Fate of Pomeranchuk effect in ultrahigh magnetic fields

Domain: physics
Published: 2026-01-12
Editorial outcome: minor_revision

Abstract
The Pomeranchuk effect is a counterintuitive phenomenon where liquid helium-3 (3He) solidifies under specific pressures, not when cooled, but when heated. This behaviour originates from the magnetic entropy of nuclear spins, suggesting a magnetic field should influence it. However, its detailed response to magnetic fields remains elusive due to the small nuclear magneton of 3He and lack of analogous fermion systems. Here, we show that an electron system also exhibit the Pomeranchuk effect, where the Fermi liquid state solidifies in a high magnetic field, unlike conventional electron systems where a field melts an electron solid into a metal. Remarkably, the electron system displays a reentrant liquid state in ultrahigh fields. These responses are explained by changes in magnetic entropy and magnetisation, extending the underlying physics to 3He. Our findings clarify magnetic-field impact on the Pomeranchuk effect and open avenues for magnetic control of chemical interactions.

Introduction
In 1950, Pomeranchuk proposed that extremely low-temperature liquid 3He could be further cooled upon solidification by applying pressure1. The nucleus of 3He consists of two protons and one neutron, making 3He a fermion. As a result, liquid 3He behaves as a Fermi liquid, exhibiting entropy proportional to temperature, S = \gamma T, where S and \gamma represent the entropy and Sommerfeld coefficient, respectively. When solidified under high pressure, localised 3He atoms with nuclear spins I = 1/2 form a paramagnet whose entropy remains nearly constant at S = R ln 2 above 20 mK (Fig. 1a). A comparison of the entropies between the solid and liquid states reveals that the slope of the melting curve (dP/dT) becomes negative below 0.32 K, as illustrated in Fig. 1b. Consequently, at a certain pressure (e.g., 30 bar), increasing the temperature results in entropically driven phase transition to a solid state, in stark contrast to the thermally driven melting observed in typical ordered systems. This counterintuitive phenomenon is known as the Pomeranchuk effect2.

The Pomeranchuk effect is expected to be influenced by magnetic fields due to the magnetic degrees of freedom associated with nuclear spins. This suggests that magnetic-field-induced melting or solidification could occur in 3He. Although previous studies investigated the melting curve of 3He in magnetic fields for its use as a low-temperature thermometry standard3,4, the observed changes were minimal because the applied fields were insufficient to polarise the nuclear spins. Hence, the detailed magnetic-field response of the Pomeranchuk effect remains elusive.

Given the universality of physics, analogous systems exhibiting the Pomeranchuk effect are expected to exist and may serve as useful platforms for exploring their magnetic-field response. In particular, electron systems provide significant advantages over 3He for studying magnetic field effects, as the Bohr magneton of an electron is 1840 times larger than the nuclear magneton of 3He, making electron systems far more sensitive to magnetic fields. For electrons, which are also fermions, a liquid-solid transition corresponds to a metal-insulator transition (MIT), where they are itinerant in the metallic state and localised in the insulating state. When MITs are driven by strong correlations originating from Coulomb repulsion, these correlations manifest as kinetic exchange interactions in the insulating phase, releasing magnetic entropy at high temperatures. As a result, conventional correlated systems do not exhibit the Pomeranchuk effect. Only a limited number of electronic systems exhibiting the Pomeranchuk-like effect have been reported5–12, yet their circumstances differ significantly from those of 3He. For instance, in the Mott transition observed in an organic material7, the strong magnetic interactions in the insulating phase mask the intrinsic magnetic-field effects. To further deepen our understanding of the innate characteristics of the Pomeranchuk effect, it has been desirable to explore another condensed-matter system that faithfully reproduces its original manifestation in 3He.

Here, we revisit the electronic states of the coordination polymer (DMe-DCNQI)2Cu, where DMe-DCNQI represents 2,5-dimethyl-N,N’-dicyanoquinonediimine. One-dimensional DMe-DCNQI columns coordinated by Cu ions form a tetragonal lattice of space group I4/a, showing metallic behaviour down to the lowest temperatures arising from a hybridised electron band composed of DCNQI \pi-electrons and Cu d-electrons13–19. External pressure or chemical substitution can shift the ground state into a Cu2+-originated paramagnetic insulator accompanied by a three-fold superlattice formation, with an intermediate range where a reentrant metal-insulator-metal transition is observed (Fig. 1d)13–15. The reentrant metallic ground state has been identified to share the same electronic state as the metallic state observed in other temperature and pressure regions20. While an earlier study has investigated the entropic relation between metallic and insulating states, which explains the emergence of the reentrant metallic phase, its link to the Pomeranchuk effect remains unexplored19. Given the paramagnetic insulator state of (DMe-DCNQI)2Cu with tiny magnetic interactions, analogous to solid 3He, the reentrant metallic state can be viewed as one manifestation of the Pomeranchuk effect. Indeed, as shown in Fig. 1c, d the temperature dependence of the entropy and phase diagram of (DMe-DCNQI)2Cu19 closely resembles those of 3He (Fig. 1a, b). To explore this connection, we examined (DMe-DCNQI)2Cu from the perspective of the electronic Pomeranchuk effect and investigated its response to magnetic fields. Chemical pressure, the counterpart of physical pressure in 3He, was introduced by mixing deuterated DMe-DCNQI (d8-DMe-DCNQI)14. The deuteration of DMe-DCNQI acts as a positive chemical pressure, as shown in Fig. 1d. Mapping the field-temperature (H-T) phase diagram of [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu alloy, we observed a field-induced solidification of electrons, suggesting the Pomeranchuk effect is suppressed under magnetic fields. In the ultrahigh-field limit, the system exhibits a reentrant metallic state, hinting at a possible emergence of a novel quantum liquid state of 3He. Our findings clarify the behaviour of Pomeranchuk electrons under magnetic fields, revealing their fate in extreme conditions.

Results
To begin with, we characterised the fundamental transport properties of the [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu alloy (x = 0.3, 0.5, and 1). Figure 2a displays the zero-field resistance of [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu as a function of temperature at ambient pressure. For the fully deuterated salt (x = 1) and x = 0.5 salt, sharp resistance jumps at T ~ 80 K and 60 K, respectively. It is accompanied by hysteresis, signifying a first-order MIT. To simplify the analysis of the external field dependence of the MIT, the transition point is defined during the cooling process unless otherwise stated. In contrast, the x = 0.3 salt, which remains metallic across all temperatures at 0 T, exhibits monotonic metallic behaviour down to 10 K. These results, consistent with the prior study14, demonstrate that the alloy systems enable access to the finite-pressure conditions under ambient pressure, which provides a basis for exploring their response under magnetic fields.

