Quantum supercritical regime with universal magnetocaloric scaling in Ising magnets

Domain: physics
Published: 2025-11-27
Editorial outcome: major_revision

Abstract
Quantum critical points ubiquitously emerge in strongly correlated systems, with their influence persisting at finite temperatures and external fields. A paradigmatic example is the quantum Ising magnet, where transverse field g controlling quantum fluctuations can expand the quantum critical point into an extended quantum critical regime. In this work, we propose a distinct quantum supercritical regime originating also from the quantum critical point but controlled by the longitudinal field h coupled to the order parameter. Through thermal tensor network simulations, we find the quantum supercritical regime is enclosed by the finite-temperature crossover boundaries T ∝ h^{zν/Δ}, where z, ν and Δ ≡ β+γ are critical exponents. We comprehend the supercritical scaling via thermal data collapse based on the derived scaling form. Amongst other intriguing phenomena in quantum supercritical regime, there exists an enhanced magnetocaloric effect characterized by a universally diverging magnetic Grüneisen ratio Γ ∝ T^{-Δ/zν}, which indicates that a small symmetry-breaking field h can generate dramatic temperature variation. We propose to observe the quantum supercritical regime in the Ising-chain compound CoNb2O6 and related quantum materials, revealing a helium-3-free pathway to millikelvin cooling via the supercritical magnetocaloric effect.

Introduction
Quantum critical point (QCP) embodies a scale-invariant many-body state that marks a continuous quantum phase transition between distinct phases of matter1–3. Nonclassical behaviors and universal scaling can be observed near the QCP, independent of microscopic details but determined by the universality class associated with global constraints such as symmetry and dimensionality. Although a QCP exists strictly at zero temperature, it profoundly shapes equilibrium and dynamical properties at finite temperatures1–3. The emergent phenomena linked to QCPs represent an intriguing topic in condensed matter physics1,4, ultracold atoms5, and quantum field theory6,7.

Specifically, a QCP extends into a quantum critical regime (QCR) at finite temperatures, where universal scaling behaviors emerge due to the intricate interplay between quantum and thermal fluctuations1–4,8–10. These finite-temperature scaling laws near QCP have been observed in various quantum materials. For instance, the Ising magnets CoNb2O6 11–21, BaCo2V2O8 22–27, and SrCo2V2O8 28–32 exhibit quantum critical behaviors governed by the (1+1)D Ising universality class. These realistic systems serve as experimental platforms for exploring quantum criticality and its emergent phenomena. Among other notable phenomena, there exists a universal magnetocaloric effect (MCE) in the QCR. Through the adiabatic demagnetization process, unlike traditional paramagnetic cooling that reaches its lowest temperatures near zero field, the quantum critical cooling achieves its minimum temperature within the QCR. The fluctuation field g—such as the transverse field in the quantum Ising model—controls quantum fluctuations and drives the QCP. A universal divergence of the magnetic Grüneisen ratio, Γ ∝ 1/T(∂T/∂g) ∝ T^{-1/zν}, with z and ν the critical exponents, has been theoretically predicted33–38 and experimentally confirmed in certain quantum magnets39–46.

Notably, the role of symmetry-breaking fields—like the longitudinal field h in the quantum Ising magnets—remains much less understood. In the ordered phase, field h couples to the order parameter and induces the first-order phase transition line that terminates at a QCP, also known as the quantum critical endpoint. The effective longitudinal field generated by interchain coupling—for instance, in Ising-chain compounds like CoNb2O6 —can give rise to massive quasiparticles described in terms of the exceptional E8 Lie algebra11,13,14,17,21,26,31. However, the universal scaling behaviors induced by the longitudinal field near such QCP remain to be systematically investigated.

We note that, in classical ferromagnetic (FM) systems, the symmetry-breaking field h can extend a finite regime above the Curie temperature, as illustrated in Fig. 1a. It is referred to as a classical supercritical regime (SR) first discovered in liquid-gas systems47,48, where the supercritical fluids possess strong thermal fluctuations. Subsequent years witnessed intensive investigations of supercritical phenomena and crossover lines in the liquid-gas systems49–51, which are very suggestive for the studies of magnetic phase transitions. We now consider that the quantum-fluctuation field g can suppress the Curie temperature Tc in an FM Ising magnet and extend it down to absolute zero (c.f., Fig. 1a). Given that, intriguing questions arise: Could there be a quantum supercritical regime (QSR) at low temperature, and whether it exhibits novel universal phenomena and distinct scaling behaviors as compared to QCR?

