ice giants, such as Uranus and Neptune, is crucial for unraveling their internal dynamics and composition. In this work, we identify a helical CH compound and examine its structural stability, static and dynamic properties, and anisotropic transport behavior under high-pressure and high-temperature conditions.

Results
We have carried out crystal structure prediction to determine the most stable phase of various compositions in the binary C-H system at 0 K. A phase is stable at its static ground state if its enthalpy of formation from either elements or any other possible compounds is negative. We identified two compounds, CH and CH2, whose static formation enthalpies (∆H) become lower than those of the elemental C–H mixture above 1000 GPa (Fig. 2a). We identified numerous candidates for the ground states of CH and CH2 (Fig. S1). Including zero-point energy (ZPE) eliminates CH2 from the convex hull, which adopts the same structure as previously reported [44–46] below 500 GPa, where it is also metastable. The CH compound is energetically stable above 1100 GPa (Fig. 2b).

While BC8 is predicted to be the lowest-enthalpy carbon phase above 1000 GPa by DFT calculations, diamond has been observed to remain stable well beyond the pressure range where BC8 is theoretically predicted to become favorable in experiments [47, 50, 51]. In light of this experimental persistence, we carried out an auxiliary comparison of formation enthalpies using diamond as an alternative carbon reference (Fig. S2). This comparison is provided for reference only and does not alter the thermodynamic convex hull, which is defined using the DFT-predicted lowest-enthalpy phases. Under this alternative reference, the relative formation enthalpy of CH is reduced, whereas CH2 remains metastable. We calculated free energy profiles, including vibrational contributions, to assess the temperature effect on CH stability (Fig. S3). Both CH and CH2 show slightly reduced stability with increasing temperature, while CH remains thermodynamically stable at 2000 GPa and 1000 K.

The CH compound exhibits a striking feature of crystallographic chirality, forming distinct right- and left-handed enantiomorphic variants (space group: P6122 and P6522). These mirror-image structures arise from the C/H helix. In the presentation of the crystal structure of P6122 phase CH (Fig. 2d,e), H-atoms are located in helical chains along the c-axis surrounded by a framework of C-atoms connected by sp3 bonds. Such a chiral framework structure was confirmed to be metastable in pure carbon at lower pressure [52]. Although the CH phase crystallizes in the P6122 space group—like AuF3 and AgF3, which share similar mass ratios—its bonding characteristics are fundamentally distinct despite the common topological framework (Fig. S4). The electronic localization function (ELF) indicates the absence of C–H bonding, with helical H chains forming quasi-one-dimensional metallic channels, unlike the electron distributions in molecular hydrogen phases [48, 49].

In the CH compound, the carbon and hydrogen atoms form cylindrical helices traced on a vertical cylinder of revolution, each maintaining a constant angle with respect to the cylinder’s axis (Fig. 2c). With compression, the radius (r) and reduced shift (h) of the C/H helix are decreasing. While the helix angles (θ), which are the angles between any helix and an axial line, are kept constant at 21◦ and 38◦ for the C/H helix, respectively. According to Hirshfeld charge analysis (Table S3), a substantial charge transfer of approximately 0.42 e− from hydrogen to carbon is observed. Crystal orbital Hamilton population (COHP) analysis (Fig. S5) reveals strong covalent bonding interactions for C–C and H–H pairs, while no significant bonding interaction is found between C and H atoms, consistent with the ELF analysis. These results indicate that the CH compound consists of two independent, covalently bonded helical sublattices of carbon and hydrogen.

Previous studies have shown that H–H interactions are remarkably weak under such conditions [53], and hydrogen-bearing compounds therefore exhibit highly anharmonic, nearly quadratic energy landscapes that facilitate large-amplitude thermal fluctuations. To investigate the dynamical behavior, we carried out first-principles molecular dynamics (FPMD) simulations within density functional theory (DFT) for a pressure range from 500 to 3000 GPa and a temperature range from 500 to 7000 K. We find a different atomic behavior by examining the mean square displacement (MSD) in these CH compounds at high temperatures (Fig. 3). With increasing temperatures, we found the following changes in behavior: At lower temperatures, the system is solid, where all atoms oscillate around their equilibrium positions, and the MSD is small with a zero slope, so that self-diffusivity D = 0. As the temperature increases beyond the melting point, the system transitions into a fluid phase where all atoms move freely, and the MSD increases with a constant slope proportional to self-diffusivity D > 0. In Fig. 3, we illustrated two different states of the CH compound in the middle temperature range. Both overall MSDs at 2000 K (Fig. 3 e) and 5000 K (Fig. 3 g) exhibit similar atomic mobile behaviors of C and H (D = 0, D > 0). Only the self-diffusivity of H increases significantly, induced by a temperature increase from 2000 K to 5000 K. Still, when we look into the trajectories at these two temperatures, H can diffuse more freely in the whole space of the C framework at 5000 K (Fig. 3 c-d), which is the so-called 3D superionic state. While at 2000 K, H is constrained in the center of the C helix chain and moves along the H helix chain (Fig. 3 a-b). In addition, the activation energies for hydrogen diffusion, extracted from the temperature dependence of diffusivity (Fig. S6), are 0.139 eV from 1000 K to 3000 K and 1.111 eV from 4000 K to 5000 K. This huge barrier energy gap between two states demonstrates the transition at 3000 K, providing mechanistic insight into hydrogen mobility and the phase transition behavior.

To analyze the difference between these two types of diffusive behaviors, we projected MSD onto different directions (x, y, z). In the 3D superionic state at 5000 K (Fig. 3h), Hx, Hy, and Hz all show a positive slope of MSD, indicating that H can diffuse freely in all directions. Hydrogen diffuses faster (∼7 times higher) along the z direction than along x and y. At 2000 K (Fig. 3f), hydrogen diffuses freely along z, while Hx and Hy show a plateau in MSD around 0.34 Å2, consistent with R2 = 0.3364 Å2, indicating a two-dimensional plastic phase where motion occurs along circular trajectories.

To elucidate this unique diffusion behavior, we have developed a mean-square-displacement (MSD) model (Eq. 7) that captures this direction-dependent motion. This model clarifies the distinction between the observed behavior and that of a conventional one-dimensional (1D) superionic phase. In a typical 1D superionic phase, hydrogen diffusion is confined to a single direction (taken here as the z-axis), where Dx = Dy = 0 and Dz > 0, yielding a diffusional dimensionality of DF = 1. In contrast, in our identified state, hydrogen exhibits not only significant diffusion along z (Dz > 0) but also substantial angular diffusivity of 5.50 s−1, resulting in an effective diffusional dimensionality of DF = 2. The key signature of this motion behaviour appears in the MSD: both the x and y components saturate at plateau values of approximately 0.34 Å2 each (summing to 0.68 Å2, which equals 2r2, Fig. 2c). This plateau behavior is characteristic of plastic crystal phases, similar to those simulations in high-pressure ammonia [39]. Although solid phases also display MSD plateaus, their amplitudes are typically much smaller than 2r2.

Thus, this state can be characterized by a combination of superionic behaviors in the z-direction and plastic behaviors in the xy-plane of H atom motion, ultimately inducing helix H chain rotation along the z-axis within the rigid C framework. Here, we refer to this state as a quasi-1D superionic state. This state corresponds to a form of confined superionicity, in which hydrogen diffusion is restricted by the geometry of the underlying scaffold—in our case, the helical carbon framework that guides hydrogen motion along helical pathways. Experiments show that most CxHy plastics dissociate into diamond and hydrogen under megabar pressures [54, 55], but such studies are limited to below 500 GPa, and no shock experiments have yet probed the TPa regime. To assess finite-size effects in DFT-MD, we constructed a large 3×3×3 supercell (324 atoms) with one C and one H vacancy and performed 12 ps simulations from 2000 to 6000 K (Fig. S7). The CH compound remains quasi-1D superionic up to 2000 K, with hydrogen diffusing through the vacancy while the carbon framework remains intact. At 4000 K, the carbon lattice remains solid but more vibrational, and hydrogen escape occurs only near 5000 K, forming a 3D superionic phase. Furthermore, we trained a machine-learning potential (MLP) for the CH compound and performed 1-ns MD simulations in a 1500-atom supercell. These long-time, large-cell MLP-MD simulations demonstrate that both the quasi-1D superionic and 3D superionic states remain intrinsically stable, even in the presence of a defect, at 1000 GPa (Fig. S10).

The quasi-1D superionic CH phase shows a close analogy to chain-melted phases [30], as both feature anisotropic diffusion constrained to quasi-one-dimensional channels within a crystalline framework. In chain-melted phases, guest atoms diffuse within the channels of a zeolite-like host framework, exhibiting a one-dimensional superionic behavior that arises from the incommensurate relationship between the guest and host lattices in pure elements. In contrast, in the quasi-1D superionic CH phase, hydrogen atoms form covalent H–H bonds, creating continuous hydrogen helices embedded within a rigid carbon lattice. Thermal excitation enables hydrogen to migrate along these helices. Importantly, whereas chain-melted phases are typically incommensurate host–guest structures, the quasi-1D superionic CH phase is fully commensurate and crystallographically chiral. Analogous to the anisotropic diffusive behaviour observed in the quasi-1D superionic state, where mobility varies distinctly along x, y, and z directions, the quasi-1D superionic CH compound also exhibits strong anisotropy of the electronic transport properties (e.g., electronic conductivity and thermal conductivity, Fig. 5) with temperature effect. With increasing temperature, the electronic conductivity decreases, while the thermal conductivity increases, and we do not observe any signature of resistivity saturation, as seen in iron or its alloys [59]. In the solid phase, the values of electronic transport properties are close to each other, indicating the CH compound is isotropic. When the CH compound transfers into the quasi-1D superionic state, the electronic conductivity remains the same in the x and y directions but is much higher in the z direction. A similar case also occurred in the thermal conductivity evolution, which has a higher value in the z direction. As the temperature increases, H begins to diffuse freely in the x and y directions, significantly affecting the slopes of their electronic and thermal conductivities. We observe clear differences between the z and xy directions in both electronic and thermal conductivities within the 3D superionic state. This anisotropy arises from the direction-dependent self-diffusivity of hydrogen (Fig. 3 h), as the strongly anisotropic carbon framework constrains hydrogen motion in the xy-plane while allowing greater mobility along the z-axis. Finally, in the fluid phase, both the electronic and thermal conductivities become isotropic again. We also performed additional calculations of the electrical transport properties for the quasi-1D superionic CH compound at pressures ranging from 2000 to 3000 GPa and temperatures of 2000 K (Fig. S12). The results confirm that, upon compression, both the electrical and thermal conductivities increase. A persistent anisotropy is observed, with the z-axis consistently exhibiting higher electrical and thermal conductivities than the xy-plane.

In addition to electronic conductivity, the thermally driven mobility of hydrogen also raises interest in its ionic conductivity, which plays an essential role in the ice layers of Uranus and Neptune [10]. Using the diffusivities of C and H, we derived the corresponding ionic conductivity of the CH compound at 1000 GPa via the Nernst–Einstein relation (Fig. S13). Notably, the ratio σel/σion decreases with increasing temperature. In particular, when CH melts at 6000 K, this ratio drops sharply, although electronic conduction remains dominant—about 70 times higher than the ionic contribution.

While the interiors of icy giant planets contain complex mixtures of C, H, N, and O rather than pure CH compounds, our results provide key insights into the intrinsic behavior of C–H systems under high-pressure and high-temperature conditions. Crystal structure predictions (Fig. S14) at 1000 GPa reveal a thermodynamically stable CN2H2 phase—consistent with recent theoretical findings [60]—whereas no stable C–O–H compounds were identified. These results suggest that nitrogen incorporation stabilizes additional stoichiometries, whereas oxygen does not play a comparable role at these pressures. Although focused on the binary C–H system as a simplified model, our study captures the essential structural and transport characteristics of C–H networks that may persist in the interiors of icy giant planets, offering a prototype for understanding charge transport and magnetic-field generation in more compositionally complex planetary ices.

The temperature-driven transitions within the quasi-1D superionic CH phase result in pronounced anisotropy in electrical and thermal conductivity, indicating a direct connection between atomic-scale structure and directional transport properties. While conventional dynamo models typically assume isotropic conductivity [61], the anisotropic charge and heat transport observed in the quasi-1D superionic phase illustrate how microscopic structural asymmetry can give rise to directional bulk transport properties under high-pressure and high-temperature conditions. More fundamentally, this work identifies a previously unrecognized transition mechanism—the quasi–one-dimensional superionic state—that provides a conceptual bridge between plastic solids and fully developed superionic matter under strong confinement. Helical confinement stabilizes an intermediate regime in which long-range structural order coexists with substantial ionic mobility, allowing superionic behavior to emerge through a topologically constrained pathway rather than isotropic sublattice melting. As a result, bonding, diffusion, and transport become strongly anisotropic and partially decoupled. Although such pressures are unlikely to occur within Uranus or Neptune, more massive exoplanets such as sub-Neptunes may approach the regime in which this phase could stabilize. We do not claim any direct planetary or magnetic modeling relevance; instead, these findings suggest a plausible microscopic mechanism for anisotropic transport in dense molecular systems and broaden the conceptual landscape for understanding material behavior under compression.

In conclusion, we have explored the C–H system under terapascal pressures and identified a ground-state CH compound with a unique helical crystal structure exhibiting both left- and right-handed chirality. Ab initio molecular dynamics simulations reveal a previously unreported quasi-1D superionic state of condensed matter—distinct from known solid, plastic, superionic, and liquid phases—in which hydrogen atoms diffuse freely along the helical axis while rotating within the transverse plane. This coupling between translational and orientational motion establishes a microscopic framework connecting superionic and plastic behaviors. The discovery of this state provides insights into the structural and transport complexity of light-element systems at high-pressure and high-temperature conditions. It suggests possible implications for anisotropic conduction in both materials science and planetary environments.

Methods
Crystal structure prediction analysis. The machine learning and graph theory assisted universal structure searcher (MAGUS) package [62–64] was used to perform crystal structure searches for C-H binary system at 0 K. We conducted extensive variable-composition searches at multiple pressures (500, 1000, 2000, and 3000 GPa), as well as fixed-stoichiometry searches for selected compositions—including pure carbon, pure hydrogen, and various C:H ratios (6:1, 4:1, 3:1, 2:1, 1:1, 2:3, 1:2, 1:3, 1:4, 1:6, etc.)—at these pressures (though not all combinations were sampled exhaustively). The full convex hull of the C-H binary system at 1000-3000 GPa is shown in Fig. S15, and the structural information is given in Section S13 of the Supplementary Information.

First-principles calculations outline. Density functional theory (DFT) calculations were performed with the Vienna ab initio simulation package (VASP) [65]. The projector augmented-wave (PAW) [66] method was employed with the generalized gradient approximation in the Perdew-Burke-Ernzerhof (PBE) [67] exchange correlation functional. The 2s22p2 and 1s1 electrons in the C and H atoms were treated as valence electrons with smaller core radii in the hard version of pseudopotentials. A plane wave energy cutoff of 1500 eV was employed along with dense Monkhorst–Pack k-point sampling grids of resolution 2π×0.025 Å. The energy and atomic forces in the structural relaxations were converged to within 10−6 eV and 0.002 eV/Å, respectively. Details of the cutoff energy and k-mesh convergence tests, and the treatment of carbon 1s orbital ionization, are given in Section S14 of the Supplementary Information.

For cross-checking the stability of the CH compound at high pressure, structural optimizations are also carried out by the CASTEP [68] code and the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) density functional [67]. The ultrasoft pseudopotentials were generated with a small core radius of 1.0 and 0.6 Bohr for C and H, respectively, using the on-the-fly (OTF) method to avoid overlapping the core radii. A full potential linearized augmented plane wave (FP-LAPW) method [69] was also carried out in the ELK code. The all-electron calculations with the Gaussian and Augmented Plane Waves (GAPW) scheme were performed in CP2K code [70]. To account for the effect of different DFT functionals, we used the plain PBE, LDA, optB88-vdW [71], and Revised PBE (RPBE) [72] to cross-check the quality of the formation enthalpy calculations. Tests of formation energy using various computational methods (OTF, SLAPW, GAPW) and DFT functionals are provided in Section S15 of the Supplementary Information. Hirshfeld charge transfer was calculated using the CP2K code [70], and chemical bonding was further analyzed by performing crystal orbital Hamilton population (COHP) calculations with the LOBSTER code [73].

Ab initio molecular dynamics simulation. AIMD simulations based on the Born-Oppenheimer approximation implemented in VASP were performed in 2×2×4 supercells for the right-hand chiral CH compound (with 192 atoms) with Γ-centered k point in the Perdew-Burke-Ernzerhof exchange correlation functional. A hard version pseudopotential and a cutoff energy of 1500 eV were adopted to ensure energy convergence of better than 10−5 eV. We adopted the canonical NVT ensemble using a Nose-Hoover thermostat [74] with SMASS=2, lasting for 12 ps with a timestep of 0.25 fs. We allowed 2 ps for thermalization and then extracted data from the last 10 ps. A single constant volume is not suitable for exploring such a wide pressure–temperature range. Accordingly, we performed a series of NVT simulations across a range of volumes rather than relying on a fixed-volume approach. The stability temperature range of each phase was determined using the heat-until-it-melts protocol. For each pressure column in the phase diagram, the corresponding volume was fixed based on zero-temperature cell relaxations performed at pressures of 500, 1000, 1500, 2000, 2500, and 3000 GPa. More details about the test of AIMD simulations can be found in Section S16 of the Supplementary Information. For comparison with the NVT ensemble results, we conducted NPT simulations of the CH compound across a temperature grid at 1000, 2000, and 3000 GPa using a Langevin thermostat [75]. The resulting phase diagram closely matches that derived from the NVT simulations (Fig. S26).