Next, we explored the effect of magnetic fields on the DCNQI family. Figure 2b presents the results of electrical resistivity (\rho) measurements for the x = 1 salt in pulsed fields generated by non-destructive pulsed magnets. The field-induced MITs are first-order phase transitions accompanied by hysteresis. The wide hysteresis indicates that nucleation associated with the phase transition requires overcoming an energy barrier, whereas the sharpness of the transition implies that domain growth develops rapidly. The transition field HMIT shifts to higher fields as the temperature increases. A similar behaviour is observed in the x = 0.5 salt as well (see Supplementary Materials).

Meanwhile, as shown in Fig. 2c, the x = 0.3 salt, which remains metallic across all temperatures at 0 T, underwent a field-induced MIT. The phase boundary, shown in Fig. 2d, exhibits a minimum in the transition field HMIT around 40 K, which corresponds to the temperature at which the entropies of the insulating and metallic states become equal (see Fig. 1c). Notably, near this temperature, the insulator-to-metal transition (IMT) during the field-descending process is absent due to the pronounced hysteresis. As illustrated in Fig. 2e, the x = 0.3 salt is metallic throughout the temperature range at 0 T. However, under a high magnetic field of 40 T, it exhibits reentrant MIM transitions, similar to the behaviour observed for the x = 0.45 salt at 0 T (inset of Fig. 2e).

To probe the field-induced MITs beyond the range accessible by non-destructive pulsed magnets, we performed impedance measurements using a single-turn coil (STC) system, extending the field range up to approximately 260 T. This technique detects resistance changes through variations in the reflection wave amplitude and phase rotation of injected RF waves20 (see Supplementary Materials, SM). Figure 2f illustrates the relative phase change \Delta\phi of RF waves as a function of the magnetic field at 98 K and 107 K, measured in pulsed fields generated by the STC system. Consistent with the electrical resistivity results in Fig. 2b, c, field-induced MITs were clearly observed as amplitude change and phase rotation of RF waves in the impedance measurements.

To push the exploration into the ultrahigh-field regime, we conducted impedance measurements on the x = 1 salt in high fields of up to 600 T using an electromagnetic flux compression (EMFC) system21 (see Methods and SM). Figures 2g, h illustrate the time dependence of the generated magnetic field and \Delta\phi during the pulsed field, respectively. When the field reached 220 T (t = 44.8 \mu s), a positive phase rotation was observed, corresponding to the field-induced MIT detected in the impedance measurements using STCs. As the applied field increased further, reaching approximately 320 T (45.4 \mu s), the phase began to revert toward its low-field values, signalling the onset of a field-induced insulator-metal transition (IMT). Shortly afterward, the RF signal collapsed (highlighted in light colour in Fig. 2h, i), notably before the coil explosion at approximately 45.9 \mu s. The RF signal collapse suggests the occurrence of the IMT. Under a large dH/dt, as produced by the EMFC system in the above-100 T region, when the sample suddenly becomes conductive during a first-order IMT, significant local eddy currents induced by the large dH/dt at higher fields cause critical damage to it. As shown in the field dependence of the phase rotation in Fig. 2g, the field-induced IMT occurs at a higher field of ~320 T in addition to the field-induced MIT at ~220 T.

We performed further resistivity and impedance measurements and constructed the H-T phase diagram for the (DMe-DCNQI)2Cu family, as shown in Fig. 3a (see Supplementary Materials for raw data and a discussion of error bars). Notably, the Pomeranchuk effect is observed within a specific range of magnetic fields. In the low-temperature region of the x = 0.3 salt, the slope of the phase boundary exhibits dH/dT < 0 between 40 T and 120 T, corresponding to the emergence of the reentrant metallic state (Fig. 2e). The slope changes sign at around 40 K, where the entropy relationship between insulating and metallic states is reversed (Fig. 1c). This indicates that the transition is driven by the excess entropy of the insulating state, representing a manifestation of the Pomeranchuk effect.

The underlying entropy behaviour can be explained as follows. Upon the transition to the insulating state, a three-fold periodic ordering of Cu ions with lattice distortion (…Cu2+Cu+Cu+…) occurs15,18,22, quenching the \pi-d hybridisation and forming an energy gap at the Fermi level. The long distance between magnetic Cu2+ ions significantly suppresses magnetic interactions, rendering the insulating state paramagnetic. The weak magnetic interactions (Curie-Weiss temperature \theta of −16 K (Fig. S3a in Supplementary Materials)) leads to a high spin entropy arising from disordered Cu2+ moments. Consequently, this degenerate spin entropy is released at low temperatures19, causing the entropy of the insulating state to exceed that of the metallic state. When a magnetic field above 120 T is applied, however, the Pomeranchuk effect is quenched. In the paramagnetic insulating state, entropy is released at higher temperatures under a higher magnetic field, following a two-level Schottky-like behaviour with an energy gap opened by the Zeeman effect. By contrast, in the metallic state, Smet is proportional to the density of states due to Fermi degeneracy, and its variation is negligible compared with the Schottky-like field dependence of Sins. As a result, the relationship between Sins and Smet reverses at high fields, leading to the disappearance of the Pomeranchuk effect.

As the entropy relationship between metallic and insulating states is reversed above 40 K (Smet > Sins), the phase boundary turns to dH/dT > 0. This behaviour is also observed in the x = 0.5 and 1 salts, which correspond to the higher-pressure regime of (DMe-DCNQI)2Cu. Here, we focus on the slopes of these phase boundaries, which contrast with typical MITs, where the boundary typically satisfies dH/dT < 023,24. In conventional electron systems, exchange interactions reduce the magnetic susceptibility of the insulating state compared with the Pauli paramagnetism of the metallic state. Thus, typical insulating states with both lower entropy and lower magnetisation (namely, Smet > Sins and Mmet > Mins) are suppressed by an applied magnetic field, leading to dH/dT < 0 in accordance with the Clausius-Clapeyron relation. In contrast, together with Smet > Sins, the observed dH/dT > 0 relationship originates from the \pi-d hybridisation at the Fermi level24,25, which gives rise to the positive magnetisation jump at the MIT (namely, Smet > Sins and Mmet < Mins)14,18,22,25–27. In the metallic state, this \pi-d hybridisation yields a three-dimensional Fermi liquid state20. While the metallic state follows Pauli paramagnetism due to Fermi degeneracy, the paramagnetic insulating state magnetises following the Brillouin function, resulting in the observed Mmet < Mins relation.