In this work, we unveil the existence of finite-temperature QSR—emanating from the QCP and extended by the symmetry-breaking field h—through thermal tensor-network calculations on quantum spin systems52–55. Figure 1b illustrates the QSR that resides in the h-T plane, orthogonal to the QCR in the g-T plane (Fig. 1c). The QSR separates two distinct ordered phases, such as the spin-up and spin-down states in the Ising model, and the field h couples to their order parameter. This leads to crossover lines characterized by a unique quantum supercritical scaling T ∝ h^{zν/Δ}. We further discover a pronounced MCE in the QSR, as shown in Fig. 1d, characterized by universally diverging Grüneisen ratio Γ ∝ T^{-Δ/zν} derived from the scaling analysis. Furthermore, we propose that quantum supercritical phenomena can be realized and investigated in the Ising-chain magnet CoNb2O6 11–13,15–20 (see Fig. 1e) and other Ising compounds22–32,56–58, and design a field-tilting process for ultra-low temperature cooling. Additionally, we discuss other quantum supercritical systems, including Heisenberg magnets with Dzyaloshinskii-Moriya (DM) interactions59–68 and heavy-fermion materials69, etc, which serve as promising quantum material coolants. Our findings establish a paradigm for exploring supercritical phenomena and demonstrate a route to extreme cooling via anisotropic quantum magnetic systems.

Results
Models
We consider a representative spin-chain compound CoNb2O6, where the Ising interactions dominate and the quantum critical properties are effectively described by a 1D transverse-field Ising model11–13,16,17,20. The Hamiltonian reads

H = J ∑_{<i,j>} Sz_i Sz_j - g ∑_i Sx_i - h ∑_i Sz_i, (1)

where the Ising coupling J ≡ 1 is the energy scale, and g (h) represents the transverse (longitudinal) field. The transverse field drives a QCP at g = 1/2 between the FM and paramagnetic phases (see Fig. 1c). Instead, the longitudinal field h breaks the Z2 symmetry and induces a first-order quantum phase transition between two different FM states.

To compute the low-temperature properties, we employ the thermal tensor-network methods52–55, particularly the linearized and tangent-space tensor renormalization group methods (see “Methods” and Supplementary Note 1). With these state-of-the-art approaches, we obtain the isentropic lines and magnetic Grüneisen ratio characterizing the MCE with high accuracy.

Quantum supercritical regime
In Fig. 2a, we present isentropic lines in the h-T plane and highlight the QSR located between the two spin-ordered phases. The QSR and its crossover lines can be identified through thermodynamic properties and spin dynamics. In Fig. 2b, the specific heat exhibits a double-peak structure, and the critical scaling C ∝ T^{d/z} (i.e., ∝ T) appears at intermediate temperatures within the QSR. However, this scaling deviates at lower temperatures below the low-T peaks of C/T, which determine the QSR crossover line in the h-T plane (see inset of Fig. 2b). These crossover lines follow the quantum supercritical scaling T ∝ h^{zν/Δ} (i.e., ∝ h^{8/15}), governed by the critical exponents z, ν and Δ of the (1+1)D Ising universality class.

Besides, the finite-temperature spin dynamics also exhibits quantum supercritical scaling. We consider the spin-lattice relaxation rate in nuclear magnetic resonance, i.e., 1/T1 ∝ S^{α}(ω=0) ∝ T^{(η+d-2)/z} = T^{-3/4} with η=1/4, d=1, and z=1 for 1D quantum Ising chain. The quantity S^{α}(0) reaches a maximum at the QSR boundary and then becomes suppressed. The positions of these rounded peaks follow the scaling relation T ∝ h^{zν/Δ}, as demonstrated in Fig. 2d and its inset.

Universal quantum supercritical cooling
In Fig. 1d, we have shown a schematic illustration of the pronounced quantum supercritical cooling, where lowest temperatures are achieved in the QSR. In Fig. 2a, we show the simulated isentropic lines near the Ising QCP and reveal a significantly enhanced MCE, where even a small symmetry-breaking field h can induce a dramatic temperature variation. To characterize the quantum supercritical cooling effect, we present the magnetic Grüneisen ratio Γ for each fixed temperature T in Fig. 2c. In the h-T plane, Γ exhibits a peak-dip structure with a sign change as the field h crosses above the QCP. In the inset, the peak locations follow the quantum supercritical scaling law and represent the QSR crossover lines in Fig. 2a.