Mean Square Displacement Diffusion coefficients (D) can be derived from the Mean Square Displacement (MSD) for each species to monitor different types of atomic motions. To calculate the MSD of a point moving freely on a circle, we describe its motion and compute the displacement over time. Assume the motion is a uniform random walk (or diffusion process). The position of the point on the circle can be described using an angle θ(t), which is measured with respect to a fixed reference point. Let the radius of the circle be R. The Cartesian coordinates of the point are:

x(t)=Rcosθ(t), y(t)=Rsinθ(t). (1)

If the point diffuses on the circle, the angular position evolves as:

θ(t)=θ(0)+ω(t), (2)

where ω(t) represents the stochastic angular displacement. For diffusion, the angular displacement satisfies:

⟨(∆θ)2⟩=2Dθ t, (3)

Where Dθ is the angular diffusion coefficient. The MSD describes the mean-squared displacement between the initial and final positions:

MSD(t)=⟨|r(t)−r(0)|2⟩. (4)

Since the particle remains on a circle of radius R, the displacement depends on the angular difference:

|r(t)−r(0)|2 =2R2[1−cos(∆θ)], (5)

where ∆θ = θ(t)−θ(0). For a random walk or Brownian motion, we have:

⟨cos(∆θ)⟩=e−Dθt. (6)

Thus, the MSD becomes:

MSD(t)=2R2[1−e−Dθt]. (7)

As t→0, MSD(t)→0 (the particle has not moved). As t → ∞, MSD(t) → 2R2 (the particle’s angular position becomes uncorrelated, effectively exploring the entire circle). This is the analytical result for the MSD of a particle undergoing free diffusion on a circle. The outcome depends on the circle’s radius R and the angular diffusion coefficient Dθ.

Transport property calculations We used a modified Spin-polarized Relativistic Korringa-Kohn-Rostoker (SPR-KKR) package [76, 77] for the transport calculations for the CH compound, randomly sampling four snapshots at high P and high T from AIMD simulations. The calculations commenced with a self-consistent-field (SCF) calculation using density functional theory (DFT) for each configuration. We used a maximum angular momentum lmax of 3 (nl = 4) with 125 k-points. Subsequently, transport calculations were performed using the converged potentials obtained from the SCF step, employing the Kubo-Greenwood method with a maximum angular momentum lmax of 3 on 216 k-points and energies [-5 kBT, 5 kBT] around the chemical potential. The values for electrical resistivity (ρ, equal to the inverse of electrical conductivity 1/σ) and thermal conductivity (κ) are expressed as follows:

σ = L11 (8)

κ = 1/eT (L22 - L12^2/L11) (9)

and L is given by:

Lij = (−1)(i+j) ∫ dε σ(ε−µ)(i+j−2) (−∂f(ε)/∂ε) (10)

At a given temperature T, f(ε) represents the Fermi-Dirac distribution function. The Kubo expression gives the energy-dependent conductivity:

σ ∝ Tr ˆjµ ImG+ ˆjν ImG+ (11)