Our results for the x = 1 salt indicate that the metallic state becomes stabilised again at ultrahigh fields, as shown in Fig. 3a. This observation suggests that in the ultrahigh-field region, the magnetisation of the metallic state is expected to exceed that of the insulating state. Considering the behaviour of the magnetisation curve, the magnetisation of the paramagnetic insulator state (Mins) follows the Brillouin function and approaches saturation in the high-field limit, as illustrated in Fig. 3b. For (DMe-DCNQI)2Cu, the magnetism in the insulating state is solely governed by the Cu2+ spins, which account for only one-third of the total Cu population14,15,26. Consequently, the saturation value is \muB/3 per mole (see Fig. S3b in SM). In contrast, in the metallic state, both the d-electrons of Cu and the \pi-electrons of DMe-DCNQI contribute to the Pauli paramagnetism, leading to a magnetisation that exceeds \muB/3 in the high-field limit. Given the electron bandwidth of approximately 1 eV28–30, the applied fields in this study remained far below this scale. Therefore, the Fermi liquid state is expected to exhibit linear magnetisation characteristic of Pauli paramagnetism (red curve in Fig. 3b) even at 600 T (Mmet ~ \chiP H, where \chiP represents the Pauli paramagnetism). This linear response, which does not saturate at \muB/3, allows the metallic state to overcome the energy difference \Delta G again (shaded region). Based on these assumptions, Smet = \gamma T (Sommerfeld electronic specific heat), Mmet = \chiP H (Pauli paramagnetism), Sins = R[ln(2 cosh(\muB \mu0 H/kBT)) + \muB \mu0 H tanh(\muB \mu0 H/kBT)/kBT] (Schottky entropy), and Mins = NA g\muB tanh(\muB \mu0 (H+Hint)/kBT) (Brillouin function), where \gamma, \mu0, \muB, kB, NA, g, and Hint represent the Sommerfeld coefficient, Bohr magneton, magnetic constant, Boltzmann constant, and Avogadro constant, g-factor, and internal field, respectively, we simulated the phase diagram phenomenologically (see Supplementary Materials for details), as indicated by the dashed curve in Fig. 3a. The simulation aligns well with the experimental data. Notably, this simulation assumes a simplified two-state model (liquid and solid), neglecting potential intermediate phases. If such phases, like liquid crystals of Pomeranchuk electrons, emerge at high fields, the experimental results should deviate from the simulation. The slight discrepancy of the data point of 109 K (310 T) might be related to the presence of such phases. However, due to technical constraints in the EMFC experiments, future theoretical studies are required for deeper discussion.

Finally, we consider the analogy of our results with 3He. Just as the field-induced solidification in (DMe-DCNQI)2Cu, the same phenomenon is also expected to occur in 3He, inherently implying the quenching of the Pomeranchuk effect at sufficiently high magnetic fields. The field-induced IMT in (DMe-DCNQI)2Cu can also be interpreted as a consequence of molecular orbital dissociation under a strong magnetic field. When a magnetic field destabilises the anti-parallel spin configuration, the system undergoes IMT. This perspective suggests a similar mechanism in 3He: if the magnetic field energy surpasses the binding energy responsible for the formation of 3He solid—namely, van der Waals interactions—a high-field liquid state of 3He might emerge. In the solid state, 3He atoms exhibit substantial zero-point vibrations even at low temperatures, with amplitudes reaching approximately 40% of the lattice spacing31. Recent thermal Hall effect measurements32,33 suggest that lattice vibrations couple directly to a magnetic field, potentially suppressing van der Waals interactions in ultrahigh fields. Such a non-trivial lattice collapse could lead to a novel quantum liquid state distinct from the conventional liquid state of 3He, presenting an intriguing avenue for future research.

In summary, we revisited the electronic states of [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu, which exhibit the Pomeranchuk effect closely resembling that of 3He. Using [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu, we successfully traced the behaviour of the Pomeranchuk effect in the high-field limit, observing its quenching in high fields and reentrant melting under extreme magnetic fields. From a phenomenological perspective, we demonstrated that this extraordinary field dependence can be understood through the relationship between the magnetisation and entropy of the Pomeranchuk electrons. By analogy to 3He, 3He may also exhibit a unique sequence of magnetic-field-induced solidification and remelting in the high-field limit.

Methods
Sample preparation
Single crystals of [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu were synthesised by the slow chemical reduction of mixture of DMe-DCNQI and d8-DMe-DCNQI using CuI in an acetonitrile solution. The mixing ratio x is determined by the ratio of the mixture DMe-DCNQI and d8-DMe-DCNQI. The typical size of the obtained crystals is about 50 \mu m \times 50 \mu m \times 1000 \mu m.

Electrical transport measurements in pulsed magnetic fields
Electrical transport measurements in pulsed magnetic fields were performed by typical ac methods below 100 T and radio-frequency impedance method above 100 T34.