When comparing universal MCEs in the QSR and QCR33, there are notable differences and similarities. As shown in Fig. 2e, the peak values Γ(h, T) are orders of magnitude larger than those of Γ in the QCR at low temperatures. This can be explained with the distinct scaling laws demonstrated in the inset, where we can identify both the quantum supercritical scaling Γ ∝ T^{-Δ/zν} (i.e., ∝ T^{-15/8}) and critical scaling Γ ∝ T^{-1/zν} (i.e., ∝ T^{-1})33–35, as confirmed by scaling analysis (see Methods). As Δ > 1 for Ising universality classes (and other conventional cases, see Table 1), the quantum supercritical scaling characterizes a more pronounced MCE. Beyond 1D chain, we also find very prominent QSR cooling in 2D Ising model (see Supplementary Note 2).

To understand the quantum supercritical scaling law, we perform data collapse on Grüneisen ratio Γ using the derived scaling form Γ = T^{-Δ/zν} ϕ_Γ(x), where x ≡ h ⋅ T^{-Δ/zν} (see scaling analysis in Methods). As shown in Fig. 2f, we obtain the scaling function ϕ_Γ(x) from an excellent collapse of the calculated Grüneisen ratio data in Fig. 2c. ϕ_Γ(x) is a parity-odd, smooth function, displaying a peak and a dip similar to Fig. 2c, but without any singularity. Given the scaling form, we have ∂Γ/∂h = T^{-2Δ/zν} ϕ'_Γ(x) = 0, indicating that the extreme values of isothermal Γ correspond to the zero points x = ±x0 of ϕ'_Γ(x), as marked by the arrows in Fig. 2f. This naturally leads to the QSR crossover lines h_cr = ±x0 ⋅ T^{Δ/zν} (see, e.g., Fig. 2a). Moreover, we find that at the crossover points, the scaling Γ = T^{-Δ/zν} ϕ_Γ(±x0) ∝ T^{-Δ/zν}, in QSR (|x| ≤ x0), ϕ_Γ(x) is approximately a linear function, i.e., Γ ⋅ T^{Δ/zν} ∝ x (see Fig. 2f). Recalling that x ≡ h ⋅ T^{-Δ/zν}, we have that Γ/h ∝ T^{-2Δ/zν}, demonstrating an even stronger divergence within the QSR and for fixed h (see Methods and Supplementary Note 3). This scaling law further underscores the enhanced quantum supercritical cooling.

Hyperscaling function and crossover surfaces
More generally, we consider the simultaneous effects of g and h fields near the QCP. In this case, the Grüneisen ratio Γ can be described as a two-variable hyperscaling function Γ = T^{-Δ/zν} ψ_Γ(x, y), with x ≡ h ⋅ T^{-Δ/zν} and y ≡ g~ ⋅ T^{-1/zν}. In Fig. 3a, we show the obtained hyperscaling function ψ_Γ(x, y) through data collapsing. The red solid lines mark the peaks and dips of ψ_Γ(x, y) by scanning x for fixed y, and each point (xi, yi) represents a crossover line in Fig. 3b. For example, the point (x0, y=0) corresponds to the QSR crossover line for g~=0 in Fig. 2a. As (xi, yi) moves, these crossover lines form two crossover surfaces in Fig. 3b, enclosing a generalized QSR bulk for g~ ≠ 0. Notably, an isothermal plane at a fixed temperature T (gray plane) intersects the crossover surfaces, resulting in two dashed lines that follow the scaling h/(g~ - g*)^Δ, with g* ≲ gc. In the zero-temperature limit, g* = gc, and the blue solid line in Fig. 1a and Fig. 3b is nothing but the ground-state supercritical crossover line h/g~^Δ proposed in ref. 71 (see details in Supplementary Note 4).