where ˆj is the current density operator, and ImG+ is the imaginary part of retarded Green function.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Please find my review attached.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "The manuscript “Thermally driven superhelical state in CH compound under planetary interior conditions” written by Liu, Cohen and Sun, proposes a computational study of a system having stoichiometry CH at high pressure and high temperature conditions. This work proposes the new CH material as relevant for planetary science, particularly to understanding the properties of ice giant planets. After reading the manuscript, I could find both positive and negative points relative to the methodology and results.\nI think that the authors have predicted an interesting system that could potentially manifest 1-D superionicity, and it is intriguing that a helix structure of carbon is stable above 1 TPa of pressure. However, there are two fundamental aspects of this work that made me concerned and that I will further develop with my comments: 1) The superhelical state seems to be just 1-D superionicity given by the confinement of the carbon lattice; and 2) the theoretical calculations appear to have significant limitations that should be addressed.\nComputational Methodology:\n- The authors do not provide any structural information about their system, apart from the two possible space groups. This made it difficult to understand the goodness of some of the parameters the authors chose. A structure should, even if theoretical, should always be provided to guarantee and help reproducibility.\n- The authors study a system starting from very high pressure, from 1 TPa to 3 TPa. At such pressures it is not possible to trust the core potentials, even if modified as the authors do in some cases. The 1s orbital of carbon will affect the intraatomic 2s and the interactions with the other carbon atoms. Therefore, all electrons calculations should be performed.\n- The k-mesh used by the authors seems very coarse. A 2π × 0.025 is surely not enough to calculate correctly the electronic structure of a metallic system and such extreme conditions of pressure. This is clear from Figure S2b and extended Figure 1. The DOSs are too spiky, which is symptomatic of an incorrect choice of the k-mesh. The authors should show also the convergence test of the k-mesh together with the cut off, for both VASP and QE.\n-The atomic radii have not been rescaled for the calculation of the pDOSs. Using the standard radii does not allow for a correct calculation and projection of the DOSs.\n- Performing NVT simulations in the range of pressure 1-3 TPa and of temperature 0-7000 K raises concerns about the validity of the approximation. Assuming that a constant volume in this case is too big of an approximation, and NPT simulations should be performed instead.\n- Finally, the crystal structure prediction is extremely unclear. The authors wrote that they used C and H as reference to calculate the enthalpies. Which phases did they use? Did they perform CSPs for C and H to check whether new phases become more stable than diamond and atomic H (I guess that these were the phases they used)? How does their prediction compare with phases like the CH2 (polyethylene) predicted in 10.1073/pnas.2026360118? Moreover, the authors should check CH4, even if it has been observed to decompose at lower pressure, but I believe that it has never been investigated above 1 TPa.\nGeneral Comments:\n- The double-helix definition is here incorrect. Contrary to structures like dna, where two carbon chains are interconnected by the ATGC bases, here the proposed system is just one row of carbon encapsulating a chain of hydrogen. In fact, Figure 2 gives a clear hint of the nature of the system: From the ELFs, it seems that the CH is formed by two independent chains, one of C and one of H, which are not bonded. Therefore, the potential superionicity, might be in reality just the melting of the H chain that is encapsulated in the C helix. As I will suggest below, the authors should calculate the charges of the atoms to better understand the nature of the system.\n- Figure 2ab: It would be helpful to rescale in meV/atom, also to better understand the magnitude of the ZPE correction on the discontinuity of the enthalpy. Moreover, the author should also show the free energy profiles.\n- On their sentence “The electronic localization functon(ELF) shows that the H helical chain forms a quasi one-dimensional metallic channel” I think it is hard to see that from that orientation. Instead, it seems that H form trimers, and not chains. Instead of having the same orientation for the two space groups in Figure 2, they should show two orientations for the same space group. The transferability from P6122 to P6522 would be accepted without showing it (or it can be shown in the SI). This will also help to understand whether the H-chain and the C-chain are bonded or not.\n- “The phonon spectrum shows CH is dynamically stable with no imaginary phonons, indicating that the CH compound can be quenched to 300 GPa (Extended Fig. 1c). The CH compound is a conventional” This conclusion should be re-evaluated once the k-mesh convergence is verified. Moreover, to justify the superconductivity statement, the authors should perform also an NPT simulation up to 160 K, their predicted Tc, to check the stability of the system above 0 K.\n- I question the interpretation that “Thus, the new state can be characterized by a combination of superionic behaviors in the z direction and plastic behaviors in the xy plane of H atomic motion, finally inducing helix H chain rotation along the z-axis in the rigid C framework. Here we named this new state, superhelical.” What would be the difference with a straight 1D tube? or zigzag? the helical shape is just the confinement of the carbon, which allows the diffusion only along one dimension. In fact, it is not plastic along x and y, because the C confinement does not allow the H to escape, while in the plastic behavior, the H stays attached to the heavier atom even if it is free to move. Therefore, the effect might be confined superionicity, which is already an interesting result, but it does not need a new name based on the shape of the scaffold, which is not related to the effect per se. However, the charge of both carbon and hydrogen should be calculated to understand if the system is truly superionic or if it is formed by two intercalated neutral frameworks (C-chain and H-chain), where one of the two (H) melts before the second one.\n-In the SI, Table ?? is missing in page 2. Moreover, Figure S3 is very puzzling. It is very surprising that c and a have the exact same behavior in and hexagonal space group. This is, in fact, possible only for cubic space groups. I believe that this is also caused by the loose k-grid.", "reviewer_4": "In the manuscript entitled « Thermally driven superhelical state in CH compound under planetary interior conditions », the authors report the results of their numerical study of the CH compounds in the multi-megabar regime. They have performed a series of ab initio calculations using DFT to examine possible stable compounds of CH at 0K, their stability based on a phonon spectrum analysis, and their properties at finite temperature. Based on their study, the authors conclude that CH forms a stable double-helix in the multimegabar regime. At low temperature, the compound is stable, but above 1000 K, it becomes “superhelical” with hydrogen diffusing along the channel formed by the carbon chain. At about 4000 K, the system becomes superionic, and ultimately melts at about 6000K. This specific geometry imprints on the transport properties, both ionic and electronic. The authors then consider possible consequences for planetary interiors such as ice giant planets, and their magnetic field.\nThe topic is of interest for multiple communities since it brings novel structure and concept for planetary physicists, but also high pressure scientists, and the warm dense matter community in general. However, the conclusion of superionicity/superlicity is in stark contrast with what experimentalists have observed with the clear formation of diamond and a phase separation of carbon and hydrogen in the megabar regime. It is quite easy to misinterpret superionicity in DFT-MD when it is actually a demixing process that should occur, since the finite-size effects tend to overstabilize the superionic structure. There are also a few shortcomings in the interpretation of the results (see details below). For all these reasons I do not believe this work can be published. I append below detailed comments to justify this recommendation and to help strengthening this work.\nDetailed comments:\n- numerous experimental studies led by Dominik Kraus’ team have shown that almost any CxHy plastics dissociate into diamond and phase separated hydrogen in the megabar regime. The conclusion from the authors regarding the superionicity of the system is thus questionable. Indeed, it is possible that finite-size effects overstabilize the homogeneous system. I would recommend to perform a few additional tests to ensure the reality of this system. The first one would be the most robust but also the most expensive. I would suggest to perform a careful thermodynamic integration to compare the homogeneous superionic system with the dissociated one. If this is too intricate to perform, it would be interesting to use the largest cell and introduce a defect with one carbon and one hydrogen removed to see if the cell remains stable. An alternative would be to start from a liquid simulation and cool it down slowly to see if a superionic phase appears. Without such a study, it is not possible to entirely justify that the superionic phase is stable, in my opinion.\n- the authors insist significantly on possible consequences for planetary interiors. This is not sufficiently justified since the composition is not pure CH in the interior. A significant proportion of oxygen and nitrogen must also be present as well as additional hydrogen. Is the double-helix still stable in this case? Militzer (PNAS 2024) showed that it was possible to form a CNH compound at high pressure. I believe that the authors need to soften their conclusion regarding the impact on the planetary interiors.\n- the authors only show one phase from their random structure search. How distant are the others in terms of enthalpy? The authors only consider a possible decomposition in CH2 but there must be many other possible compounds and compositions. I think the authors could provide more discussion on this part.\n- the authors claim that the C-chain is connected with sp3 bonds. How can they demonstrate the nature of this bond?\n- I am surprised the energy cutoff is different for the “first-principles calculations” and for the “ab initio molecular dynamics” although they must use the same pseudo-potentials. 600 eV seems really low.\n- I do not understand the purpose of section S1. The authors must explain why it is relevant to discuss superconductivity\n- did the authors consider the effects of a higher order form for the potential than the 2nd order for the phonon calculation? It is common that hydrogen bearing compound presents a quadratic form since the hydrogen bond can be weak. See the example of pure hydrogen in Monserrat et al. (PRL 2018)."}, "author_response": "We sincerely thank all reviewers for their careful evaluations, constructive criticism, and insightful suggestions, which have substantially strengthened the clarity, scope, and rigor of our manuscript. The major comments centered on three aspects: (i) clarifying the structural and dynamical nature of the predicted superhelical and superionic phases, (ii) validating our methodology through additional convergence and finite-size testing, and (iii) providing a more balanced discussion of the implications for planetary materials. Through the revisions detailed below, we have significantly improved the manuscript, offering a more rigorous, transparent, and comprehensive account of the thermally driven phase transition in CH and its potential impact on anisotropic transport in extreme environments. We highlights all revisions in the main text, summarize the main changes and provide a detailed point-by-point response to all reviewer comments below:\n• Added quasi-harmonic approximation (QHA) free-energy profiles, including diamond carbon as a reference, to verify the stability of CH (paragraph 5, page 2).\n• Improved the clarity and visual presentation of Figure 1.\n• Included the effect of anharmonicity on electron–phonon coupling (paragraph 1, page 3).\n• Added discussion of Arrhenius behavior in transport properties (paragraph 1, page 4).\n• Clarified the distinction between one-dimensional superionic behavior and the present superhelical state (paragraph 3, page 4).\n• Verified the stability of the superhelical and superionic phases by introducing a vacancy test based on machine learning potential (paragraph 4, page 4).\n• Compared the chain-melted phase with the superhelical phase (paragraph 5, page 4).\n• Added discussion on the determination of phase boundaries (paragraph 2, page 5).\n• Included pressure-dependent electronic transport properties (paragraph 1, page 6).\n• Added ionic conductivity results and their temperature dependence (paragraph 2, page 6).\n• Discussed the influence of nitrogen and oxygen incorporation on the stability of the C–H binary system (paragraph 3, page 6).\n• AddeddiscussionontheimplicationsofthesuperhelicalCHphaseinthecontextofplanetary science (paragraph 4, page 6).\n• Added k-mesh and energy-cutoff convergence tests in VASP (Section S16, Supplemental Materials).\n• Added molecular dynamics validation tests (Section S18, Supplemental Materials).\n• Added k-mesh and energy-cutoff convergence tests in Quantum ESPRESSO (Section S19, Supplemental Materials).\nThis manuscript presents a theoretical prediction of a novel high-pressure phase of CH, termed superhelical by the authors, which emerges at pressures exceeding 1000 GPa. In this new phase, carbon and hydrogen atoms arrange into intertwined helices within an I-4 structure. Upon heating, hydrogen atoms exhibit a chain-melted-like diffusion along these carbon helices. At higher temperatures, hydrogen becomes delocalized throughout the lattice, occupying the interstitial spaces of a carbon lattice, resulting in a superionic phase. Eventually, at even higher temperatures, the carbon lattice also melts, leading to a fully liquid system.\nThe authors provide a comprehensive analysis, supporting their prediction through an array of computational techniques. They report enthalpies of formation (convex hulls) at zero Kelvin across three pressures and perform DFT-based molecular dynamics simulations to study the temperature-dependent structural evolution and hydrogen transport behavior. Their heat-until-it-melts simulations along six isochores capture all reported phases and they propose a tentative phase diagram in pressure-temperature space. Furthermore, the authors compute transport properties, including thermal conductivity and electrical resistivity via Onsager coefficients, highlighting strong anisotropy with dominant transport along the Z-direction. Additionally, they present calculations of the Eliashberg spectral function, estimating a superconducting critical temperature at low temperature. If true, their findings are both fascinating and of great importance for understanding the stable phases of hydrocarbons at extreme conditions.\nHowever, their implications for planetary science are questionable. While one typically expects high-pressurephasestofavorsimple, symmetricstructures, theemergenceofthissuperhelicalmotif at pressures larger 1000 GPa is both surprising and scientifically significant. But the temperatures at which this phase appears are too low, relatively, to exist in the interior of either Uranus or Neptune where it would remain completely molten, according to the isentropes presented, if this material makes it all the way to the center of the planet somehow, which also sounds unlikely. And not even at the center of these planets the pressures are high enough to stabilize this phase, according to their predictions. Although exoplanets like sub-Neptunes could indeed reach higher internal pressures and stabilize this phase in their mantles, a clearer discussion of the compatible thermodynamic conditions of this phase in planetary interiors would substantially strengthen the manuscript. All connections to planetary magnetic fields so far in this manuscript are weak and “handwavy,” without a real justification. For instance, there is no mention about how high the electrical conductivity must be, compared to other conductive materials in these planets, to significantly change the magnetic field of the planet. Does any material, no matter how low its conductivity is, contribute to the magnetic field generation? This point must be addressed carefully if the authors pursue publication so they should either justify well or completely take away applications to planetary science.\nNevertheless, the findings of this manuscript are worthy of publication in Nature Communications, given a major revision. Below, I provide a list of issues that need to be addressed that, I believe, will increase the impact of this work.\nReply: We thank the reviewer for the positive evaluation of our work and for highlighting its potential suitability for publication in Nature Communications. We agree that the stabilization of the superhelical phase above 1000 GPa lies beyond the pressure range of Uranus and Neptune, and we have clarified this limitation in the revised Discussion. Our results are instead presented as an expanded conceptual framework for CH behavior under extreme conditions, potentially relevant to more massive exoplanets (e.g., sub-Neptunes) where such pressures may occur. The discussion of planetary magnetic fields has been reframed to remove speculative statements and instead emphasize the intrinsic anisotropic transport properties of the superhelical state. We also clarify the relative magnitudes of electrical and ionic conductivities with respect to dynamo requirements. These revisions render the planetary discussion balanced and physically grounded, while improving the overall clarity, rigor, and scope of the manuscript.\nGENERAL COMMENTS\n1. WhiletheproposedsuperhelicalCHphaseisnovelandstructurallyintriguing,particularlyin itsapparentroleasaprecursortosuperionicity,itsnaturecloselyresemblesthatofthechain-melted phase (Naden Robinson, Victor, et al. ”On the chain-melted phase of matter.” Proceedings of the National Academy of Sciences 116 (2019): 10297-10302.). In both cases, atoms exhibit anisotropic diffusion along one-dimensional channels, while the surrounding sublattice remains vibrationally stable. It would strengthen the manuscript to acknowledge this connection and discuss similarities and differences explicitly. Furthermore, given that the CH phase crystallizes in the P6122 space group, the authors should briefly comment on why similar superhelical behavior has not been reported in other binary systems adopting this symmetry, such as gold fluoride (AuF3) or silver fluoride (AgF3), considering that the mass ratio between species is comparable. Is the behavior unique to the light mass and bonding of hydrogen, or is it a more general feature of certain topologies?\nFig. R1: Electron localization function of AgF (left) and AuF (right) at 0 GPa (isovalue = 3 3 0.8). Both phases are presented in the P6 22 symmetry.\n1 3\nReply: We thank the reviewer for pointing out this, and we have revised the Discussion in the last paragraph on Page 4 to explicitly compare the superhelical state with the chain-melted phase, highlighting both the similarities and the differences. Regarding the comparison with other P6 22 1 compounds such as AuF and AgF , we note that although the topological framework is similar, 3 3 the bonding characteristics are fundamentally different, as shown in Fig. R1. In CH, carbon atoms form an sp3-bonded framework, while the hydrogen sublattice becomes partially metallized, with electrons delocalized along the hydrogen helix. By contrast, in AuF and AgF , the bonding is 3 3 strongly ionic, with fluorine atoms capturing electrons from the cationic Au or Ag sublattice.\n2. I am not sure if enumerated citations [1-3] are allowed in an abstract for Nature Communications.\nReply: We have removed the enumerated citations from the abstract to the introduction.\n3. In lines 28-29, the percentages you quote for ratios of water:methane:ammonia are extremely specific– is this a reference to “Synthetic Uranus” i.e. solar ratios of H:O:C:N? If so, the authors should be more specific in stating this assumption.\nReply: We thank the reviewer for pointing this out. The quoted ratios (H O:CH :NH = 7:4:1) 2 4 3 indeed correspond to the so-called “Synthetic Uranus” composition, derived from solar atomic abundances of H:O:C:N under the assumption that all O, C, and N are bound in water, methane, and ammonia, respectively. We have clarified this explicitly in the first paragraph of Introduction.\n4. The statement about dynamical phases in line 36, while broadly true, the superionic phase certainly does not always appear, even in multicomponent systems, nor does the plastic phase. We could add the chain-melted phase as well, so there is no clear classification of “dynamical phases.” Adding bit more of generality would be more accurate, i.e. “many compressed molecular systems containing two or more elements exhibit two or more of the following four dynamical phases...”\nReply: We revised the sentence to soften this nuance in the second paragraph of Introduction. This addition acknowledges the chain-melted phase and removes any overgeneralization regarding the universality of superionicity.\n5. In line 44, this classification of degrees of freedom is odd, or at least, not standard. How do authors arrive to DF=2 for plastic states where H is bound to a surface surrounding each C? There areperovskiteswithammoniaunitsrotatingaroundthelatticepointwheretheyareanchoredthat, in addition, exhibit molecular vibrational modes inside the crystal. So stating that there are just 2 degrees not only seems to contradict what in the literature people call degrees of freedom, but also seems inaccurate. In addition, a solid can also have diffusional degrees of freedom when there are vacancies present.\nReply: We thank the reviewer for pointing this out. In the revised manuscript, we clarify that the diffusional dimensionality (DF) values represent a conceptual measure of diffusional dimensionality—the number of spatial directions along which atoms exhibit long-range diffusion—rather than strict degrees of freedom. In plastic phases, atoms or molecular units can rotate or reorient on a spherical surface while remaining confined radially (r fixed, θ and ϕ free), giving DF = 2. In addition, solids may exhibit diffusional degrees of freedom when vacancies are present. Although vacancy-mediated hops between neighboring lattice sites can produce finite atomic mobility, the atoms remain statistically localized around their equilibrium positions over long timescales, ensuring that long-range crystalline order is maintained. Thus, the self-diffusion coefficient is effectively negligibly small. We have added this clarification to the third paragraph of the Introduction in the manuscript to prevent confusion and better contextualize the DF scheme.\n6. Fig. 1 is a clever and thought-provoking graphic. However, the axes labels need to be bigger (comparable to the main text font size) and I would emphasize (vertical axis label) that it refers to diffusional degrees of freedom, and not degrees of freedom in general, as a solid is understood to have 3N degrees of freedom. Also, the authors imply that constant volume heating results in increase in DF, and constant temperature compression decreases DF, but the latter is not discussed anywhere in your text, and it would be helpful to have a 2nd horizontal axis for pressure on this figure. As a matter of fact, there is no isotherm in this study that compression will drive a liquid→superionic→superhelical→solid transition, especially considering that the melting curve may have a negative Clapeyron slope. If this is not what the authors intended, then they should explain more clearly what they meant with this figure, as it leads to confusion.\nReply: We thank the reviewer for these constructive suggestions. In the revised Figure 1, the axis labels have been enlarged to match the main text font size, and the vertical axis is now explicitly labeled “Diffusional dimensionality” to avoid confusion with the total atomic degrees of freedom (3N). A secondary horizontal axis for pressure has been added to illustrate the qualitative trend along isochores. We also clarified in the caption that the primary horizontal axis represents temperature under constant-volume conditions, and that the secondary axis provides an approximate correspondence to pressure. To prevent misunderstanding, we clarify that the constant-volume heating path (black) represents the simulation scheme adopted in this study, whereas the constant-temperature compression path (gray) illustrates an opposite trend in DF variation compared with the simulated heating isochores.\n7. Inline59,thediscussionofanisotropicdiffusion,particularlygiventhefocusonsuperionicity, would be improved by a discussion of J. Li’s 2023 paper on one dimensional superionicity in silica-water compounds (Li, Junwei, et al. ”Silica-water superstructure and one-dimensional superionic conduit in Earth’s mantle.” Science Advances 9.35 (2023): eadh3784.)\nReply: We thank the reviewer for pointing out this relevant study. We have now incorporated a discussion of Li et al. (2023) that emphasizes the broader significance of anisotropic diffusion for transport phenomena under extreme conditions.\n8. In line 101, a similar metallic channel formed by chains of H was reported by Wang 2024 - https://pubs.acs.org/doi/10.1021/acs.jpclett.4c01223. They find that this electronic distribution leads to superconductivity at low temperature but breaks down at high temperature – is this consistent with your findings for C-H?\nReply: We performed an additional molecular dynamics simulation for 12 ps at 1000 GPa and 200 K, as shown in Fig. R2. The CH compound remains in the solid phase with only minor atomic vibrations under these conditions. Considering that the highest superconducting transition temperature (T ) of the CH compound is 160 K, superconductivity is unlikely to be affected by the c high-temperature electronic distribution.\n9. For Fig. 2a, it would be nice (but not necessary) to show the CH2 enthalpies at 3000 K, even if they are above the convex hull. Also the 2 “CH2” should be a subscript. Fig. 2c is not referred to explicitely anywhere in the text.\nReply: We have now updated Figure 2a accordingly, although CH remains above the convex hull. 2 In addition, Figure 2c is now explicitly referenced in the main text.\n10. Fig. 3 is a fantastic figure but your labeling could be improved. The alternating use of different color keys is a bit confusing, and it would be helpful to explicitly label on the figure, or at least in the figure caption, which phase corresponds with which panel. While you did describe this in great detail in the text, for those just browsing through the figures what is happening here is not super clear.