Generation of pulsed magnetic fields
Pulsed magnetic fields were generated using in-house pulsed magnets at the Institute for Solid State Physics, University of Tokyo. For magnetic fields below 100 T, two types of non-destructive pulsed magnets were employed. The first generated a maximum field of 60 T with a pulse duration of 36 ms, while the second achieved 88 T with a pulse duration of 3 ms. For the generation of magnetic fields exceeding 100 T, destructive pulsed magnets, single-turn coils (STCs) and electromagnetic flux compression (EMFC) systems21, were utilised. The STCs generated pulsed magnetic fields ranging from 135 to 260 T within 4−6 \mu s, while the EMFC system achieved fields as high as over 600 T. Detailed magnetic field profiles used in this study are provided in the Supplementary Materials.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The work by N. Matsuyama et al. presents ultrahigh magnetic field characterization of an interesing organic conductor, where field induced metal-insulator transitions and the unsual relationship between critical fields and temperatures have been established. The results support the explantion of entropy driven Pomeranchuk effect, analogous to that happens in 3He. The experiment is intriguing and the physics unveiled is significant. I recommend its publication after some revisons.\n1. The material studied should be described clearly at the beginning of the main text, at least in the last paragraph of introduction. Besides the basic information about crystal structures and electronic bands (mentioned a little in the second paragraph in Discussion) , the authors should introduce in brief several kinds of explanations of the MIT and reentrant metallic state. In particular, one of them (ref. 15) is highly relevant in regards of entropy measurements. This work is an independent examination of the explanation of Pomeranchuk effect, which should be presented in a clearer background of recent progresses.\n2. The minimum transition B field for x=0.3 is found at 40 K. Is this temperature related to the extreme point of the melting curve in Fig. 1b? The hysteresis in Fig. 2b-d should be discussed in details. Can more information be inferred?\n3. The measurement of x=0.45 that have direct observation of reentrant metallic states in the low temperature regime has been in lack. Any low temperature limitation for the high field measurement? It will be a strong connection between the presented two ends, i.e., x=1 and 0.3. In addition, the data points in Fig. 3 seem to be more than those in Fig. 2. The raw data should be added in SI.\n4. The manuscript repeatedly refers to the analogy with 3He across multiple sections, leading to redundancy and dilution of the key message. Conversely, important aspects such as the modeling behind Fig. 3b are presented rather briefly, without adequate explanation of the assumptions or derivations. A more compact and focused structure—concentrating on the core physics rather than reiterating the heuristic analogy—would benefit the clarity and impact of the work.\n5. While the manuscript is generally well-written and grammatically correct, several aspects of the scientific writing could benefit from clarification and greater precision. For instance, the phrase “an anomalous entropy relation” appears repeatedly but is not clearly defined. It would help to explicitly state the direction of the entropy difference between phases to avoid ambiguity. Additionally, in several instances, key modeling assumptions (e.g., specific forms of heat capacity or magnetization) are introduced only implicitly. Expanding these descriptions would enhance the transparency and reproducibility of the theoretical analysis.\nOverall, the work deserves to be published in Nature Communications after properly addressing the above concerns.", "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": "In this manuscript, the authors revisit the electronic properties of the coordination polymer [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu for x = 0.3, 0.45, and 1 through the analogy with Pomeranchuk effect in He3. According to the data shown in the manuscript, the Pomeranchuk effect indeed appears for the x = 0.45 salt at zero field, and for the x = 0.3 salt within the range ~30 T < μ0H< ~120 T. However, for the x = 1 salt, no evidence supports the Pomeranchuk effect based on the standard definition (liquid system solidifies upon heating) and they interpret the ultrahigh magnetic field data as evidence of a re-entrant liquid state. This behavior is phenomenologically demonstrated using the relationship between magnetization and entropy of the electrons. However, based on the data and discussions presented in the current form, I can not suggest publication. The main concerns are as follows:\n1. The authors report their one remarkable finding that the x = 1 salt displays a re-entrant liquid state in ultrahigh magnetic fields. This conclusion is hardly supported due to the excessively noisy high-field data, making it difficult to distinguish the IMT signal from the noise. I also have concerns regarding magnetic field induced heating effect. The manuscript is presented assuming the sample temperature is fixed during the measurements. For sure it is very challenging to conduct experiments under this high magnetic field. However, it is highly possible that the sample is heated up during the rapid field ramping up process. Then “re-entrant metallic state” is just the high-temperature phase and not very interesting. Consequently, the title of the manuscript, “Fate of Pomeranchuk effect in ultrahigh magnetic fields,” is misleading.\n2. According to the authors, the metal-insulator transition for x=1 sample is accompanied by a structural phase transition. While here, they want to claim a purely electronic Pomeranchuk effect. This needs to be more clearly addressed.\n3. What is the magnetic response for the x = 0.45 salt, which shows the re-entrant metallic state at low temperatures?\nOther minor suggestions and points needed to be addressed:\n1. No page number is labeled.\n2. Page 3, “Only a few examples of electrons exhibiting the Pomeranchuk like effect in solids have been reported”. In recent years, the Pomeranchuk-like effect has been observed in more 2D systems, including TMDCs moiré systems (Nature 597, 350-354 (2021), Phys. Rev. X 12, 041015 (2022), ) and rhombohedral multilayer graphene system (Nature 640, 355–360 (2025)).\n3. Page 4, “Chemical pressure, the counterpart of physical pressure in 3He, was introduced by mixing deuterated DMe-DCNQI (d8-DMe-DCNQI)”, How does the chemical pressure P depend on the d8-DMe DCNQI fraction x?\n4. In Page 6, “In the lower-temperature region of the x = 0.3 salt, the slope of the phase boundary exhibits dH/dT < 0, coinciding with the emergence of the reentrant metallic state.” This is a bit confusion, coinciding with the reentrant metallic state in x=0.45 salt?\n5. In Figure 2a, why is the ρ-T curve not smooth for x = 0.45?\n6. In Figure 2b, 2c, and 2d, what do the light-colored curves represent? It does not seem to have been mentioned anywhere in the manuscript. It’s likely the down-ramped curve. Would the hysteresis cause by temperature difference? Why aren't the lighter-colored curves shown for the two middle panels of Figure 2c?\n7. Page 4, “These results indicate that the region of the insulating phase in the pressure temperature phase diagram expands toward the lower-pressure side under magnetic fields”, authors could draw a schematic diagram for this statement.\n8. Page 5, “Figure 2d illustrates the relative phase change Δφ of RF waves as a function of the magnetic field at 100 K and 110 K”, the temperatures labeled in Figure 2d are 98 K and 107 K, respectively.\n9. In Figure 3a, the authors should state how the data uncertainties were determined.\n10. The authors interpret the noise in Fig. 2f and 2g as the damage to the sample caused by eddy currents. Is there further evidence of sample damage? Is it feasible to reduce dH/dt? Do the authors have the data for decreasing magnetic field conditions in the EMFC system?", "reviewer_4": "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."