Quantum supercriticality in realistic materials
In Fig. 4, we calculate the universal scaling in CoNb2O6 and demonstrate that even a small longitudinal field can induce significant temperature variations extending into the deeply sub-Kelvin regime. Remarkably, starting from an initial temperature 1.8 K, CoNb2O6 cools to 280 mK under a very small change of longitudinal field, while from 1.3 K, it achieves an ultralow temperature 52 mK, as shown in Fig. 4a. The pronounced sensitivity to longitudinal magnetic fields facilitates the implementation of the field-tilting process for ultra-low temperature cooling. Additionally, we discuss other quantum supercritical systems, including Heisenberg magnets with Dzyaloshinskii-Moriya (DM) interactions59–68 and heavy-fermion materials69, etc, which serve as promising quantum material coolants. Our findings establish a paradigm for exploring supercritical phenomena and demonstrate a route to extreme cooling via anisotropic quantum magnetic systems.

Methods
Thermal tensor network methods
To accurately simulate the low-temperature properties of quantum lattice models, we employ the linearized tensor renormalization group (LTRG)52,53 and tangent-space tensor renormalization group (tanTRG) methods55,80. For Ising chains discussed in the main text, LTRG is exploited to simulate the infinite-chain system, and the calculated results exhibit excellent convergence and high accuracy. For example, with a bond dimension of D=200 and Trotter step of τ=0.01, the relative error of the ground-state energy is less than 10^{-3}. For 2D Ising models, we use the tanTRG method on 12 sites (see Supplementary Notes 1 and 2).

Critical scaling analysis of Γg
Scaling functions serve as powerful tools for investigating universal behaviors and scaling laws. Here, we employ scaling forms to investigate the quantum critical cooling induced by transverse fields near the QCP. Previous studies have shown that the QCP, i.e., Γg(gc, T) ∝ T^{-1/zν} 33–38. However, for the transverse-field Ising chain, the Grüneisen ratio remains a constant, i.e., Γg(gc, T) = 1/2 at the self-dual QCP38. This non-diverging Γg(gc, T) at QCP is thus no longer appropriate for characterizing the quantum critical cooling in Ising chain.

To capture the pronounced low-temperature cooling effect clearly visible in the isentropic lines of Fig. 5a, we propose that the peak/dip values of Γg exhibit the characteristic T^{-1/zν} scaling, even near such a self-dual QCP. From the scaling form of the free energy, the Grüneisen ratio Γg = T^{-1/zν} ϕ_Γg(y), with y ≡ g~ ⋅ T^{-1/zν} and ϕ_Γg(y) is the scaling function. Consequently, ∂Γg/∂g~ = T^{-2/zν} ϕ'_Γg(y) = 0, which defines the extrema of isothermal Γg, which naturally define the crossover lines. For instance, considering the peak/dip condition y = ±y0, the peak locations follow universal scaling g_cr = ±y0 T^{1/zν} ∝ T^{1/zν} (see inset of Fig. 5b), while the peak values satisfy Γg = T^{-1/zν} ϕ_Γg(±y0) ∝ T^{-1/zν} (see Fig. 2h within the main text).

Supercritical scaling analysis of Γh
Here, we conduct the scaling analysis of the quantum supercritical MCE induced by longitudinal fields. In the vicinity of a QCP, the singular part of entropy has a universal form S = T^{d/z} ψ_S(h T^{-Δ/zν}, g~ T^{-1/zν}) 10. For g~ = 0, the hyperscaling function reduces to a single-variable scaling function ψ_S(x, 0) → ϕ_S(x), with x ≡ h ⋅ T^{-Δ/zν}. Moreover, the isothermal entropy derivative (∂S/∂h)_T = T^{d/z - Δ/zν} ϕ'_S(x) is parity odd and the specific heat Ch = T(∂S/∂T)_h = (d/z) T^{d/z} [ϕ_S(x) - x ϕ'_S(x) Δ/dν] is parity even. As Grüneisen ratio Γh = (∂S/∂h)_T / Ch = T^{-Δ/zν} ϕ_Γh(x), where ϕ_Γh(x) = ϕ'_S(x) / [(d/z) ϕ_S(x) - x ϕ'_S(x) Δ/dν]. As shown in Fig. 2f, the peak/dip locations of Γh correspond to the extrema of ϕ_Γh(x), which naturally define the crossover lines h_cr = ±x0 T^{Δ/zν} ∝ T^{Δ/zν}, where the peak/dip values satisfy Γh = T^{-Δ/zν} ϕ_Γh(±x0) ∝ T^{-Δ/zν}. As in the parity-even function, if we assume ϕ_Γh(x) ≪ 1 for small x, we can further simplify the expression, where we can further expand ϕ_S(x) = ∑_{i=0} A_i x^{2i} for small x and arrive at