\nReply: We have revised Fig. 3 to include explicit phase labels directly, and simplified the color keys to avoid confusion.\n11. Regarding the computed electronic band structures, does the roughly equivalent band contributions of C and H have any significance?\nReply: The roughly equivalent band reflects the comparable projection weights of the atomic orbitals involved in the hybridized bands.\n12. In line 147, “a lot” is quite informal. Use “significantly”\n13. Do you have any sense of why you observe a negative Clapeyron slope above 2000 GPa? Can this be related to any change in electronic structure or bonding regimes? Also, why would you expect that this negative dT/dP slope leads to a decrease in the melting line? This does not seem to be informed by your simulations; if there is a precedent for this behavior it would be helpful for you to give some references which informed your choice to show a decreasing melting line.\nReply: In our fixed-volume molecular dynamics heating runs, the pressure increases steadily with temperature until melting begins, at which point the measured pressure exhibits a pronounced decrease above 2000 GPa. This behavior is a direct signature that the nascent liquid is denser than the solid (i.e., ∆V = v − v < 0): as a liquid fraction develops, the mixture’s average liq sol specific volume decreases at fixed total V, which relieves the accumulated thermal pressure and yields the observed pressure drop. By the Clapeyron relation dT /dP = ∆V/∆S (with ∆S > 0 for m melting), a negative ∆V implies dT /dP < 0, i.e., a melting line that slopes down with increasing m pressure, consistent with prior reports of anomalous melt behavior and negative Clapeyron slopes in other dense systems (e.g., compressed hydrogen (Helled et al.(Nature Review Physics (2020))) and certain alkali metals (Naden Robinson et al. (PNAS 116, 10297 (2019)))). Microscopically, we interpret this densification as arising from increased electronic delocalization and reorganization of C–H bonding in the liquid under extreme compression, producing a more closely packed, higher-compressibility fluid. We have also included this detailed derivation of the Clapeyron slope in the Supplemental Materials.\n14. There is something grammatically odd with the sentence in line 219. and the reason is not provided. “Because there is anisotropy...”\nReply: The sentence has been revised for clarity.\n15. In line 223, authors wrote “isotopic” instead of “isotropic”; I would also adjust this to say “resistivity and thermal conductivity become isotropic again.”\nReply: The term “isotopic” has been corrected to “isotropic”.\nFig. R3: Electronic transport properties of the superhelical CH compound. The simulations were carried out under 1000, 2000, and 3000 GPa and 2000 K. a Electronic conductivity and b thermal conductivity are represented in x, y, z directions, as well as the average.\n16. Did the authors calculate these values for any other pressures? I realize this is a lot of work so while not necessary, it would certain add valuable information on how these transport properties evolve with pressure.\nReply: We have performed additional calculations of the transport properties of the superhelical CH compound at 2000–3000 GPa and 2000 K (Fig. R3). The results confirm that, upon compression, both the electrical and thermal conductivities increase. A persistent anisotropy is observed, with the z-axis consistently exhibiting higher electrical and thermal conductivities than the xy-plane. These findings, now discussed in the revised manuscript and detailed in the Main Text, underscore the highly directional nature of charge and heat transport in this phase.\n17. In line 245 authors say “in other words” twice. The authors could avoid saying this at all and just be specific enough the first time.\nReply: We revised the sentence.\n18. Itwouldbeveryinterestingandlikelyrequireatrivialamountofyourtimetocalculateionic conductivities and compare them to electronic conductivities – this is an important question regarding ionic transport vs electronic transport influences on dynamo generation. Sigma(electronic) /sigma(ionic) have very different ratios for water and ammonia – so it would be both interesting and important to see how superionic carbohydrides behave in this regard.\nFig. R4: Ionic transport properties of CH compound at around 1000 GPa with heating up. (a) The diffusivity derived from mean square displacement of hydrogen and corresponding (b) ionic conductivity are represented in x, y, z directions, as well as the average. (c) Total ionic conductivity contributed from both H and C atoms and comparison to electronic conductivity. (d) Ratio of ionic to electronic conductivity (σ /σ ).\nel ion\nReply: We thank the reviewer for this insightful suggestion regarding the comparison of ionic and electronic conductivities. We have now performed these calculations, which indeed provide crucial insights into charge transport mechanisms in superionic carbohydrides. Following the reviewer’s suggestion, we calculated ionic conductivities using the Nernst-Einstein relation and compared them directly with electronic conductivities. Hydrogen exhibits distinct diffusion patterns across different phases – solid (D ≈ 0), superhelical (D = D = 0, D > 0), superionic (D > 0, D > 0, x y z x y D > 0), and fluid states – while carbon remains immobile until the fluid phase (Fig. R4 a). z The directional ionic conductivity reflects this progression, with z-direction conduction dominating in the superhelical phase (1000-3000 K) and full three-dimensional conduction emerging in the superionic phase above 4000 K (Fig. R4 b and c). Most significantly, the ratio σ /σ decreases el ion with temperature. In particular, when CH melts at 6000 K, the ratio drops sharply, although electronic conduction remains dominant (Fig. R4 d). The comparable magnitudes of ionic and electronic conductivities at extreme conditions imply that electronic transport may substantially contribute to magnetic field generation in planets containing superionic carbohydrides. We believe this analysis substantially strengthens our manuscript by providing a comprehensive picture of charge transport in these exotic materials.\n19. The manuscript reports calculations of the electronic band structure, Eliashberg spectral function and a corresponding superconducting critical temperature at 300 GPa. However, based on the authors’ own convex hull analysis, the superhelical CH phase is not thermodynamically stable below 1000GPa. Thisraisesthequestionofwhysuperconductivityisevaluatedatapressurewhere the phase is presumably metastable or inaccessible. The authors should either clarify the rationale for selecting 300 GPa or revise the superconductivity analysis to match the actual pressure range where the phase is stable. Without such justification, the physical relevance of the result remains unclear and the choice of a pressure of 300 GPa, far away from the other pressures at which this material was analyzed, seems random.\nReply: To address the reviewer’s concern more comprehensively, we have now also performed electron-phonon coupling calculations at 1000 GPa, where the phase is thermodynamically stable (Table R2). These new results (included in the Supplemental Materials) reveal significantly lower T values at higher pressures. We have revised the manuscript to address these points clearly and c to emphasize that the 300 GPa calculations represent an investigation of metastable, potentially synthesizable phases rather than the ground state.\n20. The manuscript does not explain how the phase boundaries in Figure 4 were determined. It is unclear how the authors determine which points lie on the phase boundaries and which ones do not. Is the temperature on the phase boundary just an average of the temperatures above and below it? And which functional form was used to fit it?\nReply: The phase boundaries were constructed through the following procedure: For each fixed volume, we identified the transition temperature by bracketing the range between the highest temperature at which the initial phase remains stable and the lowest temperature at which the new phase forms during our heat-until-melting simulations. These bracketing temperatures and corresponding pressures were then averaged to define the boundary point at each pressure. Finally, a cubic polynomial fit was applied to generate smooth phase boundaries across the pressure range. We have now explicitly described this methodology in the revised manuscript and expanded the figure caption to clarify that these boundaries represent approximate transitions based on our simulation data.\n9\n21. Themethodtodeterminetherangeofstabilityofeachphase(solid,superhelical,superionic, liquid) appears to be the heat-until-it-melts approach, if I understand correctly. This means that a super cell of given, fixed volume is heated up gradually until a phase transition is observed to occur duringtheDFT-MD.Whilethismethodcancapturequalitativetrends, itisknowntooverestimate the stability range of initial phases due to superheating and lack of coexistence dynamics. In the absenceofGibbsfreeenergycalculations,theproposedtransitiontemperaturesmaynotreflecttrue thermodynamic phase boundaries. The authors, therefore, should at least perform some two-phase coexistencesimulationstoestablishtherobustnessofthesuperhelicalphase(halfsuperhelical+half superionic to show that H chains form out of the H liquid) and at least one coexistence simulation betweentheliquidandsuperionicphase(halfliquid+halfsuperionic)toshowatleasttheformation of the C chains. Performing these simulations below and above the phase boundaries will provide a proof that the transition temperatures are not hugely overestimated by the heat-until-it-melts approach.\nFig.R5: Two-phase simulations of the CH compound at ∼1000 GPa. Twoinitialsupercells are constructed: (b) half superhelical (left) and half superionic (right), and (e) half superionic (left) and half fluid (right). Each panel represents an MD simulation under high pressure and temperature: (a) superhelical, (c) superionic, (d) coexistence of superionic and liquid, and (f) fully fluid. The dashed red box highlights the hydrogen diffusion channels characteristic of the superhelical phase, where hydrogen motion is confined, whereas in the superionic phase, hydrogen diffuses beyond these channels. The dashed red line marks the phase boundary in the two-phase coexistence simulation.\nReply: We agree with the reviewer that the heat-until-it-melts method may slightly overestimate transition temperatures. To address this concern, we have now performed two-phase coexistence simulations for (i) the superhelical–superionic and (ii) the superionic–liquid interfaces at fixed volume. Ineachcase,moleculardynamicssimulationswerecarriedoutfor12pswithatemperature interval of 100 K across the phase boundary. First, we started from a coexistence configuration of halfsuperhelicalandhalfsuperionicphaseswithX=0.5(Fig.R5b)andthenequilibratedthesystem at 3100 K. The coexistence evolved into a pure superhelical phase, where hydrogen atoms migrate 10 along the channels (see dashed box in Fig. R5a). Upon heating to 3200 K, the system transformed into the superionic phase, in which hydrogen atoms diffuse across the channels (Fig. R5c). No evidence of two-phase coexistence between the superhelical and superionic phases was detected in our simulations. Second, we initialized a two-phase configuration consisting of half superionic and half liquid phase with X=0.5 (Fig. R5e) and then equilibrated the system at 4700 K. The system maintainedcoexistencebetweenthesuperionicandfluidphases, duringwhichthesuperionicregion gradually expanded to X=0.68 (Fig. R5d). Upon heating to 4900 K, the entire system transformed into the liquid phase (Fig. R5f). The melting points identified from the two-phase coexistence simulationsaresubstantiallylower—byabout600–800K—thanthoseobtainedfromtheheat-until- it-meltsmethod,indicatingthatthelatterdoesnothugelyoverestimatethetransitiontemperatures.\nTable. R1: Thermodynamic data from simulations performed using different supercell sizes (number of atoms N), k-point meshes, simulation durations, exchange–correlation functionals, and a machine-learning potential (MLP), illustrating their influence on the motion behavior of the CH compoundatapproximately1000GPaand2000K.EachFPMDtrajectorycorrespondstoanNVT MD run of at least 12 ps, with the first 2 ps discarded for thermalization, from which pressure (P) and temperature (T) are extracted. For the MLP calculations, NPT MD was performed with a 1500-atom supercell for 1 ns (1000 ps).\nfunctional supercell N kmesh time (ps) P (GPa) T (K) D (˚A2/ps) State z\nPBE 2×2×2 96 1×1×1 10 1044 1977 1.52 superhelical\nPBE 2×2×3 144 1×1×1 10 1043 1980 1.30 superhelical\nPBE 2×2×4 192 1×1×1 10 1041 1979 1.25 superhelical\nPBE 3×3×2 216 1×1×1 10 1042 1978 1.40 superhelical\nPBE 2×2×4 192 1×1×1 50 1041 1979 1.32 superhelical\nPBE 3×3×3 324 1×1×1 10 1033 1979 1.19 superhelical\nPBE 2×2×4 192 2×2×2 10 1032 1979 1.21 superhelical\nRPBE 2×2×4 192 1×1×1 10 1043 1974 1.50 superhelical\noptB88 2×2×4 192 1×1×1 10 1045 1988 1.17 superhelical\nLDA 2×2×4 192 1×1×1 10 1034 1990 1.41 superhelical\nMLP 5×5×5 1500 - 1000 1000 2000 1.40 superhelical\nMLP 5×5×10 3000 - 20 1000 2000 1.36 superhelical\n22. While the authors report that the superhelical structure remains stable upon increasing the simulation cell size, they do not evaluate whether this impacts the diffusion coefficient, which is known to be strongly system-size dependent, especially in anisotropic or quasi-1D systems. Since oneofthekeyfindingsofthepaperisdirectionalhydrogendiffusion,itisimportanttoverifythatthe reporteddiffusionbehaviorisconvergedwithrespecttocellsize. Theauthorsshouldincludeabrief analysis of how the mean square displacement (MSD) and resulting diffusion coefficients vary with system size (e.g., between the 2x2x4 and 3x3x3 cells), or at least comment on the expected impact 11 and limitations. This would help validate the transport conclusions presented in the manuscript.\nReply: Toaddresssystem-sizeeffects,weperformedadditionalDFT-MDsimulationsupto3×3×3 supercells at ∼1000 GPa and ∼2000 K, as shown in the Table. R1. The resulting hydrogen diffusivities (D ) along the z-axis showed excellent agreement within 10% above 2×2×4 (192 z atoms), indicating that the reported directional diffusion is well converged with respect to cell size.\n23. One of the isochores presented in the phase diagram of Figure 4, starting at approximately 500 GPa, lies outside the predicted thermodynamic stability range of the superhelical CH phase. More generally – is CH stable in the low P range you give in your phase diagram? This is quite important to the discussion of significance for Uranus and Neptune. Given that the authors’ own convex hull calculations suggest this phase becomes stable only above 1000 GPa, it is unclear why this lower-pressure isochore is included in the same phase diagram. The authors should clarify the purpose of including simulations at pressures below the stability field, and explicitly acknowledge whether these points correspond to metastable persistence of the superhelical structure. To avoid confusion, a clear visual boundary should be added to the phase diagram separating regions where the structure is thermodynamically stable from those where it is merely metastable. This will help readers interpret the extent and relevance of each phase more accurately.\nReply: Weremovedtheisochoresbelow1000GPaintherevisedfigureandaddedadashedbound- ary in Figure 4 to separate the metastable region (P < 1100 GPa) from the thermodynamically stable superhelical phase.\n24. There seems to be a disconnection between what the authors show in their MSD plots and the model they discuss in the Methods section. The authors present a model for the mean square displacement (MSD) in the superhelical phase, showing that the trajectory of the hydrogen atoms should follow the form MSD = 2R2[1 − e−Dt] for the superhelical state, where R is the confinement radius of the helix and D the angular diffusion coefficient. However, this expression is not actually used anywhere in the text nor tested against the simulation data. Why do the authors bother deriving it if they are not going to use it? The authors should validate whether this MSD theory applies to the simulations they have generated. That is, the MSD should converge to 2R2, with R obtained from the cell geometry, and the numerical DFT-MD simulations should obey the exponential growth up to this asymptotic value. A fit to the data could even provide a value for D, which was not discussed. Additionally, it would strengthen the manuscript to plot diffusion coefficients as a function of temperature, especially to explore Arrhenius-like behavior. This would help quantify the transition from quasi-1dimensional diffusion in the superhelical state to 3-dimensional diffusion in the superionic regime. Since the slope of the MSD is proportional to 2D in one dimension and to 6D in three dimensions, the trend for D is not clear. In addition, Arrheniustrendsareoftenusedtoextractactivationbarriers,whichcouldaddvaluablemechanistic insight into the phase behavior and hydrogen mobility in this system.\nReply: We thank the reviewer for this valuable suggestion, and we added more discussion on this point. In Fig. 2c, we show the evolution of the H-helix radius with pressure and applied the MSD model to compare with the plateau value of the superhelical MSD in Fig. 3f. For each component in the x and y directions, the MSD can be expressed as MSD = R2[1 − e−Dt]. The plateau value of 0.34 ˚A2 closely matches R2 = 0.3364 ˚A2. Using this model, we fitted the DFT-MD data H and derived an angular diffusivity of 5.50 s−1. In addition, the activation energies for hydrogen diffusion, extracted from the temperature dependence of diffusivity (Fig. R6), are 0.139 eV in the superhelicalstatenear1000GPaand1.111eVinthesuperionicstate. Thishugebarrierenergygap between two states demonstrates the transition from quasi-1D diffusion in the superhelical phase to 3D diffusion in the superionic phase, providing mechanistic insight into hydrogen mobility and the phase transition behavior.\n25. The authors do not specify the reference phases used in the enthalpy of formation calculations. In particular, it is unclear whether the CH enthalpy was computed relative to diamond or to the BC8 phase of carbon, and which phases of molecular hydrogen were used as the H reference. This is a critical detail, as the thermodynamic stability of CH hinges on the choice of these elemental reference states. While BC8 is predicted to be the lowest-enthalpy carbon phase above 1000 GPa by DFT calculations, there is no experimental evidence for its formation, even under dynamic compressions up to 2 TPa. Notably, both Lazicki et al. (Nature, 2021, http://dx.doi.org/10.1038/s41586-020-03140-4) and Nguyen-Cong et al. (J. Phys. Chem. Lett., 2024, http://dx.doi.org/10.1021/acs.jpclett.3c03044) show that diamond remains metastable far beyond the pressure at which BC8 is expected to become favorable. Given this experimental uncertainty, the authors should re-evaluate and report how the enthalpy of formation of CH changes when the comparison is made using diamond instead of BC8 as the carbon reference. This would give readers a clearer understanding of the robustness of the predicted stability and the sensitivity of the convex hull to reference phase assumptions.\nReply: We thank the reviewer for highlighting this crucial point. In our convex hull construction, we used the diamond and BC8 phases as references for carbon, and the Cs-IV (I4 /amd) and 1 oC12 (Cmcm) phases as references for hydrogen. We performed full cell relaxations and included zero-point energy corrections from phonon calculations for all these reference phases relative to the CH and CH compounds to evaluate their formation energies. Experimentally, diamond has been 2 observed to remain stable well beyond the pressure range where BC8 is theoretically predicted to become favorable. In the revised manuscript, we therefore recalculated the enthalpy of formation (Fig. R7) using diamond as the carbon reference and found that diamond C can increase the stabilityoftheCHcompound, whiletheCH2compoundremainsmetastableabovetheconvexhull. This updated analysis is presented in the Supplemental Materials and discussed in the main text, providing a clearer understanding of the reference-phase sensitivity in our stability calculations.\n26. Theyauthorsalsomentionedadiscontinuityin∆H(Fig. 2)generatedbyaphasetransition in hydrogen, but they do not mention which phase of hydrogen it is. They should describe more clearly which phases they are comparing.\nReply: Here, we used the Cs-IV (I4 /amd) phase below 1600 GPa and the oC12 (Cmcm) phase 1 above that pressure as references for hydrogen. Notably, the OC12 phase becomes dynamically unstable below 1600 GPa.\nTable. R2: Summary of calculated T values at 300 and 1000 GPa with/without anharmonic effect c from SSCHA.\npressure degauss µ∗ λ ω (K) TMc (K) TAD (K) log c c\nharmonic 300 0.004 0.10 1.383 1367.540 143.754 160.866\nanharmonic 300 0.004 0.10 0.925 1325.944 117.119 123.682\nharmonic 1000 0.004 0.10 0.284 2597.122 0.699 0.706\nanharmonic 1000 0.004 0.10 0.287 1695.905 0.752 0.759\n27. The Eliashberg spectral function is derived under assumptions (harmonic phonons, weak coupling) that may not hold in a quasi-1D, high-pressure system. The authors already showed that 14 the effect of the zero point energy is large, as it modifies the stability field significantly. How valid is the Eliashberg the authors apply it at 300 GPa, so far away from the range of stability that they predict this structure has?\nFig. R8: Anharmonic effect. The phonon spectrum of the CH compound at a 300 GPa and b 1000 GPa, as well as diamond C and Cs-IV phase H at 1000 GPa, showing the harmonic (blue dashed) and anharmonic (orange) results. The SSCHA minimization already includes four phonon scatterings at the lowest order perturbation theory.\nReply: We thank the reviewer for this insightful comment. We additionally performed electron–phonon coupling calculations at 1000 GPa and accounted for anharmonic effects for the phonon spectrum in the SSCHA code. The SSCHA minimization already includes four phonon scatterings at the lowest order perturbation theory. All compounds remain dynamically stable when anharmonic effects are included, although the phonon frequency ranges of CH and H narrow slightly due to the strong anharmonicity of hydrogen at these high pressures (Fig. R8). We further estimate the formation energy of CH, relative to elemental C and H at 1000 GPa, to be 5 meV per atom with anharmonic corrections—lower than the 10 meV per atom obtained within the harmonic approximation. Compared to the harmonic case, anharmonicity lowers the estimated superconducting transition temperature from 160 K to 123 K (Table R2). We have clarified these points in the revised manuscript and added the SSCHA results for direct comparison.\n15\n28. In Fig. 5, I am very enthusiastic that the authors went to the effort to calculate transport properties for this material, because this is so seldom done. However, why do you show resistivity and not conductivity? For those looking to your paper to get a sense of how your calculated electrical transport properties fit into dynamo theory, conductivity is much more accessible. In addition, it would be valuable to have ionic contributions because mobile atoms will certainly affect the overall conductivity. The authors could do so with a simple Nernst-Einstein model. Sigma(electronic)/sigma(ionic) have very different ratios for water and ammonia – so it would be both interesting and important to see how superionic carbohydrides behave in this regard.\nReply: We thank the reviewer for this insightful comment. In the revised Fig. 5, we now report the corresponding electrical conductivity σ (in S/m), derived using σ = 1×108/ρ, which is more directly relevant for planetary dynamo considerations. Furthermore, we have estimated the ionic contribution using the Nernst–Einstein relation based on DFT-MD hydrogen trajectories. The electronic-to-ionic conductivity ratio, σ /σ , is further discussed in Comment 18. e i\n29. Did the authors calculate these transport properties at any other pressures? I realize this is a lot of work so while not necessary, it would certainly add valuable information on how these transport properties evolve with pressure.\nReply: This comment is the same as Comment 16.\n30. In the conclusions, I strongly disagree with the statement that they authors are providing a “unifying framework” to connect microscopic atomic dynamics with macroscopic field generation. This cannot be farther from it. The authors do not provide estimate of the strength of the magnetic field generated (through the magnetic Reynolds number or something similar), nor they perform a model of planetary interior. What is worse, the interior conditions for Uranus and Neptune are too hot to generate this phase, as CH will likely be superionic or liquid at their interior. Larger exoplanets will likely be hotter, so it is even less likely that this phase plays any role in their magnetic fields, especially considering that it needs a pressure that is too high to even form. Thus, the connection to planetary science and planetary magnetic fields is, at best, weak.\nReply: We thank the reviewer for this important clarification and agree that our work does not directly model planetary magnetic field generation or quantify field strength through magnetic Reynolds numbers or interior modeling. In the revised manuscript, we have further softened our concluding statements to avoid any implication of a macroscopic dynamo framework. Our intent is now limited to highlighting that the superhelical CH phase exhibits strong anisotropy in charge and heat transport, which represents a microscopic property of potential relevance to materials under extreme planetary conditions. We explicitly acknowledge that the pressure–temperature range required for stabilizing this phase likely exceeds those of Uranus and Neptune. Therefore, any connection to planetary magnetic fields should be regarded as speculative.\n31. The last paragraph of section S1 of the supplementary material (lines 27 onwards) would be valuable in the main text – my main concern regarding overlapping frozen cores was answered here but it would be better to not have to go to the supplemental for such a question.\nReply: WehavemovedthediscussionofoverlappingfrozencoresandthechoiceofDFTfunctionals from the Supplemental Materials to the main text for greater clarity and accessibility.\n32. WhilethecomparisonofthedifferentpseudopotentialsinFig. S9isavaluableandnecessary part of this manuscript, it would be valuable to add it to the convergence tests of Table S1. Does the superhelical state prevails when changing the functional? It is possible that some functionals tend to overbind the system and prevent the transition (or at least shift it to higher temperatures).