}, "author_response": "We thank the Reviewers for their insightful comments on our paper, which have substantially elevated the quality of our manuscript. In response to their valuable feedback, we have implemented the following revisions, which are intended to address the Reviewers' concerns. We thank Reviewers #2 and #4 for their time and effort in co-reviewing our manuscript.\n----------------------------------------------------------------------------------------------------------\nResponse to Reviewer #1\nThe work by N. Matsuyama et al. presents ultrahigh magnetic field characterization of an interesing organic conductor, where field induced metal-insulator transitions and the unsual relationship between critical fields and temperatures have been established. The results support the explantion of entropy driven Pomeranchuk effect, analogous to that happens in 3He. The experiment is intriguing and the physics unveiled is significant. I recommend its publication after some revisons.\n[reply] We thank the first Reviewer for rating our work highly and raising valuable suggestions that are helpful to improve our manuscript. We reply point by point to all the comments below and indicate the revised parts of the manuscript.\n[Comment 1] The material studied should be described clearly at the beginning of the main text, at least in the last paragraph of introduction. Besides the basic information about crystal structures and electronic bands (mentioned a little in the second paragraph in Discussion) , the authors should introduce in brief several kinds of explanations of the MIT and reentrant metallic state. In particular, one of them (ref. 15) is highly relevant in regards of entropy measurements. This work is an independent examination of the explanation of Pomeranchuk effect, which should be presented in a clearer background of recent progresses.\n[reply] We thank the Reviewer for their helpful advice. As pointed out, it is important to clarify the present material at the beginning of the manuscript. We have revised the introduction to include a more detailed explanation of the fundamental properties of the materials, as well as the MIT and the re-entrant metallic behaviour. In addition, as the Reviewer noted, Ref. 15 is an essential prior study for the discussion of entropy. We have now carefully incorporated it into the revised version as part of the background of the advances presented in this work.\n[Change]\n(line 103) “One-dimensional DMe-DCNQI columns coordinated by Cu ions form a tetragonal lattice of space group I4 /a, showing metallic behaviour down to the lowest temperatures arising from a hybridised electron band composed of DCNQI π-electrons and Cu d-electrons. External pressure or chemical substitution can shift the ground state into a Cu2+-originated paramagnetic insulator accompanied by a three-fold superlattice formation, with an intermediate range where a reentrant metal-insulator-metal transition is observed (Fig. 1d)”\n(line 110) “While an earlier study has investigated the entropic relation between metallic and insulating states, which explains the emergence of the reentrant metallic phase, its link to the Pomeranchuk effect remains unexplored.”\n[Comment 2] The minimum transition B field for x=0.3 is found at 40 K. Is this temperature related to the extreme point of the melting curve in Fig. 1b? The hysteresis in Fig. 2b-d should be discussed in details. Can more information be inferred?\n[reply] We thank the Reviewer for the insightful comment. As the Reviewer pointed out, the minimum in the transition field at 40 K is related to the minimum in the melting curve on the P–T phase diagram shown in Fig. 1b. At 40 K, the entropies of the liquid and solid states become equal, resulting in a zero slope of the melting curve according to the Clausius–Clapeyron relation, since the transition entropy ΔS is zero at this temperature (see the top panel of Fig. 1b). Consequently, the minimum transition field also appears at 40 K in the H–T phase diagram. This correspondence provides strong support for the thermodynamic interpretation of our system.\nRegarding the observed hysteresis in Figs. 2b–d, it is a characteristic feature of nucleation and growth in first-order phase transitions. Although the transition is very sharp, indicating rapid domain growth, nucleation requires overcoming an energy barrier, which leads to the observed hysteresis. In the revised manuscript, we have added a brief explanation. We note that a detailed discussion of hysteresis is challenging because the field-sweep rate at the phase transition differs among the measurement datasets.\n[Change]\n(line 146) “The phase boundary, shown in Fig. 2d, exhibits a minimum in the transition field H MIT around 40 K, which corresponds to the temperature at which the entropies of the insulating and metallic states become equal (see Fig. 1c)”\n(line 140) “The field-induced MITs are first-order phase transitions accompanied by hysteresis. The wide hysteresis indicates that nucleation associated with the phase transition requires overcoming an energy barrier, whereas the sharpness of the transition implies that domain growth develops rapidly.”\n[Comment 3] The measurement of x=0.45 that have direct observation of reentrant metallic states in the low temperature regime has been in lack. Any low temperature limitation for the high field measurement? It will be a strong connection between the presented two ends, i.e., x=1 and 0.3. In addition, the data points in Fig. 3 seem to be more than those in Fig. 2. The raw data should be added in SI.\n[reply] Our samples for x = 0.45 were unfortunately thin and fragile, which prevented us from performing pulsed-field measurements at low temperatures. As the temperature was decreased, the sample underwent two MITs, until it cracked. It is indeed unfortunate that the field effect on re-entrant metallic states is lacking. However, the electronic state of re-entrant metallic state has been shown to possess no difference compared with the metallic state seen in high-temperature or low-pressure regions (Solid State Commun. 93, 203 (1995).). Thus, we do not think the lack of the low-temperature data for x = 0.45 weakens our discussion.\nWhereas we could not investigate the low-temperature region with our x = 0.45 sample, we succeeded in conducting additional measurements on an x = 0.5 salt at high temperatures. We believe these data provide further information on the field response of the system, supporting the connection between the x = 0.3 and x = 1 salts. In addition, we have added Fig. 2e, which shows the 40 T data for x = 0.3, clearly indicating a similarity to the x = 0.45 data at 0 T. Therefore, the x = 0.45 data have been moved into the inset of Fig. 2e, and the x = 0.5 data has been added to Figs. 2a and 3a.\nAs for the data points shown in the phase diagram (Fig. 3a), we have summarised the raw data in another section of our Supplementary Materials.\n[Change]\n(Fig. 2a, 2e, and 3a)\n(Fig. S4 in SM) We added a new section 5 in our Supplementary material.\n[Comment 4] The manuscript repeatedly refers to the analogy with 3He across multiple sections, leading to redundancy and dilution of the key message. Conversely, important aspects such as the modeling behind Fig. 3b are presented rather briefly, without adequate explanation of the assumptions or derivations. A more compact and focused structure—concentrating on the core physics rather than reiterating the heuristic analogy—would benefit the clarity and impact of the work.\n[reply] We appreciate this comment as well. We have revised our manuscript to include a more detailed explanation of our simulation, which was previously mentioned in our Supplementary Material. Additionally, we have separated the discussion on the experimental results on (DMe-DCNQI) Cu systems and shortened the part discussing the analogy to 3He.