\nReply: We appreciate the reviewer’s suggestion. We have now included the influence of different exchange–correlation functionals on the MD simulations of the superhelical state in a 2×2×4 supercell (see Table S1). The results show that the superhelical CH phase remains stable across all tested functionals, with only minor variations in the hydrogen diffusivity along the z-axis (D ). In z addition, we performed heat-until-it-melts simulations (Fig. R9) for the CH compound at approximately 1000 GPa using these functionals. Notably, in the optB88 functional simulation, hydrogen exhibits a much lower diffusivity in the x and y directions, indicating an expansion of the superhelical region at 4000 K. This analysis demonstrates the robustness of the predicted superhelical phase with respect to the choice of functional.\n33. I presume that, complying with Nature policies, the authors will provide all the related material available, such as the CIF/POSCAR files of their structures, as well as videos of the molecular dynamics in the superhelical phase. That would be appreciated by the readers.\nReply: We confirm that, in accordance with Nature Communications policies, we provided all relevant atomic structure information of the predicted structures, as well as MD trajectory videos of the superhelical and superionic phase.\nThe manuscript “Thermally driven superhelical state in CH compound under planetary interior conditions” written by Liu, Cohen and Sun, proposes a computational study of a system having stoichiometry CH at high pressure and high temperature conditions. This work proposes the new CH material as relevant for planetary science, particularly to understanding the properties of ice giant planets. After reading the manuscript, I could find both positive and negative points relative to the methodology and results.\nI think that the authors have predicted an interesting system that could potentially manifest 1-D superionicity, and it is intriguing that a helix structure of carbon is stable above 1 TPa of pressure. However, there are two fundamental aspects of this work that made me concerned and that I will further develop with my comments: 1) The superhelical state seems to be just 1-D superionicity given by the confinement of the carbon lattice; and 2) the theoretical calculations appear to have significant limitations that should be addressed.\nReply: We thank the reviewer for these insightful comments. We agree that the new state is constrained by the carbon framework, within which hydrogen diffuses along quasi-one-dimensional channels. However, these channels are helical rather than linear, forcing hydrogen atoms to move in spiral trajectories. To clarify this distinction, we developed a mean-square-displacement (MSD) model capturing direction-dependent diffusion. Unlike a conventional 1D superionic state—where D = D = 0 and D > 0 (DF = 1)—the superhelical state exhibits both axial diffusion (D > 0) x y z z and angular diffusivity of 5.50 s−1, yielding an effective DF = 2. The MSD shows plateaus of ∼0.34 ˚A2 in x and y (total 2R2 ), characteristic of rotationally mobile hydrogen as in two dimen- H sionalplastic-crystalphasesinxy plane. Toaddresstheoreticallimitations, weconductedextensive validation, including convergence of k-mesh, cutoff energy, and exchange–correlation functionals; system-size and convex-hull tests; and an explicit treatment of the C 1s core. These extensive validation tests provide strong confidence in the reliability and robustness of our results.\nComputational Methodology:\n- The authors do not provide any structural information about their system, apart from the two possible space groups. This made it difficult to understand the goodness of some of the parameters the authors chose. A structure should, even if theoretical, should always be provided to guarantee and help reproducibility.\nReply: We have now provided the full crystallographic information for the CH phase in the Supplemental Materials.\n- The authors study a system starting from very high pressure, from 1 TPa to 3 TPa. At such pressures it is not possible to trust the core potentials, even if modified as the authors do in some cases. The 1s orbital of carbon will affect the intraatomic 2s and the interactions with the other carbon atoms. Therefore, all electrons calculations should be performed.\nReply: We thank the reviewer for raising the concern regarding core-electron ionization. Previous studiesbyBethkenhagenet al. (Phys.Rev.Research2020)andCl´erouinet al. (Phys.Rev.E2022) explored the equivalence between pressure- and structure-defined ionization in hot dense carbon. These works demonstrated that 1s ionization in carbon occurs only at much higher pressures (P∼ 237.6 TPa) and temperatures (T ∼ 100 eV) than those investigated in the present study. Moreover, we also performed band-structure calculations using the CP2K code, which can explicitly include all-electron effects, including the carbon 1s orbital. The resulting band structure is nearly identical to that obtained without accounting for the 1s orbital (Fig. R10). Relative to diamond carbon and Cs-IV hydrogen, the CH compound has a formation energy of 21 meV per atom at 1000 GPa, consistent with our FP-LAPW result (Fig. S21 in Supplemental Materials). Therefore, we confirm that the 1s electrons remain tightly bound and are not stripped in our simulations. The electronic structure and bonding characteristics are thus governed entirely by valence electrons, and no core ionization effects are expected to influence our results.\nFig. R10: Band structure of the CH compound at 1000 GPa. Calculated using CP2K with and without inclusion of the carbon 1s orbital.\nFig. R11: Density of states of CH compound at 1000 GPa in VASP by varying kmesh.\n- The k-mesh used by the authors seems very coarse. A 2π× 0.025 is surely not enough to calculate correctly the electronic structure of a metallic system and such extreme conditions of 19 pressure. This is clear from Figure S2b and extended Figure 1. The DOSs are too spiky, which is symptomatic of an incorrect choice of the k-mesh. The authors should show also the convergence test of the k-mesh together with the cut off, for both VASP and QE.\nFig. R12: Kmesh test of inner energy for a C, b H, and C CH compound by varying cutoff energy at 1000 GPa in VASP. d Band structure comparison between two kmesh of 0.025×2π and 0.015×2π.\nFig. R13: Encut test of inner energy for a C, b H, and C CH compound by varying cutoff energy at 1000 GPa in VASP. d Band structure comparison between two encut of 1500 eV and 2000 eV.\nReply: We thank the reviewer for pointing out the issue regarding the spiky density of states (DOS). Upon checking, we found that the spiky DOS was caused by using a small smearing width of 0.02 eV, which is indeed too low for metallic CH. We have now employed a more appropriate smearing parameter of σ = 0.2 eV, resulting in a smooth DOS (Fig. R11). We also performed detailed convergence tests for the VASP calculations by varying the k-mesh (Fig. R12) and energy cutoff (Fig. R13). The total energy differences are below 1 meV between k-meshes of 0.025×2π and 0.015×2π, and between cutoff energies of 1500 eV and 2000 eV. Furthermore, the corresponding band structures remain essentially unchanged under these variations. Therefore, we conclude that a cutoff energy of 1500 eV and a k-mesh of 0.025×2π are sufficient to ensure well-converged results for our calculations. For the QE convergence test, we varied both the plane-wave cutoff energy and the kmesh to assess the internal energy of the CH compound (Fig. R14). Our calculations converged with a cutoff energy of 100 Ry and a 16×16×16 k-mesh.\nFig. R14: Convergence test of internal energy for the CH compound at 1000 GPa in QE. a Variation with plane-wave cutoff energy. b Variation with k-point mesh, using a cubic grid of N ×N ×N and varying N.\n-The atomic radii have not been rescaled for the calculation of the pDOSs. Using the standard radii does not allow for a correct calculation and projection of the DOSs.\nReply: We rescaled the atomic radius with LORBIT = 11 for the pDOS calculation in VASP.\n- Performing NVT simulations in the range of pressure 1-3 TPa and of temperature 0-7000 K raises concerns about the validity of the approximation. Assuming that a constant volume in this case is too big of an approximation, and NPT simulations should be performed instead.\nReply: A single constant volume is not suitable for exploring such a wide pressure–temperature range. Accordingly, we performed a series of NVT simulations across a range of volumes rather than relying on a fixed-volume approach. The stability temperature range of each phase was determined using the heat-until-it-melts protocol. For each pressure column in the phase diagram, the corresponding volume was fixed based on zero-temperature cell relaxations performed at pressures of 500, 1000, 1500, 2000, 2500, and 3000 GPa. With increasing temperature, the equilibrium pressureofeachcellincreasesslightly; at6000K,thepressureshiftislessthan10%. Wealsoperformed a set of NPT simulations at selected pressures and temperatures to construct the phase diagram (Fig. R15). The main structural transitions and diffusion behaviors observed in the NPT simulations are consistent with those obtained from the NVT simulations. These results are discussed in the revised manuscript, demonstrating that our simulations provide a reliable description of the phase evolution of the high-pressure superhelical CH system.\nFig. R15: Proposed P–T phase diagram of the chiral CH compounds obtained from NPT-MD simulations. Each point represents an independent simulation performed at 1000, 2000, and 3000 GPa and temperatures up to 7000 K. The simulations are denoted by four symbols: blue circles, green diamonds, orange squares, and red triangles, corresponding to the solid, superhelical, superionic, and fluid states, respectively. The black dashed line along the x-axis separates the region where the structure is thermodynamically stable from that where it is metastable.\n- Finally, the crystal structure prediction is extremely unclear. The authors wrote that they used C and H as reference to calculate the enthalpies. Which phases did they use? Did they perform CSPs for C and H to check whether new phases become more stable than diamond and atomic H (I guess that these were the phases they used)? How does their prediction compare with phases like the CH2 (polyethylene) predicted in 10.1073/pnas.2026360118? Moreover, the authors should check CH4, even if it has been observed to decompose at lower pressure, but I believe that it has never been investigated above 1 TPa.\nReply: In constructing the convex hull, we used the diamond and BC8 phases as reference structures for carbon, and the Cs-IV (I4 /amd) and oC12 (Cmcm) phases as references for hydrogen. 1 We also performed CSP calculations for pure C and H, but did not identify any new phases more stable than the chosen references in the 1–3 TPa range. For the CH compound, our calculations 2 predict the lowest-energy Cmcm phase, consistent with previous studies (Lewis J. Conway, PNAS, 2021; Anastasia S. Naumova, JPCC, 2019; Hanyu Liu, JPCL, 2016). In our CSP calculations, we explored a wide range of compositions from C H to CH , but only CH and CH lie close to the 6 6 2 convex hull. Even CH lies more than 0.1 eV/atom above the convex hull (Fig. R16). C-rich com- 4 pounds are significantly less stable, lying more than 0.4 eV/atom above the hull at 1000 GPa, while H-rich compounds exhibit smaller excess energies of approximately 0.15 eV/atom. With increasing pressure, CH and CH become slightly more stable and approach the convex hull (within ∼0.05 4 6 eV/atom at 3000 GPa). Our results indicate that CH decomposes into CH and H above 1 TPa, 4 confirming that CH remains the preferred stoichiometry under the studied conditions. We revised Method discription and added discussion on this in Supplemental Materials.\nFig. R16: Convex hull of C–H compounds at a 1000 GPa, b 2000 GPa, and c 3000 GPa. All candidate structures obtained from our crystal structure prediction (CSP) calculations are shown, without including zero-point energy (ZPE) corrections.\nGeneral Comments:\n-Thedouble-helixdefinitionishereincorrect. Contrarytostructureslikedna,wheretwocarbon chains are interconnected by the ATGC bases, here the proposed system is just one row of carbon encapsulating a chain of hydrogen. In fact, Figure 2 gives a clear hint of the nature of the system: From the ELFs, it seems that the CH is formed by two independent chains, one of C and one of H, which are not bonded. Therefore, the potential superionicity, might be in reality just the melting of the H chain that is encapsulated in the C helix. As I will suggest below, the authors should calculate the charges of the atoms to better understand the nature of the system.\nReply: We have revised the text to clarify that the structure consists of a carbon framework encapsulating a hydrogen chain to avoid misleading “double helix”. To further elucidate its electronic structure, we performed Hirshfeld charge analysis using the CP2K code for the CH compound at 1000 GPa. The results reveal a substantial charge transfer of approximately 0.42 e− from H to C.\n- Figure 2ab: It would be helpful to rescale in meV/atom, also to better understand the magnitude of the ZPE correction on the discontinuity of the enthalpy. Moreover, the author should also show the free energy profiles.\nReply: We have updated Figure 2(a,b) to display energies in units of meV per atom. In addition, we now provide free-energy profiles incorporating vibrational contributions to illustrate the temperature-dependent stability of the CH compound (Fig. R17). Because the CH system transforms into a superhelical state above 1000 K, the free-energy profile below this temperature was constructed using the quasi-harmonic approximation. The results show that both CH and CH 2 become slightly less stable as the temperature increases.\nFig. R17: Free energy profile of C–H compounds at a 1000 GPa, b 2000 GPa, and c 3000 GPa, constructed at temperatures of 0 K, 500 K, and 1000 K.\n- On their sentence “The electronic localization function(ELF) shows that the H helical chain forms a quasi one-dimensional metallic channel” I think it is hard to see that from that orientation. Instead, it seems that H form trimers, and not chains. Instead of having the same orientation for the two space groups in Figure 2, they should show two orientations for the same space group. The transferability from P6122 to P6522 would be accepted without showing it (or it can be shown in the SI). This will also help to understand whether the H-chain and the C-chain are bonded or not.\nReply: We have revised the Fig. 2e and f to include two crystallographic orientations—⟨100⟩ and ⟨001⟩—of the same P6 22 phase. This modification more clearly reveals that the hydrogen atoms 1 form quasi-one-dimensional channels within the helical carbon framework.\n-“ThephononspectrumshowsCHisdynamicallystablewithnoimaginaryphonons, indicating that the CH compound can be quenched to 300 GPa (Extended Fig. 1c). The CH compound is a conventional” This conclusion should be re-evaluated once the k-mesh convergence is verified. Moreover, to justify the superconductivity statement, the authors should perform also an NPT simulation up to 160 K, their predicted Tc, to check the stability of the system above 0 K.\nReply: WehaveverifiedtheconvergenceofthephononspectrafortheCHcompoundat1000GPa with respect to the k-point mesh. As shown in Fig. R18, the phonon spectra obtained using k- meshes of 16×16×16 and 24×24×24 are identical, confirming that our chosen mesh of 16× 16×16issufficientlydenseforaccuratephononcalculationsinQuantumESPRESSO.Additionally, we performed NPT simulations up to 200 K to verify the structural stability near the predicted superconducting transition temperature, T (Fig. R2). Both the atomic trajectories and the mean- c squared displacement (MSD) analysis confirm that the CH compound remains in a solid phase at 200 K, thereby validating the relevance of our superconductivity analysis under these conditions.\nFig. R18: Phonon spectrum of C–H compounds at 1000 GPa by varying kmesh of a 16×16×16 and b 24×24×24 in QE.\n- I question the interpretation that “Thus, the new state can be characterized by a combination of superionic behaviors in the z direction and plastic behaviors in the xy plane of H atomic motion, finally inducing helix H chain rotation along the z-axis in the rigid C framework. Here we named this new state, superhelical.” What would be the difference with a straight 1D tube? or zigzag? the helical shape is just the confinement of the carbon, which allows the diffusion only along one dimension. In fact, it is not plastic along x and y, because the C confinement does not allow the H to escape, while in the plastic behavior, the H stays attached to the heavier atom even if it is free to move. Therefore, the effect might be confined superionicity, which is already an interesting result, but it does not need a new name based on the shape of the scaffold, which is not related to the effect per se. However, the charge of both carbon and hydrogen should be calculated to understand if the system is truly superionic or if it is formed by two intercalated neutral frameworks (C-chain and H-chain), where one of the two (H) melts before the second one.\nReply: We thank the reviewer for this careful and insightful analysis. We agree that the newly identified state is confined by the carbon framework, within which hydrogen diffuses along quasi-one-dimensional channels. These channels, however, are not strictly linear but follow a helical geometry, causing the hydrogen atoms to move along spiral trajectories. To elucidate this unique diffusion behavior, we have developed a mean-square-displacement (MSD) model (Method section) that captures this direction-dependent motion. This model clarifies the distinction between the observed behavior and that of a conventional one-dimensional (1D) superionic state. In a typical 1D superionic phase, hydrogen diffusion is confined to a single direction (taken here as the z-axis), where D = D = 0 and D > 0, yielding a diffusional dimensionality of DF = 1. In contrast, x y z in our identified state, hydrogen exhibits not only significant diffusion along z (D > 0) but also z substantial angular diffusivity of 5.50 s−1, resulting in an effective diffusional dimensionality of DF = 2. The key signature of the new motion behaviour appears in the MSD: both the x and y components saturate at plateau values of approximately 0.34 ˚A2 each (summing to 0.68 ˚A2, which equals 2R2). This plateau behavior is characteristic of plastic crystal phases, similar to H those observed in high-pressure simulations of ammonia. Although solid phases also display MSD plateaus, their amplitudes are typically much smaller than 2R2. We fully agree with the reviewer that this new state corresponds to a form of confined superionicity, in which hydrogen diffusion is restricted by the geometry of the underlying scaffold—in our case, the helical carbon framework that guides hydrogen motion along helical pathways. We have clarified this point in the main text byexplicitlyreferringtothisnewlyidentifiedstateasthesuperhelicalphase, therebydistinguishing it from the 1-D superionic state. Furthermore, our Hirshfeld charge analysis by CP2K reveals a substantial charge transfer of approximately 0.42 e− from hydrogen to carbon. Therefore, the CH compound is a truly superionic phase at high temperatures.\n-In the SI, Table ?? is missing in page 2. Moreover, Figure S3 is very puzzling. It is very surprising that c and a have the exact same behavior in and hexagonal space group. This is, in fact, possible only for cubic space groups. I believe that this is also caused by the loose k-grid.\nReply: We have corrected the missing table reference in the Supplemental Materials and updated the figure (now Fig. S15) accordingly. The previous apparent coincidence in the lattice parameter c/a ratio resulted from the wide y-axis range used in the earlier plot. After narrowing the range, the c/a ratio is clearly seen to be around 0.992. We also performed additional cell relaxations using larger cutoff energies and denser k-point meshes, and the c/a ratio remained unchanged.\nIn the manuscript entitled ’hermally driven superhelical state in CH compound under planetary interior conditions’, the authors report the results of their numerical study of the CH compounds in the multi-megabar regime. They have performed a series of ab initio calculations using DFT to examine possible stable compounds of CH at 0K, their stability based on a phonon spectrum analysis,andtheirpropertiesatfinitetemperature. Basedontheirstudy,theauthorsconcludethat CH forms a stable double-helix in the multimegabar regime. At low temperature, the compound is stable, but above 1000 K, it becomes “superhelical” with hydrogen diffusing along the channel formed by the carbon chain. At about 4000 K, the system becomes superionic, and ultimately melts at about 6000K. This specific geometry imprints on the transport properties, both ionic and electronic. Theauthorsthenconsiderpossibleconsequencesforplanetaryinteriorssuchasicegiant planets, and their magnetic field.\nThe topic is of interest for multiple communities since it brings novel structure and concept for planetary physicists, but also high pressure scientists, and the warm dense matter community in general. However, the conclusion of superionicity/superlicity is in stark contrast with what experimentalists have observed with the clear formation of diamond and a phase separation of carbon and hydrogen in the megabar regime. It is quite easy to misinterpret superionicity in DFT-MD when it is actually a demixing process that should occur, since the finite-size effects tend to overstabilize the superionic structure. There are also a few shortcomings in the interpretation of the results (see details below). For all these reasons I do not believe this work can be published. I append below detailed comments to justify this recommendation and to help strengthening this work.\nReply: We thank the reviewer for the thoughtful and critical assessment of our work. We fully acknowledge the experimental evidence showing that C–H systems tend to dissociate into diamond and hydrogen in the megabar regime. Our own static calculations also confirm that no C–H compound remains stable between 500 and 1000 GPa. We removed the isochores below 1000 GPa from the phase diagram to avoid potential misunderstandings. However, existing experiments are currently limited to pressures up to ∼300 GPa, whereas the new CH compound we report emerges above 1000 GPa—conditions that have not yet been experimentally accessible. Our study, therefore, aims to extend the theoretical understanding of C–H chemistry into this ultra-high-pressure regime, complementing rather than contradicting current experimental observations. To ensure that the predicted superhelical behavior is not an artifact of finite-size effects or simulation details, we performed additional defect-stability tests, large-scale molecular dynamics, and two-phase coexistence simulations. All these independent checks consistently demonstrate that the superhelical and superionic phases persist above 1000 GPa, supporting their intrinsic thermodynamic stability in this previously unexplored pressure–temperature range.\nDetailed comments: - numerous experimental studies led by Dominik Kraus’ team have shown that almost any CxHy plastics dissociate into diamond and phase separated hydrogen in the megabar regime. The conclusion from the authors regarding the superionicity of the system is thus questionable. Indeed, it is possible that finite-size effects overstabilize the homogeneous system. I would recommend to perform a few additional tests to ensure the reality of this system. The first one would be the most robust but also the most expensive. I would suggest to perform a careful thermodynamic integration to compare the homogeneous superionic system with the dissociated one. If this is too intricate to perform, it would be interesting to use the largest cell and introduce a defect with one carbon and one hydrogen removed to see if the cell remains stable. An alternative would be to start from a liquid simulation and cool it down slowly to see if a superionic phase appears. Without such a study, it is not possible to entirely justify that the superionic phase is stable, in my opinion.\nWeappreciatethereviewer’sconstructivesuggestionsandagreethatfinite-sizeeffectscansome- times overstabilize ordered or homogeneous configurations in DFT-MD simulations. Following the reviewer’s recommendations, we performed several additional tests to confirm the robustness of the predicted superionic CH phase. First, we constructed a 3×3×3 supercell (324 atoms) and introduced a vacancy by removing one carbon and one hydrogen atom (Fig. R19). DFT-MD simulations indicate that the CH compound remains in a superhelical state at 2000 K, with the carbon framework preserved and hydrogen continuing its helical diffusion within the channels. A small number of hydrogen atoms near the vacancy can spill into the cavity created by the missing carbon, but their motion remains locally confined rather than freely diffusive. At 4000 K, the CH compound remains in a superhelical state but exhibits stronger vibrations. The hydrogen atoms begin to move freely throughout the lattice at 5000 K, marking the onset of the superionic state. These simulations demonstrate that the CH compound retains both the structural integrity of the carbonframeworkandthecharacteristichydrogendiffusionbehavioreveninthepresenceofapoint defect, highlighting the intrinsic robustness of the superhelical phase.\nFig. R19: Atomic trajectories of the CH compound at a, 2000 K; b, 4000 K; c, 5000 K; and d, 6000 K. Carbon and hydrogen atoms are shown in brown and silver, respectively. The dashed blue box marks the region where a carbon atom is removed and the channel structure is disrupted.\nDue to the limited accessible box size (324 atoms) and simulation time (12 ps) in FPMD, we further trained a machine-learning potential (MLP) for the CH compound using the DeepMD framework implemented in the LAMMPS package. The training dataset consists of DFT-MD trajectories spanning 500–3000 GPa and sampling all relevant states—solid, superhelical, superionic, and liquid. The final root-mean-square errors (RMSE, Fig. R20) are 0.008 eV/atom for energies and 0.477 eV/˚Afor forces, enabling large-scale MD simulations. We assessed finite-size effects using supercells from 324 to 3000 atoms and computed the hydrogen mean-squared displacement at 2000 K and 4000 K under 1000 GPa (Fig. R21a,b). Hydrogen diffusivity converges for system sizes above 1500 atoms with only minor fluctuations. Additionally, radial distribution functions (RDFs; Fig. R21c,d) for C–C, C–H, and H–H pairs show excellent agreement between MLP-MD (scatter) and DFT-MD (lines) for both superhelical and superionic trajectories, confirming the reliability of the MLP-MD simulations with 1500-atom supercells. Thus, we introduced a vacancy by removing one carbon and one hydrogen atom in a 5×5 ×5 supercell (1500 atoms) and performed MLP-MD simulations of the CH compound at 1000 GPa. The carbon framework remains rigid at both 2000 K and 4000 K, even after 1 ns of simulation (Fig. R22). At 2000 K, the hydrogen atoms are constrained to diffuse along the helical channel. At 4000 K, the hydrogen can diffuse freely in the xy plane and jump from one channel to another. These long-time, large-cell MLP-MD simulations demonstrate that both the superhelical and superionic states are intrinsically stable at 1000 GPa. Movies of the defect-containing superhelical and superionic trajectories have been included in the Supplementary Information.\nFig. R20: Training error of the machine-learning potential (MLP) for the CH compound. Comparison between DFT reference data and predictions from our MLP model. The agreement is excellent for a potential energies and for forces along the b x, c y, and d z directions, demonstrating the reliability of the machine-learning simulations.\nFig. R22: MD snapshots of the CH compound using the machine-learning potential (MLP) at 1000 GPa. Snapshots of the superhelical state at 2000 K (a, b) and the superionic state at 4000 K (c, d) are shown at 1 nanosecond, viewed along the ⟨001⟩ and ⟨100⟩ directions in the unwrapped simulation cell. Carbon and hydrogen atoms are shown in brown and light yellow, respectively.\n- the authors insist significantly on possible consequences for planetary interiors. This is not sufficiently justified since the composition is not pure CH in the interior. A significant proportion of oxygen and nitrogen must also be present as well as additional hydrogen. Is the double-helix still stable in this case? Militzer (PNAS 2024) showed that it was possible to form a CNH compound at high pressure. I believe that the authors need to soften their conclusion regarding the impact on the planetary interiors.\nFig. R23: Ternary convex hull for a C–N–H and b C–O–H systems at 1000 GPa from crystal structure prediction. Stable compositions lying on the convex hull are marked with red outlines. Thecolorscaleshowstheenergydifferencerelativetotheconvexhull, whereyellowcolorsrepresent stable compounds and blue colors represent metastable compounds.\nReply: We thank the reviewer for this thoughtful comment. We agree that the interiors of icy giants contain complex mixtures of O, N, C, and H rather than pure CH compounds. To address this, we performed extensive crystal structure predictions for the ternary C–N–H and C–O–H systems at 1000 GPa (Fig. R23). We found that a new CN H compound lies on the convex hull, 2 2 indicating thermodynamic stability, which is consistent with the findings of Militzer (PNAS, 2024). In contrast, no stable compounds were identified in the C–O–H system under the same conditions. We acknowledge that the ice layers of icy giants are composed of complex mixtures where multiple compounds and states may coexist. Our study focuses on the binary C–H system as a simplified modeltoelucidatetheintrinsicstructuralandtransportpropertiesofCHcompoundsunderextreme pressures. While this simplification does not capture the full chemical complexity of planetary interiors, it provides valuable insights into the electrical and ionic conductivities of C–H phases, 31 whichmaycontributetothegenerationofplanetarymagneticfields. Accordingly, wehavesoftened the discussion on planetary implications and emphasized that our results primarily provide insight into the possible behavior and transport properties of C–H compounds under extreme pressure and temperature conditions.\n- the authors only show one phase from their random structure search. How distant are the others in terms of enthalpy? The authors only consider a possible decomposition in CH2 but there must be many other possible compounds and compositions. I think the authors could provide more discussion on this part.\nFig. R24: Pressure-dependent enthalpy. a, Enthalpy of CH relative to the Cmce phase. b, Enthalpy of CH relative to the P3m1 phase.\n- the authors claim that the C-chain is connected with sp3 bonds. How can they demonstrate the nature of this bond?\nReply: In our CH compound, the carbon atoms adopt the same chiral framework structure as reported in the previous study by Chris J. Pickard et al. (Phys. Rev. B, 2010), in which all carbon atoms exhibit sp3 hybridization. To illustrate this more clearly, we present the carbon framework of the CH compound along with the corresponding electron localization function (ELF) in Fig. R25. Each carbon atom is coordinated by four neighboring carbon atoms, and the corresponding bond angles range from 104◦ to 124◦. Although slightly distorted from the ideal tetrahedral angle of 109.5◦, this angular distribution is fully consistent with a distorted sp3-bonded network 32 characteristic of chiral carbon frameworks. We have uploaded the CIF files for all of these phases.\nFig. R25: Electron localization function of C chain in CH compound at 1000 GPa (isovalue = 0.8).\n- I am surprised the energy cutoff is different for the “first-principles calculations” and for the “ab initio molecular dynamics” although they must use the same pseudo-potentials. 600 eV seems really low.\nReply: Weappreciatethereviewer’sconcern. Inourstaticcalculations(includingstructuralrelax- ation, self-consistent field, band structure, and phonon calculations), we employed a hard version of the pseudopotential, which requires a large plane-wave cutoff energy of 1500 eV. However, in the molecular dynamics simulations, using such a hard pseudopotential is computationally expensive. Therefore,weinitiallyusedastandard(softer)versionofthepseudopotentialwithacutoffenergyof 600 eV to accelerate the simulations. To ensure full consistency, we have now repeated all MD simulations using the hard pseudopotential with a cutoff energy of 1500 eV in the revised manuscript. The results show only minor effects on atomic diffusivity, and the overall phase diagram remains essentially unchanged.\n- I do not understand the purpose of section S1. The authors must explain why it is relevant to discuss superconductivity\nReply: We have moved the superconductivity results to the Supplemental Materials, as they pertain to physical properties that are not directly relevant to the atomic dynamics and structural behavior at high temperatures.\n- did the authors consider the effects of a higher order form for the potential than the 2nd order for the phonon calculation? It is common that hydrogen bearing compound presents a quadratic form since the hydrogen bond can be weak. See the example of pure hydrogen in Monserrat et al. (PRL 2018).\nReply: We thank the reviewer for this insightful comment. We agree that the harmonic phonons may be partially violated in quasi-1D systems under extreme pressures. We also performed phonon calculations with anharmonic effects with the SSCHA method at 300 and 1000 GPa (Fig. R8). The SSCHA minimization already includes four phonon scatterings at the lowest order perturbation theory. The CH compound remains dynamically stable when anharmonic effects are included, although the phonon frequency range is slightly reduced at both pressures. We further estimate the formation energy of CH, relative to elemental C and H at 1000 GPa, to be 5 meV per atom with anharmonic corrections—lower than the 10 meV per atom obtained within the harmonic approximation. Compared to the harmonic case, anharmonicity lowers the estimated superconducting transition temperature from 160 K to 123 K (Table R2). We have clarified these points in the revised manuscript and added the SSCHA results for direct comparison.\nWe thank the reviewer for the thorough assessment of our manuscript and for the valuable suggestions. In response, we have carefully revised the manuscript to address all comments and concerns, improving the clarity, rigor, and consistency of the analysis. Below, we provide a detailed point-by-point response to each comment.\n• Replaced the term “superhelical” with the more physically precise description “quasi–one-dimensional superionic” throughout the manuscript.\n• Revised Fig. 2(d,e) to clarify the structural features discussed in the text.\n• Added a new Fig. S5 to the Supplemental Materials and crystal orbital Hamilton population (COHP) analysis in the last paragraph on page 2 to provide a quantitative characterization of chemical bonding.\n• Reworded the discussion of the diamond carbon reference in the 4th paragraph on page 2 to clarify its purpose and limitations.\n• Added an explicit note regarding the two-phase coexistence method in the caption of Fig. 4 and in the second paragraph on page 5.\n• Reworded the discussion of planetary isentropes in the third paragraph on page 5 to reflect updated models and to clarify their implications.\n• Highlighted the significance of the newly identified transport and dynamical behavior in the fourth paragraph on page 6.\nThe authors have satisfactorily addressed my comments as well as those raised by the other referees. However, the revisions were not consistently highlighted in blue, which made it difficult to distinguish newly added text from the original version, as both appear largely in black. I would encourage the authors to ensure consistent use of revision highlighting in future submissions to facilitate review.\nPrecisely because of this, I could not find where is that “Figure 2c is now explicitly referenced in the main text.” The authors do not indicate where, nor the quote the text the added in the manuscript to make this reference explicit.\nReply: Wethanktherefereeforthepositiveassessmentandforraisingthesepoints. Inthecurrent revisedmanuscript, allchangesarenowconsistentlyhighlightedinbluetoclearlydistinguishnewly added or modified text. Regarding Fig. 2c, it is explicitly referenced in the 4th paragraph of the RESULTS section and 2nd paragraph of page 4. We regret that this was not clearly indicated in our previous response and have now clarified its location for ease of review.\nThe authors would greatly benefit from discussing and incorporating insights from the recently published study by Militzer (2025, ApJ, 990, 20): “Ab Initio Entropy Calculations of Water Predict the Interiors of Uranus and Neptune to Be 15%–30% Colder than Previous Models.” This work suggests that the interiors of Uranus and Neptune are significantly colder than previously estimated (below ∼4000 K), which increases the potential relevance and stability window of the proposed superhelical CH phase under planetary conditions. This information was not available when I submitted my first report and substantially changes the interpretation, as it contradicts my earlier statement that the interiors of Uranus and Neptune would be too hot to host this phase. I recommend that the authors add the updated isentropes to Figure 4 or discuss them explicitly in the text, as doing so would strengthen the planetary implications of their findings and bring the discussion up to date with the latest models.\nReply: We thank the referee for highlighting the recent study by Militzer (ApJ 2025) and for noting its important implications for ice-giant interior temperatures. We have now incorporated this work in both phase diagram (Fig. 4 and its caption) and 3rd paragraph in page 5.\nThetwo-phasesimulationsareagreatadditiontothemanuscript. Asexpected,theyyieldlower melting temperatures than those obtained with the heat-until-it-melts approach originally used by theauthors. However,thephasediagraminFigure4hasnotbeenupdatedaccordingly. Theauthors report that “upon heating to 4900 K, the entire system transformed into the liquid phase” in their two-phase simulations near 1000 GPa, yet in Figure 4 this point (∼1000 GPa, 5000 K) remains classified as superionic. I understand that fully revising the superionic-liquid phase boundary for consistency would require re-running additional two-phase simulations, which is not strictly necessary at this point. Instead, I suggest that the authors acknowledge this discrepancy explicitly inthecaptionofFig. 4,forexamplebyaddinganotesuchas: ”Whitelinesindicatephasetransition boundariesobtainedbyaveragingthebracketingtemperaturesandpressuresderivedfromtheheat- until-it-melts approach. Actual melting temperatures may be 600-800 K lower (see Supplementary Section S11).” A functional fit that is more physical than a 3rd order polynomial is the Simon- 2 Glazel fit, which could be useful for readers: http://link.aps.org/doi/10.1103/PhysRevB.65.052102\nReply: We have explicitly acknowledged this discrepancy in the caption of Fig. 4 and refitted the phase boundary with Simon–Glatzel approach.\nIn the Supplementary Material, Section S5, the authors likely intend the title “Analogous configurations” rather than “Analogy configuration.” The ELF comparisons between AgF3 and AuF3 presentedinthissectionareuseful, butthecentralpurposeofthisanalysisshouldbetorelatethese structures to CH. To make this connection clearer, the authors should include a third supercell in Fig. S5 showing the ELF of CH in the same orientation (replicate vertically, if necessary). Fur- thermore, in the newly added paragraph in the manuscript, the authors state that “The electronic localization function (ELF) indicates the absence of C–H bonding, with helical H chains forming quasi-one-dimensional metallic channels, unlike the electron distributions in molecular hydrogen phases...” However, no corresponding ELF plot for CH is shown to support this claim. Including such a figure would make this statement demonstrably clear and back up the overall argument.\nReply: We have corrected the title of Section S4 to “Analogous configurations.” To strengthen the connection to CH, we have added the ELF of a 1×1×3 supercell of the CH compound to Fig. S4, shown in the same crystallographic orientation as AgF and AuF . 3 3\nAll other concerns I raised have been carefully addressed by the authors to my satisfaction. I supportthepublicationofthismanuscriptaftertheauthorshaveincorporatedtheminorsuggestions I listed above.\nReply: We thank the referee for the careful evaluation of our manuscript and for the positive assessment. We appreciate the support for publication and have incorporated all of the minor suggestions noted above.\nI was pleased to see that the authors carefully read all my comments and tried to answer to all of them. The authors addressed all my concerns regarding the level of theory and the calculations, and I have almost no more comment on those aspects.\nHowever, I still have a few concerns regarding some of the discussed results:\nReply: We thank the referee for the careful evaluation of our manuscript and for the positive assessmentofourresponsestothemethodologicalandcomputationalcomments. Below,werespond in detail to the remaining points regarding the interpretation and discussion of the results.\n(1) The choice of the superhelical state terminology is likely to become misleading. I have read the previous comment by the author, and I still believe that the confined superionicity in a 1-D channel, and the fact that this channel has the structure of a helix, are two different things. The helical motif is structural, while the different melting temperature between carbon and hydrogen is the phase transition to superionic. If the carbon motif was different, like a large nanotube, that would probably change the transition temperature, but the two systems (nanotube and helix) would still be two pseudo 1-D superionic systems. In fact, the plateau in x and y described by the authors might still be obtained in motifs that are not helical. Therefore, I remark that the two aspects, carbon motif and type of superionic transition, are different, and that mixing them in one term – superhelix – would not have any real meaning. I understand the intention to find a term to describe the 1-D superionicity – 2-D plasticity, or pseudo 1-D superionicity. However, that should probably not be related to the motif.\nReply: We thank the referee for this thoughtful and important clarification and agree with the underlying physical distinction being emphasized. We no longer use the term “superhelical” to denote the phase itself. Instead, we refer to it as a quasi–one-dimensional superionic state, while describing the helical carbon framework explicitly and separately as a structural feature of the host lattice.\n(2) It is not clear to me how C can be in a sp3 configuration and still show an helix structure. If so, it should have a near tetrahedral coordination, with two bonds and two lone pairs. However, in that case, the lone pairs should possess one un-paired electron each, and therefore, being a radical. Did the authors check for spin polarized solutions? Or are the carbons bonded in through inter-helix C-C bonds?\nReply: We thank the referee for raising this important point regarding the bonding nature of carbon in the helical framework. In CH, carbon and hydrogen form independent helical sublattices. Withinthecarbonhelices, eachcarbonatomisbondedtotwonearest-neighborcarbonatomsalong the helix and to two additional carbon atoms via inter-helix C–C bonds, resulting in an sp3-bonded carbon framework. This is explicitly demonstrated in revised Fig. 2d and e, where we show a larger supercell that clearly reveals the presence of inter-helix C–C bonding.\n(3) The authors wrote “According to Hirshfeld charge analysis (Table S3) performed with the CP2K code [53], a substantial charge transfer of approximately 0.42 e- from hydrogen to carbon is revealed, suggesting that the CH compound consists of two covalently bonded helical chains of carbon and hydrogen.” This is somehow unclear. How can a charge transfer be connected to covalent bond? The two things are not connected. To check for covalent bonds, the authors should calculate quantities related to the orbitals overlap (e.g. ICOHP or ICOOP), and not charge transfers. In fact, if hydrogen loses electrons, the covalency should decrease. In this context, the ELFs give a better idea of covalency, even if it is more qualitative; but there is no obvious connection, in this case, between charge transfer and covalency. Therefore, I do not think that the added sentence is necessarily correct (or complete) from a chemical and bonding point of view.\nFig. R1: Projected COHP analysis of the CH compound at 1000 GPa, showing C–C, H–H, and C–H pairs. The dashed vertical line denotes the energy of the top of the valence band.\nReply: We thank the referee for this important clarification. Following this comment, we added discussion on orbitls overlap (COHP) and reworded the sentence in 4th paragraph of RESULTS section. To characterize the chemical bonding more rigorously, we have performed crystal orbital Hamilton population (COHP) analysis using the LOBSTER code (Fig. R1). The projected COHP (pCOHP) results show strong covalent bonding interactions for C–C and H–H pairs, while no significant bonding interaction is found between C and H atoms, consistent with the ELF analysis. In particular, we observe pronounced negative −pCOHP features in the energy range ∼-4 to -8 eV below the top of the valence band, indicating antibonding character for C–H interactions. This is consistent with the absence of C–H covalent bonds and supports a picture in which carbon and hydrogen form distinct sublattices rather than covalently bonded chains.\n(4) The authors mention the superconductivity at 300 GPa, but their compound is unstable already at 1000 GPa. What is the Gibbs free energy difference, compared to C and H, or any other stable phase, at that pressure? Does it make sense to calculate superconductivity for a very unstable phase?\nReply: We have added free-energy profiles at 300 GPa in Fig. S3. At this pressure, the CH compound is energetically unstable relative to the reference diamond carbon phase and the C2/c- 24 hydrogen phase. In the revised manuscript, we have therefore moved the superconductivity calculations for this metastable phase from the main text to the Supplemental Materials.\n(5) General comment, not strictly related to something done by the authors, but more to what was proposed by one of the reviewers. The authors had to add the following sentences in the SI and main text “Given this experimental uncertainty, we re-evaluated the enthalpy of formation of CH by using diamond, rather than BC8, as the carbon reference (Fig. S2). This comparison shows that adopting diamond as the reference phase increases the relative stability of the CH compound,” and then “While BC8 is predicted to be the lowest-enthalpy carbon phase above 1000 GPa by DFT calculations, diamond has been observed to remain stable well beyond the pressure range where BC8 is theoretically predicted to become favorable in experiments [47–49]. Given this experimental uncertainty, we also recalculated the enthalpy of formation (Fig. S2) using the diamond phase as the reference for carbon. We found that diamond C can increase the stability of the CH compound, while the CH2 compound remains metastable above the convex hull. We calculated free energy profiles, including vibrational contributions, to assess the temperature effect on CH stability (Fig. S3). Both CH and CH2 show slightly reduced stability with increasing temperature, while CH remains thermodynamically stable at 2000 GPa and 1000 K.” The choice of an unstable reference state will automatically make anything else more stable, but that does not make it thermodynamically stable (not on the real convex hull). However, proposing of using a reference state, like diamond at 2 TPa, just because it was not observed in one experiment in which a carbon based material was shocked-compress for 2 ns, is conceptually wrong from the computational point of view, which instead, should take the most stable known phase as reference for the calculation of the enthalpy and the Gibbs free energy. Otherwise, they should do something similar also for hydrogen, since the Cs-IV type of structure has never been observed to date, which again, would be wrong.\nReply: We thank the referee for this important conceptual clarification and fully agree with the concern raised. we have accordingly restructured and reworded the discussion involving diamond carbon in 2nd paragraph of RESULTS section and in Section S2 of Supplemental Materials. The comparisonusingdiamondcarbonhasbeenretainedonlyasalimitedsensitivityanalysis, explicitly framed to illustrate how experimental uncertainties in the persistence of carbon polymorphs might affect relative formation enthalpies, without implying equilibrium stability. With these revisions, the manuscript now adheres strictly to the correct computational thermodynamic definition of stability, and avoids any misleading interpretation arising from the choice of a metastable reference phase.\nTherevisedmanuscriptincludessignificantnewmaterialswithaseriesofadditionalcalculations. Itisclearthattheauthorswenttogreatlengthtoincludeallthenumeroussuggestionsofthereferees andtoreplytotheircritics. Theadditionalcalculationsseemtosupportthefindingsoftheauthors regardingthestabilityofthissuper-helicalphase, precursortothesuper-ionicphase. Iacknowledge the effort in the rebuttal from the authors. I find the discussion regarding the conductivity and the diffusion properties quite interesting from a conceptual point-of-view.\nHowever, the predicted super-helical phase is only achievable above 1 TPa. Below this pressure, CxHy mixtures dissociate into pure carbon and hydrogen (confirmed by the authors calculations as well as experimentally). I do not see any planets that would reach the temperature and pressure conditions necessary for the super-helical phase to exist (either they are two low in pressure or to high in temperature). We can also assume that the large scale convection would force the composition to significantly change as eddies are exposed to lower pressure and that only very carbon rich inner layers would be present at such high pressures. The pathways to reach the necessary conditions to obtain this super-helical phase are extremely narrow. For this reason I do not think it can have any significance for planetology. But this is also true for the synthesis of this compound experimentally. No experiment is at the moment able to reach this kind of conditions unless maybe ramp compressions at MJ-class laser facilities. But then you need to make sure to be fast enough to prevent any demixing to occur as you go in the 100 GPa regime yet keep everything cool enough to prevent the temperature to increase. Again, this seems very unrealistic and I do not think the high-pressure community will be enthusiastic regarding the results of this work.\nAlthough the results are definitely interesting from a physics/conceptual point of view and bring some interesting mechanisms to go from a solid phase into a superionic phase, I do not think the results are of sufficient significance to be published in Nature Communications. I would recommendthepublicationinamorespecializedjournal. Iftheauthorschoosetodoso,Ihaveafew recommendations in order to ensure the consistency of their presentation and the reproducibility of their work.\nReply: We thank the referee for the careful re-evaluation of the manuscript and for recognizing the substantial effort invested in the revisions, as well as the conceptual interest of the results, particularly regarding transport and diffusion mechanisms. We also appreciate the referee’s candid assessment concerning physical realizability, planetary relevance, and experimental feasibility.\nWe fully agree that the super-helical phase is stabilized only at ultra-high pressures (>1 TPa) andthatbelowthisregimeC–Hmixturesdissociateintoelementalcarbonandhydrogen, consistent withbothourcalculationsandexistingexperimentalevidence. Wefurtheragreethatsuchpressure– temperatureconditionsareunlikelytobebroadlyrealizedinplanetaryinteriors,andthatlarge-scale convection, compositional stratification, and thermal effects would strongly limit the persistence of mixed C–H phases. Likewise, we acknowledge that experimental realization would be extremely challenging and, at present, speculative.\nHowever, we respectfully disagree with the implication that these constraints diminish the scientificsignificanceoftheworkatthelevelrequiredforNature Communications. Thecentralcontri- 7 bution of this manuscript is not the prediction of a practically realizable planetary or experimental phase, but rather the identification of a previously unrecognized transition mechanism—the quasi- 1Dsuperionicstate—thatprovidesaconceptualbridgebetweenconventionalplasticandsuperionic matter under extreme confinement.\nSpecifically, this work demonstrates that helical confinement can give rise to a distinct intermediate regime characterized by coupled structural order and ionic mobility, that superionic behavior can emerge through a topologically constrained pathway rather than simple sublattice melting, and that bonding, transport, and diffusion can become strongly anisotropic and decoupled in light-element systems under extreme compression. These insights are general and transferable, extending beyond the specific C–H system studied here, and are relevant to a broad community spanning condensed-matter physics, high-pressure chemistry, and materials science.\nIn response to the referee’s concerns, we have substantially revised the manuscript to make this distinction explicit in 4th paragraph of page 6. The revised version removes claims of direct planetaryornear-termexperimentalrelevance, clearlypresentsthequasi-1Dsuperionicphaseasan extreme-condition limiting case, and foregrounds the transition mechanism and transport physics as the primary contribution, with realizability discussed only as contextual motivation.\nRegarding journal scope, while we respect the referee’s recommendation for a more specialized venue, we believe that the revised manuscript now clearly articulates a broad conceptual advance aligned with the mission of Nature Communications: to publish work that introduces new physical mechanisms of general interest, even when realized under extreme or currently inaccessible conditions.