\n[Change]\n(line 227) “Based on these assumptions, S = γT (Sommerfeld electronic specific heat), M = met met χ H (Pauli paramagnetism), S = R[ln(2cosh(μ μ H/k T))+μ μ H tanh(μ μ H/k T)/k T] P ins B 0 B B 0 B 0 B B (Schottky entropy), and M = N gμ tanh(μ μ (H+H )/k T) (Brillouin function), where γ, μ , ins A B B 0 int B B μ , k , N , g, and H represent the Sommerfeld coefficient, Bohr magneton, magnetic constant, 0 B A int Boltzmann constant, and Avogadro constant, g-factor, and internal field, respectively, we simulated the phase diagram phenomenologically (see Supplementary Materials for details), as indicated by the dashed curve in Fig. 3a.”\n(The paragraph starting at line 239 has been shortened.)\n[Comment 5] While the manuscript is generally well-written and grammatically correct, several aspects of the scientific writing could benefit from clarification and greater precision. For instance, the phrase “an anomalous entropy relation” appears repeatedly but is not clearly defined. It would help to explicitly state the direction of the entropy difference between phases to avoid ambiguity. Additionally, in several instances, key modeling assumptions (e.g., specific forms of heat capacity or magnetization) are introduced only implicitly. Expanding these descriptions would enhance the transparency and reproducibility of the theoretical analysis.\n[reply] We appreciate the Reviewer’s comments on the ambiguity in our original manuscript. We revised our Discussion section, especially taking special care to eliminate ambiguous terms.\n[Change]\n(line 199) “(S > S )”, (line 205) (namely, S > S and M > M ), (line 207) S > S , (line met ins met ins met ins met ins 209) (namely, S > S and M < M )\nmet ins met ins\n(line 225) “M ~ χ H, where χ represents the Pauli paramagnetism”\nmet P P\n(line 227) “Based on these assumptions, S = γT (Sommerfeld electronic specific heat), M = met met χ H (Pauli paramagnetism), S = R[ln(2cosh(μ μ H/k T))+μ μ H tanh(μ μ H/k T)/k T] P ins B 0 B B 0 B 0 B B (Schottky entropy), and M = N gμ tanh(μ μ (H+H )/k T) (Brillouin function), where γ, μ , ins A B B 0 int B B μ , k , N , g, and H represent the Sommerfeld coefficient, Bohr magneton, magnetic constant, 0 B A int Boltzmann constant, and Avogadro constant, g-factor, and internal field, respectively, we simulated the phase diagram phenomenologically (see SM for details), as indicated by the dashed curve in Fig. 3a.”\n[Comment 6] Overall, the work deserves to be published in Nature Communications after properly addressing the above concerns.\n[reply] We appreciate all the insightful comments from the Reviewer and believe our revised manuscript is now ready for publication in Nature Communications.\n----------------------------------------------------------------------------------------------------------\nResponse to Reviewer #3\nIn this manuscript, the authors revisit the electronic properties of the coordination polymer [(DMe-DCNQI)1-x(d8-DMe-DCNQI)x]2Cu for x = 0.3, 0.45, and 1 through the analogy with Pomeranchuk effect in He3. According to the data shown in the manuscript, the Pomeranchuk effect indeed appears for the x = 0.45 salt at zero field, and for the x = 0.3 salt within the range ~30 T < μ0H< ~120 T. However, for the x = 1 salt, no evidence supports the Pomeranchuk effect based on the standard definition (liquid system solidifies upon heating) and they interpret the ultrahigh magnetic field data as evidence of a re-entrant liquid state. This behavior is phenomenologically demonstrated using the relationship between magnetization and entropy of the electrons. However, based on the data and discussions presented in the current form, I can not suggest publication.\n[reply] We sincerely appreciate the Reviewer’s thorough review and constructive comments. We agree that the concerns raised by the Reviewer are essential for the proper interpretation of our results. Following the suggestions, we have revised the manuscript to clarify our interpretation. We hope that these revisions will facilitate a better understanding of our findings.\nAs Reviewer #3 pointed out, the x = 1 salt itself does not exhibit the Pomeranchuk effect. However, given its continuous connection in the phase diagram with the x = 0.3 and 0.45 salts, which do exhibit the Pomeranchuk effect, the results for the x = 1 salt are also important for a comprehensive understanding of an “electron system exhibiting the Pomeranchuk effect.” This continuous connection is supported by a past dHvA experiment, in which the re-entrant metallic state has been shown to possess no difference compared with the metallic state observed in the high-temperature or low-pressure region (Solid State Commun. 93, 203 (1995)).\nTo solidify the discussion of this connection, we additionally conducted measurements on the x = 0.5 salt at high temperatures, as it was challenging to investigate our x = 0.45 salt. Indeed, the field response shows a continuous change with the variation in the fraction x. We believe these data provide further information on the field response of the system, supporting the connection between the x = 0.3 and x = 1 salts.\nRegarding the evidence for a re-entrant liquid state in the x = 1 salt, we have explained in detail why our interpretation is correct and why the evidence is sufficient to conclude the re-entrant behaviour in our reply to [Comment 1].\n[Change]\n(Fig. 3a) We have added the x = 0.5 data.\n[Comment 1] The authors report their one remarkable finding that the x = 1 salt displays a re-entrant liquid state in ultrahigh magnetic fields. This conclusion is hardly supported due to the excessively noisy high-field data, making it difficult to distinguish the IMT signal from the noise. I also have concerns regarding magnetic field induced heating effect. The manuscript is presented assuming the sample temperature is fixed during the measurements. For sure it is very challenging to conduct experiments under this high magnetic field. However, it is highly possible that the sample is heated up during the rapid field ramping up process. Then “re-entrant metallic state” is just the high-temperature phase and not very interesting. Consequently, the title of the manuscript, “Fate of Pomeranchuk effect in ultrahigh magnetic fields,” is misleading.\n[reply] We appreciate the comment pointing out the unclear description of the noisy data and the heating effect in the previous manuscript.\nWe here reply to the question regarding the observed noise in the EMFC measurements. In single-turn coil measurements, significant noise is often observed due to the electromagnetic noise produced by the high-voltage switching of capacitors. However, we believe that the only possible origin of the noise is sample destruction in the EMFC measurements, based on two main reasons.\nFirst, the data before 45.3 μs exhibit no significant noise, and the noise starts at 45.3 μs, before the coil explosion at 45.9 μs, which indicates that the noise is unrelated to the collapse of the experimental setup due to the explosion. Second, EMFC measurements are typically tolerant of extrinsic noise before the coil explosion (e.g. see Nature Commun. 14, 1744 (2023)., Phys. Rev. B 105, L251105 (2022).). Although electrical measurements using destructive single-turn coils are inherently challenging and the data obtained are often noisy, in EMFC measurements the sample space remains electrically stable and quiet, with minimal mechanical vibration, until the explosion of the coil, because dH/dt is extremely slow at the initial stage of field generation. Once the sample is destroyed, however, the circuit suddenly becomes open, causing the open terminal to act as an antenna collecting electrical noise. Consequently, the phase of the signal becomes unstable, and noise is expected to occur.