\nWe are grateful for the referee’s forthright and insightful comments, which have helped us sharpen both the framing and the scientific message of the work, and we believe the revised manuscript now addresses these concerns directly.\nDetailed comments:\n- Numerical details are missing. The nature of the machine learning potential is not provided. What descriptors were used? Is a linear relation assumed? What are the hyper-parameters?\nReply: We thank the referee for pointing out the need to clarify the numerical details of the machine-learning potential. We have now added a full description of all relevant hyperparameters, including network architectures, cutoff radii, loss-function weights, and training schedules in the DeepMD framework in the Section S7 of Supplemental Materials.\n- The RMSE figure show some significant outliers on the forces and I believe they are the reason of the discrepancy on the height of the peaks in the pair distribution functions. I think that the MLIPs can only be considered as qualitative, and not sufficiently accurate to offer a real prediction.\nReply: We thank the referee for raising this important point regarding force outliers and the reliability of the machine-learning potential (MLP). In response, we have further refined and validated the DeepMD potential. As shown in the updated Fig. S8, the MLP now achieves an energy RMSE of 5 meV/atom and a force RMSE of ∼0.33 eV/˚A, which are well within—and in fact improve upon—the accuracy range commonly accepted for MLPs applied to strongly compressed, high-temperature systems. 8 With this improved force accuracy, the previously noted discrepancy in the height of the peaks in the pair distribution functions has been resolved (Fig. S9 in Supplemental Materials). The MLP-based PDFs are now in good quantitative agreement with those obtained from direct ab initio molecular dynamics at representative state points, confirming that residual force outliers do not affect the structural characterization.\nIn this work, the MLP is employed to access the long time and length scales required to characterize thermodynamic trends, diffusion coefficients, and transport mechanisms that are beyond the reach of direct first-principles simulations. Within this intended scope, an energy error of a few meV/atom and a force RMSE of approximately 0.3 eV/˚A provide quantitatively reliable predictions of phase stability, ionic mobility, and dynamical behavior under extreme conditions. While we agree that sub-percent-level variations in PDF peak amplitudes should not be over-interpreted, the improved agreement with DFT confirms that the main conclusions of this work are robust.\nWe have revised the manuscript accordingly to report the updated error metrics, additional moleculardynamicsvalidationtests,representativestructuralsnapshots,andsupplementarymovies. These additions document the improved agreement between MLP-based and first-principles pair distributionfunctions,andclearlydelineateboththecapabilitiesandthelimitationsofthemachine- learning potential used in this study.\n- It is unclear what charge the authors chose in their calculations of the ionic conductivity.\nReply: In the Nernst–Einstein conductivity estimates, we used nominal effective charges of +4e to carbon and +1e to hydrogen. These values are not intended to represent formal oxidation states but serve as a standard approximation to estimate transport trends, with conductivity dominated by hydrogen diffusion. We have clarified this assumption in Section S10 of Supplemental Materials.\n- The authors wrote in their rebuttal: “The melting points identified from the two-phase coexistence simulations are substantially lower—by about 600–800 K—than those obtained from the heat-until-it-melts method, indicating that the latter does not hugely overestimate the transition temperatures.” It seems to me that then heat-until-it-melts simulations do overestimate the transition temperatures.\nReply: WehaveaddedthefollowingnotetothecaptionofFig.4: “Theactualmeltingtemperatures may be 600–800 K lower, as indicated by two-phase coexistence simulations (see Section S8 of the Supplemental Materials).” Wehavealsorewordedthecorrespondingsentenceinthe2ndparagraph on page 5 to ensure consistency and clarity.\n- Fig S1 and S2 seem to be in contradiction with S3. In the latter, CH2 has a positive excess enthalpy at 0K, which is not the case on the other figures. It cast shadows on the results presented by the authors.\nReply: Fig. S1 represents enthalpies of different phases in CH and CH compound. Fig. S2 shows 2 relative formation enthalpies referenced to diamond carbon rather than the lowest-enthalpy BC8 carbonphase,followingaspecificrequestbyanotherrefereetoassessthesensitivityoftheresultsto the choice of carbon reference. As a result, adopting diamond as the reference phase systematically shifts the relative enthalpies and can make compounds such as CH appear more stable, even 2 thoughthisdoesnotimplytruethermodynamicstability. Wehavenowaddedanexplicitdiscussion 9 clarifying the different reference choices and their implications in Section S2 of the Supplemental Materials. By contrast, Fig. S3 presents the physically relevant convex-hull analysis, including vibrational contributions, constructed using the lowest-enthalpy reference phases. We emphasize thatFig.S2isintendedsolelyasacomparativereferenceillustratingtheeffectofthechosencarbon phase, whereas Fig. S3 reflects the actual thermodynamic stability with convex hull.\n- Fig 4 indicates the super-helical phase as stable at 1000 GPa, but figures S1-S3 seem to say otherwise.\nReply: Fig. S1–S3 present static and vibrational free-energy analyses, showing that CH becomes thermodynamicallystableonlyabove∼1100GPawhenzero-pointenergyandfinite-temperaturevi- brationalcontributionsareincludedwithinthequasiharmonicapproximation(QHA).At1000GPa, CH remains dynamically stable, with no imaginary phonon modes, but is thermodynamically metastable. To eliminate potential ambiguity, we have added a dashed vertical line along the pressure axis in Fig. 4 to explicitly distinguish the metastable regime from the thermodynamically stable pressure range. The revised figure and caption now clearly indicate that CH is thermody- namically stable only above ∼1100 GPa, whereas the region around 1000 GPa corresponds to a metastable yet dynamically stable phase.\nReviewer 1\nThe authors have addressed all my concerns. I have no further comments. I recommend this manuscript for publication as is.\nReply: We thank the reviewer for the positive assessment and recommendation for publication.\nReviewer 4\nThe resubmitted manuscript, supplementary material and rebuttal include satisfactory replies to all of the concerns I raised in my previous report. I appreciate the effort put in by the authors and I believe that the manuscript is now much stronger scientifically. I am still very cautious about any applicability for the planetary interiors as the dynamics is going to significantly modify the composition even in the deepest regions. However, the conceptual advances brought by the present work and the tools developed to characterize the new phase are useful for the high-pressure community as well as for the condensed matter community.\nI thus recommend the publication of the current manuscript without further correction.\nReply: We thank the reviewer for the careful reassessment and for the positive evaluation of the revised manuscript. We fully acknowledge the reviewer’s caution regarding direct applicability to planetary interiors, and we are grateful for the recommendation for publication.\n1"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Please find my review attached.", "reviewer_4": "The revised manuscript includes significant new materials with a series of additional calculations. It is clear that the authors went to great length to include all the numerous suggestions of the referees and to reply to their critics. The additional calculations seem to support the findings of the authors regarding the stability of this super-helical phase, precursor to the super-ionic phase. I acknowledge the effort in the rebuttal from the authors. I find the discussion regarding the conductivity and the diffusion properties quite interesting from a conceptual point-of-view.\nHowever, the predicted super-helical phase is only achievable above 1 TPa. Below this pressure, CxHy mixtures dissociate into pure carbon and hydrogen (confirmed by the authors calculations as well as experimentally). I do not see any planets that would reach the temperature and pressure conditions necessary for the super-helical phase to exist (either they are two low in pressure or to high in temperature). We can also assume that the large scale convection would force the composition to significantly change as eddies are exposed to lower pressure and that only very carbon rich inner layers would be present at such high pressures. The pathways to reach the necessary conditions to obtain this super-helical phase are extremely narrow. For this reason I do not think it can have any significance for planetology. But this is also true for the synthesis of this compound experimentally. No experiment is at the moment able to reach this kind of conditions unless maybe ramp compressions at MJ-class laser facilities. But then you need to make sure to be fast enough to prevent any demixing to occur as you go in the 100 GPa regime yet keep everything cool enough to prevent the temperature to increase. Again, this seems very unrealistic and I do not think the high-pressure community will be enthusiastic regarding the results of this work.\nAlthough the results are definitely interesting from a physics/conceptual point of view and bring some interesting mechanisms to go from a solid phase into a superionic phase, I do not think the results are of sufficient significance to be published in Nature Communications. I would recommend the publication in a more specialized journal. If the authors choose to do so, I have a few recommendations in order to ensure the consistency of their presentation and the reproducibility of their work.\nDetailed comments:\n- Numerical details are missing. The nature of the machine learning potential is not provided. What descriptors were used? Is a linear relation assumed? What are the hyper-parameters?\n- The RMSE figure show some significant outliers on the forces and I believe they are the reason of the discrepancy on the height of the peaks in the pair distribution functions. I think that the MLIPs can only be considered as qualitative, and not sufficiently accurate to offer a real prediction.\n- It is unclear what charge the authors chose in their calculations of the ionic conductivity.\n- The authors wrote in their rebuttal: “The melting points identified from the two-phase coexistence simulations are substantially lower—by about 600–800 K—than those obtained from the heat-until-it-melts method, indicating that the latter does not hugely overestimate the transition temperatures.” It seems to me that then heat-until-it-melts simulations do overestimate the transition temperatures.\n- Fig S1 and S2 seem to be in contradiction with S3. In the latter, CH2 has a positive excess enthalpy at 0K, which is not the case on the other figures. It cast shadows on the results presented by the authors.\n- Fig 4 indicates the super-helical phase as stable at 1000 GPa, but figures S1-S3 seem to say otherwise."}, "author_response": "We thank the reviewer for the thorough assessment of our manuscript and for the valuable suggestions. In response, we have carefully revised the manuscript to address all comments and concerns, improving the clarity, rigor, and consistency of the analysis. Below, we provide a detailed point-by-point response to each comment.\n• Replaced the term “superhelical” with the more physically precise description “quasi–one-dimensional superionic” throughout the manuscript.\n• Revised Fig. 2(d,e) to clarify the structural features discussed in the text.\n• Added a new Fig. S5 to the Supplemental Materials and crystal orbital Hamilton population (COHP) analysis in the last paragraph on page 2 to provide a quantitative characterization of chemical bonding.\n• Reworded the discussion of the diamond carbon reference in the 4th paragraph on page 2 to clarify its purpose and limitations.\n• Added an explicit note regarding the two-phase coexistence method in the caption of Fig. 4 and in the second paragraph on page 5.\n• Reworded the discussion of planetary isentropes in the third paragraph on page 5 to reflect updated models and to clarify their implications.\n• Highlighted the significance of the newly identified transport and dynamical behavior in the fourth paragraph on page 6.\nThe authors have satisfactorily addressed my comments as well as those raised by the other referees. However, the revisions were not consistently highlighted in blue, which made it difficult to distinguish newly added text from the original version, as both appear largely in black. I would encourage the authors to ensure consistent use of revision highlighting in future submissions to facilitate review.\nPrecisely because of this, I could not find where is that “Figure 2c is now explicitly referenced in the main text.” The authors do not indicate where, nor the quote the text the added in the manuscript to make this reference explicit.\nReply: Wethanktherefereeforthepositiveassessmentandforraisingthesepoints. Inthecurrent revisedmanuscript, allchangesarenowconsistentlyhighlightedinbluetoclearlydistinguishnewly added or modified text. Regarding Fig. 2c, it is explicitly referenced in the 4th paragraph of the RESULTS section and 2nd paragraph of page 4. We regret that this was not clearly indicated in our previous response and have now clarified its location for ease of review.\nThe authors would greatly benefit from discussing and incorporating insights from the recently published study by Militzer (2025, ApJ, 990, 20): “Ab Initio Entropy Calculations of Water Predict the Interiors of Uranus and Neptune to Be 15%–30% Colder than Previous Models.” This work suggests that the interiors of Uranus and Neptune are significantly colder than previously estimated (below ∼4000 K), which increases the potential relevance and stability window of the proposed superhelical CH phase under planetary conditions. This information was not available when I submitted my first report and substantially changes the interpretation, as it contradicts my earlier statement that the interiors of Uranus and Neptune would be too hot to host this phase. I recommend that the authors add the updated isentropes to Figure 4 or discuss them explicitly in the text, as doing so would strengthen the planetary implications of their findings and bring the discussion up to date with the latest models.\nReply: We thank the referee for highlighting the recent study by Militzer (ApJ 2025) and for noting its important implications for ice-giant interior temperatures. We have now incorporated this work in both phase diagram (Fig. 4 and its caption) and 3rd paragraph in page 5.\nThetwo-phasesimulationsareagreatadditiontothemanuscript. Asexpected,theyyieldlower melting temperatures than those obtained with the heat-until-it-melts approach originally used by theauthors. However,thephasediagraminFigure4hasnotbeenupdatedaccordingly. Theauthors report that “upon heating to 4900 K, the entire system transformed into the liquid phase” in their two-phase simulations near 1000 GPa, yet in Figure 4 this point (∼1000 GPa, 5000 K) remains classified as superionic. I understand that fully revising the superionic-liquid phase boundary for consistency would require re-running additional two-phase simulations, which is not strictly necessary at this point. Instead, I suggest that the authors acknowledge this discrepancy explicitly inthecaptionofFig. 4,forexamplebyaddinganotesuchas: ”Whitelinesindicatephasetransition boundariesobtainedbyaveragingthebracketingtemperaturesandpressuresderivedfromtheheat- until-it-melts approach. Actual melting temperatures may be 600-800 K lower (see Supplementary Section S11).” A functional fit that is more physical than a 3rd order polynomial is the Simon- 2 Glazel fit, which could be useful for readers: http://link.aps.org/doi/10.1103/PhysRevB.65.052102\nReply: We have explicitly acknowledged this discrepancy in the caption of Fig. 4 and refitted the phase boundary with Simon–Glatzel approach.\nIn the Supplementary Material, Section S5, the authors likely intend the title “Analogous configurations” rather than “Analogy configuration.” The ELF comparisons between AgF3 and AuF3 presentedinthissectionareuseful, butthecentralpurposeofthisanalysisshouldbetorelatethese structures to CH. To make this connection clearer, the authors should include a third supercell in Fig. S5 showing the ELF of CH in the same orientation (replicate vertically, if necessary). Fur- thermore, in the newly added paragraph in the manuscript, the authors state that “The electronic localization function (ELF) indicates the absence of C–H bonding, with helical H chains forming quasi-one-dimensional metallic channels, unlike the electron distributions in molecular hydrogen phases...” However, no corresponding ELF plot for CH is shown to support this claim. Including such a figure would make this statement demonstrably clear and back up the overall argument.\nReply: We have corrected the title of Section S4 to “Analogous configurations.” To strengthen the connection to CH, we have added the ELF of a 1×1×3 supercell of the CH compound to Fig. S4, shown in the same crystallographic orientation as AgF and AuF . 3 3\nAll other concerns I raised have been carefully addressed by the authors to my satisfaction. I supportthepublicationofthismanuscriptaftertheauthorshaveincorporatedtheminorsuggestions I listed above.\nReply: We thank the referee for the careful evaluation of our manuscript and for the positive assessment. We appreciate the support for publication and have incorporated all of the minor suggestions noted above.\nI was pleased to see that the authors carefully read all my comments and tried to answer to all of them. The authors addressed all my concerns regarding the level of theory and the calculations, and I have almost no more comment on those aspects.\nHowever, I still have a few concerns regarding some of the discussed results:\nReply: We thank the referee for the careful evaluation of our manuscript and for the positive assessmentofourresponsestothemethodologicalandcomputationalcomments. Below,werespond in detail to the remaining points regarding the interpretation and discussion of the results.\n(1) The choice of the superhelical state terminology is likely to become misleading. I have read the previous comment by the author, and I still believe that the confined superionicity in a 1-D channel, and the fact that this channel has the structure of a helix, are two different things. The helical motif is structural, while the different melting temperature between carbon and hydrogen is the phase transition to superionic. If the carbon motif was different, like a large nanotube, that would probably change the transition temperature, but the two systems (nanotube and helix) would still be two pseudo 1-D superionic systems. In fact, the plateau in x and y described by the authors might still be obtained in motifs that are not helical. Therefore, I remark that the two aspects, carbon motif and type of superionic transition, are different, and that mixing them in one term – superhelix – would not have any real meaning. I understand the intention to find a term to describe the 1-D superionicity – 2-D plasticity, or pseudo 1-D superionicity. However, that should probably not be related to the motif.\nReply: We thank the referee for this thoughtful and important clarification and agree with the underlying physical distinction being emphasized. We no longer use the term “superhelical” to denote the phase itself. Instead, we refer to it as a quasi–one-dimensional superionic state, while describing the helical carbon framework explicitly and separately as a structural feature of the host lattice.\n(2) It is not clear to me how C can be in a sp3 configuration and still show an helix structure. If so, it should have a near tetrahedral coordination, with two bonds and two lone pairs. However, in that case, the lone pairs should possess one un-paired electron each, and therefore, being a radical. Did the authors check for spin polarized solutions? Or are the carbons bonded in through inter-helix C-C bonds?\nReply: We thank the referee for raising this important point regarding the bonding nature of carbon in the helical framework. In CH, carbon and hydrogen form independent helical sublattices. Withinthecarbonhelices, eachcarbonatomisbondedtotwonearest-neighborcarbonatomsalong the helix and to two additional carbon atoms via inter-helix C–C bonds, resulting in an sp3-bonded carbon framework. This is explicitly demonstrated in revised Fig. 2d and e, where we show a larger supercell that clearly reveals the presence of inter-helix C–C bonding.\n(3) The authors wrote “According to Hirshfeld charge analysis (Table S3) performed with the CP2K code [53], a substantial charge transfer of approximately 0.42 e- from hydrogen to carbon is revealed, suggesting that the CH compound consists of two covalently bonded helical chains of carbon and hydrogen.” This is somehow unclear. How can a charge transfer be connected to covalent bond? The two things are not connected. To check for covalent bonds, the authors should calculate quantities related to the orbitals overlap (e.g. ICOHP or ICOOP), and not charge transfers. In fact, if hydrogen loses electrons, the covalency should decrease. In this context, the ELFs give a better idea of covalency, even if it is more qualitative; but there is no obvious connection, in this case, between charge transfer and covalency. Therefore, I do not think that the added sentence is necessarily correct (or complete) from a chemical and bonding point of view.\nFig. R1: Projected COHP analysis of the CH compound at 1000 GPa, showing C–C, H–H, and C–H pairs. The dashed vertical line denotes the energy of the top of the valence band.\nReply: We thank the referee for this important clarification. Following this comment, we added discussion on orbitls overlap (COHP) and reworded the sentence in 4th paragraph of RESULTS section. To characterize the chemical bonding more rigorously, we have performed crystal orbital Hamilton population (COHP) analysis using the LOBSTER code (Fig. R1). The projected COHP (pCOHP) results show strong covalent bonding interactions for C–C and H–H pairs, while no significant bonding interaction is found between C and H atoms, consistent with the ELF analysis. In particular, we observe pronounced negative −pCOHP features in the energy range ∼-4 to -8 eV below the top of the valence band, indicating antibonding character for C–H interactions. This is consistent with the absence of C–H covalent bonds and supports a picture in which carbon and hydrogen form distinct sublattices rather than covalently bonded chains.\n(4) The authors mention the superconductivity at 300 GPa, but their compound is unstable already at 1000 GPa. What is the Gibbs free energy difference, compared to C and H, or any other stable phase, at that pressure? Does it make sense to calculate superconductivity for a very unstable phase?\nReply: We have added free-energy profiles at 300 GPa in Fig. S3. At this pressure, the CH compound is energetically unstable relative to the reference diamond carbon phase and the C2/c- 24 hydrogen phase. In the revised manuscript, we have therefore moved the superconductivity calculations for this metastable phase from the main text to the Supplemental Materials.\n(5) General comment, not strictly related to something done by the authors, but more to what was proposed by one of the reviewers. The authors had to add the following sentences in the SI and main text “Given this experimental uncertainty, we re-evaluated the enthalpy of formation of CH by using diamond, rather than BC8, as the carbon reference (Fig. S2). This comparison shows that adopting diamond as the reference phase increases the relative stability of the CH compound,” and then “While BC8 is predicted to be the lowest-enthalpy carbon phase above 1000 GPa by DFT calculations, diamond has been observed to remain stable well beyond the pressure range where BC8 is theoretically predicted to become favorable in experiments [47–49]. Given this experimental uncertainty, we also recalculated the enthalpy of formation (Fig. S2) using the diamond phase as the reference for carbon. We found that diamond C can increase the stability of the CH compound, while the CH2 compound remains metastable above the convex hull. We calculated free energy profiles, including vibrational contributions, to assess the temperature effect on CH stability (Fig. S3). Both CH and CH2 show slightly reduced stability with increasing temperature, while CH remains thermodynamically stable at 2000 GPa and 1000 K.” The choice of an unstable reference state will automatically make anything else more stable, but that does not make it thermodynamically stable (not on the real convex hull). However, proposing of using a reference state, like diamond at 2 TPa, just because it was not observed in one experiment in which a carbon based material was shocked-compress for 2 ns, is conceptually wrong from the computational point of view, which instead, should take the most stable known phase as reference for the calculation of the enthalpy and the Gibbs free energy. Otherwise, they should do something similar also for hydrogen, since the Cs-IV type of structure has never been observed to date, which again, would be wrong.\nReply: We thank the referee for this important conceptual clarification and fully agree with the concern raised. we have accordingly restructured and reworded the discussion involving diamond carbon in 2nd paragraph of RESULTS section and in Section S2 of Supplemental Materials. The comparisonusingdiamondcarbonhasbeenretainedonlyasalimitedsensitivityanalysis, explicitly framed to illustrate how experimental uncertainties in the persistence of carbon polymorphs might affect relative formation enthalpies, without implying equilibrium stability. With these revisions, the manuscript now adheres strictly to the correct computational thermodynamic definition of stability, and avoids any misleading interpretation arising from the choice of a metastable reference phase.\nThe resubmitted manuscript, supplementary material and rebuttal include satisfactory replies to all of the concerns I raised in my previous report. I appreciate the effort put in by the authors and I believe that the manuscript is now much stronger scientifically. I am still very cautious about any applicability for the planetary interiors as the dynamics is going to significantly modify the composition even in the deepest regions. However, the conceptual advances brought by the present work and the tools developed to characterize the new phase are useful for the high-pressure community as well as for the condensed matter community.\nI thus recommend the publication of the current manuscript without further correction.\nReply: We thank the reviewer for the careful reassessment and for the positive evaluation of the revised manuscript. We fully acknowledge the reviewer’s caution regarding direct applicability to planetary interiors, and we are grateful for the recommendation for publication.\n1"}]