\nAs for the mechanism of sample destruction, we speculate that eddy-current heating induced by the time-varying magnetic field would destroy our sample together with the sharp first-order IM transition. The non-uniform growth of metallic domains during the transition causes an uneven distribution of eddy currents and resulting local heating, eventually destroying the sample. For the MI transition in the low-field region, we were able to measure the resistance without sample breakage. Since the insulating state has much higher resistivity, insulating domain growth would not lead to an increase in eddy currents.\nNext, we would like to respond to the second concern regarding the heating effect. As discussed above, we also considered heating as a possible source of sample destruction. The estimated temperature increase during the period when the sample is metallic is around 10 K (see Section 4 in Supplementary Material for details). However, since the insulating state, with eight orders of magnitude higher resistivity, would be free from heating by eddy currents, we assume that the isothermal condition holds in the insulating phase, namely the field range of 220-340 T.\nTherefore, we can conclude that the re-entrant phase originates not from heating, but from the magnetic-field-induced transition.\nAs for the concern on the sample temperature, the details are described in our Supplementary Materials (see the section .4).\n[Change]\n(line 170) “Under a large dH/dt, as produced by the EMFC system in the above-100 T region, when it suddenly becomes conductive during a first-order IMT, significant local eddy currents induced by the large dH/dt at higher fields cause critical damage to the sample.”\n(Section 4 in Supplementary Materials) “It should be noted that the sample breakage originates from heating induced by metallisation during the insulating–metal transition. If the phase transition proceeded uniformly, it would not result in breakage, as the discussion above indicates that only heating on the order of 10 K would occur. However, the phase transition in this system is first-order and accompanied by a large hysteresis. When a metallic domain begins to form through nucleation, the current density becomes concentrated within that domain, generating a substantial local heating effect. Damage to this region then causes the current density to increase further in the next domain undergoing metallisation, ultimately leading to sample breakage.”\n[Comment 2] According to the authors, the metal-insulator transition for x=1 sample is accompanied by a structural phase transition. While here, they want to claim a purely electronic Pomeranchuk effect. This needs to be more clearly addressed.\n[reply] Not only for the x = 1 sample, the transition between metallic and insulating states in (DMe-DCNQI)₂Cu systems is accompanied by a structural transition. The three-fold superlattice formation is a common characteristic of the insulating state of DCNQI–Cu salts (Bull. Chem . Soc. Jpn., 73, 515 (2000).). We constructed our discussion assuming the metallic and insulating states, which can be switched with each other by changing the external magnetic field. Depending on the strength of the external magnetic field, the favoured state is either metallic or insulating, which is reproduced based on our thermodynamic simulation shown in the manuscript. Magnetic fields can directly affect the spin degree of freedom, which acts as a driving force for the transition between metallic and insulating states. Since the structural phase transition is a resultant accompaniment of the electronic transition and is not the essence of this phase transition, the main logic would not be altered by including a structural phase transition.\nWhile the discussion itself is preserved in its original form, to clarify the fundamental properties of the investigated system, we have revised our introduction. We hope this resolves the Reviewer’s concern.\n[Change]\n(line 103) “One-dimensional DMe-DCNQI columns coordinated by Cu ions form a tetragonal lattice of space group I4 /a, showing metallic behaviour down to the lowest temperatures arising from a hybridised electron band composed of DCNQI π-electrons and Cu d-electrons. External pressure or chemical substitution can shift the ground state into a Cu2+-originated paramagnetic insulator accompanied by a three-fold superlattice formation, with an intermediate range where a re-entrant metal-insulator-metal transition is observed (Fig. 1d)”\n[Comment 3] What is the magnetic response for the x = 0.45 salt, which shows the re-entrant metallic state at low temperatures?\n[reply] Our samples for x = 0.45 were unfortunately thin and fragile, which prevented us from performing pulsed-field measurements at low temperatures. As the temperature was decreased, the sample underwent two MITs, until it cracked. It is indeed unfortunate that the field effect on re-entrant metallic states is lacking. However, the electronic state of re-entrant metallic state has been shown to possess no difference compared with the metallic state seen in high-temperature or low-pressure regions (Solid State Commun. 93, 203 (1995).). Thus, we do not think the lack of the low-temperature data for x = 0.45 weakens our discussion.\nWhereas we could not investigate the low-temperature region with our x = 0.45 sample, we succeeded in conducting additional measurements on an x = 0.5 salt at high temperatures. We believe these data provide further information on the field response of the system, supporting the connection between the x = 0.3 and x = 1 salts. In addition, we have added Fig. 2e, which shows the 40 T data for x = 0.3, clearly indicating a similarity to the x = 0.45 data at 0 T. Therefore, the x = 0.45 data have been moved into the inset of Fig. 2e, and the x = 0.5 data has been added to Figs. 2a and 3a.\n[Change]\n(line 130) “For the fully deuterated salt (x = 1) and x = 0.5 salts, sharp resistance jumps at T ~ 80 K and 60 K, respectively.”\n(line 149) “As illustrated in Fig. 2e, the x = 0.3 salt is metallic throughout the temperature range at 0 T. However, under a high magnetic field of 40 T, it exhibits reentrant MIM transitions, similar to the behaviour observed for the x = 0.45 salt at 0 T (inset of Fig. 2e).”\n(Fig. 2a, 2e, and 3a)\n[Comment 4] Other minor suggestions and points needed to be addressed:\n1. No page number is labeled.\n2. Page 3, “Only a few examples of electrons exhibiting the Pomeranchuk like effect in solids have been reported”. In recent years, the Pomeranchuk-like effect has been observed in more 2D systems, including TMDCs moiré systems (Nature 597, 350-354 (2021), Phys.Rev. X 12, 041015 (2022), ) and rhombohedral multilayer graphene system (Nature 640, 355–360 (2025)).\n3. Page 4, “Chemical pressure, the counterpart of physical pressure in 3He, was introduced by mixing deuterated DMe-DCNQI (d8-DMe-DCNQI)”, How does the chemical pressure P depend on the d8-DMe DCNQI fraction x?\n4. In Page 6, “In the lower-temperature region of the x = 0.3 salt, the slope of the phase boundary exhibits dH/dT < 0, coinciding with the emergence of the reentrant metallic state.”This is a bit confusion, coinciding with the reentrant metallic state in x=0.45 salt?\n5. In Figure 2a, why is the ρ-T curve not smooth for x = 0.45?\n6. In Figure 2b, 2c, and 2d, what do the light-colored curves represent? It does not seem to have been mentioned anywhere in the manuscript. It’s likely the down-ramped curve. Would the hysteresis cause by temperature difference? Why aren't the lighter-colored curves shown for the two middle panels of Figure 2c?\n7. Page 4, “These results indicate that the region of the insulating phase in the pressure temperature phase diagram expands toward the lower-pressure side under magnetic fields”, authors could draw a schematic diagram for this statement.\n8. Page 5, “Figure 2d illustrates the relative phase change Δφ of RF waves as a function of the magnetic field at 100 K and 110 K”, the temperatures labeled in Figure 2d are 98 K and 107 K, respectively.\n9. In Figure 3a, the authors should state how the data uncertainties were determined.\n10. The authors interpret the noise in Fig. 2f and 2g as the damage to the sample caused by eddy currents. Is there further evidence of sample damage? Is it feasible to reduce dH/dt? Do the authors have the data for decreasing magnetic field conditions in the EMFC system?