Decision letter
NATURE COMMUNICATIONS: REVIEWER REPORT
Date: December 10, 2025
Manuscript Number: NCOMMS-25-45429A
Title: “Thermally driven superhelical state in CH compound under planetary interior conditions”
Authors: Cong Liu, R. E. Cohen, Jian Sun
The authors have satisfactorily addressed my comments as well as those raised by the other referees.
However, the revisions were not consistently highlighted in blue, which made it difficult to distinguish
newly added text from the original version, as both appear largely in black. I would encourage the authors
to ensure consistent use of revision highlighting in future submissions to facilitate review.
Precisely because of this, I could not find where is that “Figure 2c is now explicitly referenced in the main
text.” The authors do not indicate where, nor the quote the text the added in the manuscript to make this
reference explicit.
The authors would greatly benefit from discussing and incorporating insights from the recently published
study by Militzer (2025, ApJ, 990, 20): “Ab Initio Entropy Calculations of Water Predict the Interiors of
Uranus and Neptune to Be 15%–30% Colder than Previous Models.” This work suggests that the
interiors of Uranus and Neptune are significantly colder than previously estimated (below ~4000 K), which
increases the potential relevance and stability window of the proposed superhelical CH phase under
planetary conditions. This information was not available when I submitted my first report and substantially
changes the interpretation, as it contradicts my earlier statement that the interiors of Uranus and Neptune
would be too hot to host this phase. I recommend that the authors add the updated isentropes to Figure
4 or discuss them explicitly in the text, as doing so would strengthen the planetary implications of their
findings and bring the discussion up to date with the latest models.
The two-phase simulations are a great addition to the manuscript. As expected, they yield lower melting
temperatures than those obtained with the heat-until-it-melts approach originally used by the authors.
However, the phase diagram in Figure 4 has not been updated accordingly. The authors report that “upon
heating to 4900 K, the entire system transformed into the liquid phase” in their two-phase simulations
near 1000 GPa, yet in Figure 4 this point (~1000 GPa, 5000 K) remains classified as superionic. I
understand that fully revising the superionic-liquid phase boundary for consistency would require re-
running additional two-phase simulations, which is not strictly necessary at this point. Instead, I suggest
that the authors acknowledge this discrepancy explicitly in the caption of Fig. 4, for example by adding a
note such as: “White lines indicate phase transition boundaries obtained by averaging the bracketing
temperatures and pressures derived from the heat-until-it-melts approach. Actual melting temperatures
may be 600-800 K lower (see Supplementary Section S11).” A functional fit that is more physical than a
3rd order polynomial is the Simon-Glazel fit, which could be useful for readers:
http://link.aps.org/doi/10.1103/PhysRevB.65.052102
In the Supplementary Material, Section S5, the authors likely intend the title “Analogous
configurations” rather than “Analogy configuration.” The ELF comparisons between AgF₃ and AuF₃
presented in this section are useful, but the central purpose of this analysis should be to relate these
structures to CH. To make this connection clearer, the authors should include a third supercell in Fig.
S5 showing the ELF of CH in the same orientation (replicate vertically, if necessary). Furthermore, in the
newly added paragraph in the manuscript, the authors state that “The electronic localization function
(ELF) indicates the absence of C–H bonding, with helical H chains forming quasi-one-dimensional
metallic channels, unlike the electron distributions in molecular hydrogen phases…” However, no
corresponding ELF plot for CH is shown to support this claim. Including such a figure would make this
statement demonstrably clear and back up the overall argument.
All other concerns I raised have been carefully addressed by the authors to my satisfaction. I support the
publication of this manuscript after the authors have incorporated the minor suggestions I listed above.
2