\n[reply]\n1. We put the page numbers in our manuscript.\n2. We thank the Reviewer for bringing up the latest research on the Pomeranchuk effect observed in 2D materials. We have added the references to our manuscript and revised the text accordingly.\nIn contrast to these 2D systems, we believe that the coordination polymer, in which the Pomeranchuk effect arises from π–d hybridization, constitutes a new class of materials worthy of investigation.\n3. We have revised the introduction to provide more fundamental information about the investigated materials. As mentioned there, deuterium substitution corresponds to positive chemical pressure (Bull. Chem . Soc. Jpn. 73, 515 (2000).).\nThe effective pressure arising from selective deuteration of the methyl group in (DMe-DCNQI)₂Cu salts was first reported to increase linearly with the number of deuterium atoms in each methyl group. Later, it was revealed that alloy systems of (DMe-DCNQI)₂Cu and the fully deuterated (d8-DMe-DCNQI)₂Cu can also reproduce the low-pressure region of the P–T phase diagram of the (DMe-DCNQI)₂Cu system, including the region where insulating and re-entrant metallic ground states are obtained (see Solid State Commun. 85, 831-835 (1993).). We have also revised Fig. 1d to better highlight the chemical pressure arising from the alloy effect.\n4. We intended to highlight that the Pomeranchuk effect is observed in the H-T phase diagram in the low-temperature regime, where dH/dT > 0. However, as the Reviewer pointed out, the discussion in the original form was somewhat confusing. We have revised our manuscript to avoid this confusion and make the discussion more straightforward.\n5. Indeed, the response of x = 0.45 does not appear smooth, showing a gradual increase before the discontinuous jump. A similar gradual increase was also observed in x = 1 salts, although it is not clearly visible in the figure. This characteristic can also be seen in past resistivity results (e.g., Fig. 2 in Solid State Commun. 85, 831 (1993)). As the precursor-like behaviour depends on the sample, sample quality, such as inhomogeneity induced by alloying, may play a role; however, the underlying cause of this gradual increase has yet to be elucidated.\n6. As Reviewer #3 pointed out, the lightly coloured curves represent the behaviour for the field-descending part. For some curves in Fig. 2c, the hysteresis was too large, and the insulating state survived down to zero field. We have added this explanation to the figure caption.\n7. We agree with your suggestion. We also expect that we could illustrate the phase boundary shift, as displayed in Fig. R1. However, in this study we investigated only the samples with x = 0.3, 0.5, and 1, and thus it is difficult to draw it accurately. Moreover, in the high-field limit, the field-induced re-entrant behaviour should result in a reversed change. Therefore, we have omitted the text to avoid potential misinterpretation.\nFig. R1 Schematic of the P-T Phase boundary at 0 T (blue) and high magnetic fields (red).\n8. We thank the Reviewer for pointing out the inconsistency in our notification. We have corrected and made them consistent in the temperature values of these two measurements.\n9. The sample temperature has some uncertainty due to eddy-current heating. As discussed in our Supplementary Materials, this heating effect is especially severe in experiments with destructive magnets. Based on the estimated temperature rise during field generation, we included a bar to indicate this uncertainty. To clarify, we added a sentence guiding the readers to the Supplementary Materials for details on the sample heating estimation.\nAs for the uncertainty in the magnetic field value in EMFC experiments, it arises from the possible deviation of the sample from the coil centre. According to past experimental results, a deviation of only 3 mm can result in a maximum field decrement of ~100 T (Rev. Sci. Instrum. 85, 036102 (2014)).\n10. Since the whole experimental setup around the coil is destroyed and lost after the EMFC measurements, it is quite challenging to further verify the sample damage. However, in single-turn coil measurements, where the setup can be preserved even after field generation, we indeed had several examples in which the sample was damaged and broken after experiencing rapid dH/dt. These experimental observations empirically support our discussion on sample damage related to the noise in Figs. 2f and 2g.\nRegarding the idea of slowing down field generation, this is unfortunately impossible. As shown in Supplementary Material 1, dH/dt shows a steep increase above the 100 T region in EMFC experiments. Although the pulsed field generated by single-turn coils shows a slowing down of dH/dt near the maximum field, the maximum field that single-turn coils can reach is not sufficient to observe the reentrant MIM transition. Finally, as we mentioned in our Supplementary Materials, field-descending data cannot be obtained in EMFC measurements due to the destruction of the setup at the maximum field.\n[Change]\n2. (Ref 10) Li, T., Jiang, S., Li, L. et al. Continuous Mott transition in semiconductor moiré superlattices. Nature 597, 350-354 (2021).\n(Ref. 11) Zhang, M., Zhao, X., Watanabe, K. et al., Tuning Quantum Phase Transitions at Half Filling in 3L−MoTe /WSe Moiré Superlattices Phys.Rev. X 12, 041015 (2022).\n2 2\n(Ref. 12) Holleis, L., Xie, T., Xu, S., et al. Fluctuating magnetism and Pomeranchuk effect in multilayer graphene. Nature 640, 355–360 (2025).\n3. (line 120) “The deuteration of DMe-DCNQI acts as a positive chemical pressure, as shown in Fig. 1d.” , (Fig. 1d) The right vertical axis.\n4. (line 180) “In the lower-temperature region of the x = 0.3 salt, the slope of the phase boundary exhibits dH/dT < 0 between 40 T and 120 T, corresponding to the emergence of the reentrant metallic state (Fig. 2e).”\n6. (line 383) “Dark-coloured curves correspond to data obtained during field-ascending sweeps, while light-coloured curves correspond to data obtained during field-descending sweeps. In some cases, due to large hysteresis, the insulating state persists down to zero field.”\n8. (line 157) “Figure 2f illustrates the relative phase change Δφ of RF waves as a function of the magnetic field at 98 K and 107 K, measured in pulsed fields generated by the STC system.”\n9. (line 396) “Bars attached to data points represent uncertainties in temperature and magnetic field. Temperature uncertainty arises from heating during a field generation (see Supplementary Material), whereas magnetic field uncertainty reflects the electromagnetic noise in STC experiments (see Supplementary Material for raw data) and possible deviation from the field centre due to sample length in EMFC experiments.”\n(Ref. 34) Nakamura, D., Sawabe, H., Takeyama, S. Experimental evidence of three-dimensional dynamics of an electromagnetically imploded liner. Rev. Sci. Instrum. 85, 036102 (2014).\nWe express our sincere gratitude to the Reviewers for their valuable comments, which have significantly enhanced the quality of our manuscript. We look forward to hearing from all the Reviewers regarding our resubmission and remain available to address any additional queries or remarks they may have."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have fully addressed my concerns. Now I recommend its publication in the current form.", "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": "In the response letter and revised version of the manuscript, the authors have thoroughly responded to my previous comments. They have improved the text and figures to avoid confusion. The additionally measurements on the x = 0.5 salt at high temperatures provide further insight into the field response of the system, supporting the connection between the x = 0.3 and x = 1 salts. Now, I recommend this manuscript for publication.", "reviewer_4": "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."}, "author_response": ""}]

Decision letter