Γh = -(∂S/∂h)_T / [T(∂S/∂T)_h] ∝ T^{d/z - Δ/zν} / T^{d/z} A_0 x ∝ T^{-Δ/zν} x. (2)

This is consistent with the results in Fig. 2f of the main text, where Γh has a universal form Γh = T^{-Δ/zν} ϕ_Γh(x), with ϕ_Γh(x) ∝ x for small x. Furthermore, when fixing the longitudinal field h and varying temperature T, from Eq. (2), we find a scaling law Γh/h ∝ T^{-2Δ/zν} (see numerical results in the Supplementary Note 3).

Heisenberg-Dzyaloshinskii-Moriya (HDM) model
Here, we introduce the HDM system that also hosts the quantum supercritical regime. The Hamiltonian of the HDM model is H = J ∑_i (Sx_i Sx_{i+1} + Sy_i Sy_{i+1} + λ Sz_i Sz_{i+1}) - g ∑_i Sx_i - h ∑_i (-1)^i Sx_i. Here, a uniform field g = (B/J) cos α and a staggered field h = (B/J) sin α are involved, where α = atan(D/J) and the easy-plane anisotropy λ = 1/sqrt(1+(D/J)^2) < 1. The simulated MCE results of the HDM chain can be found in Supplementary Note 5.

In the easy-plane XXZ model, the field g controls quantum fluctuations, and a Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition occurs at g = 0; while h breaks the Z2 symmetry and drives quantum supercriticality. According to the sine-Gordon model, the QSR crossovers follow T ∝ h^{1/(2-p)} 60,81, where 1/4 (XY limit) ≤ p ≤ 1/2 (Heisenberg limit) is the scaling dimension. Note that, as the exponent Δ is not defined for the BKT transition, its supercritical exponent p~ = 2-p is peculiar for the 1D HDM chain considered here. In Fig. 4b of the main text, we present the results of Γh ∝ 1/T^{p~}, where a quantum supercritical scaling Γh ∝ T^{-3/2} is observed for small D = 0.1 (λ ≃ 0.995 close to the Heisenberg limit).

Critical exponents of typical universality classes
The existence of universality classes with unique critical exponents is a fundamental aspect of QCPs not limited to Ising systems. A thorough examination of common universality classes, as listed in Table 1, reveals that Δ > 1 consistently holds. This ensures that the MCE of QSR is always significantly enhanced compared to that of QCR. In Table 1, for the conventional (d+1)D universality classes at dimensions d ≤ 2 (below the upper critical dimension), scaling relations yield Δ/zν = (d+3-η)/2. This results in quantum supercritical exponents of (2-η/2) for d=1 and (5/2-η/2) for d=2. Given that η/2 is usually a small positive value, this implies Δ/zν ≈ (d+3)/2. Consequently, we expect the Grüneisen ratio Γh to exhibit a significantly stronger divergence in 2D lattices than in 1D chains, e.g., Δ/zν ≃ 2.482 (2D) versus 1.875 (1D) for Ising systems. Furthermore, our numerical simulations can capture this distinct scaling behavior, as detailed in Supplementary Note 2.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors define new quantum supercritical regime, where -- as they show – small variations of the symmetry-breaking field induce large variations in temperature. They are demonstrated by tensor network simulations and their universal features are derived based on scaling analysis, showing universality of the phenomenon. They could have potential applications in new cooling techniques. Large temperature sensitivity was observed in experiments on Ising magnets and this is where the manuscript findings could find potential applications. I find the paper suitable for Nature Communications after the following points are addressed in the manuscript:\n1) The Ising magnets are Ising only in first approximation. There are extra terms in the Hamiltonian to which the near-critical system can be quite sensitive. For instance, the universality class can change from E_8 to D_8 in 1D magnets in the symmetry-breaking field. Do the experimental predictions of the manuscript survive these perturbations?\n2) Impressive tensor network calculations in 1D and on a finite-width cylinder were performed. Fig. 1 SM demonstrates convergence in D. How about convergence in the cylinder’s width W? Is W=10 enough for convergence? Near the critical point correlation range should be long ranged making any results sensitive to W.", "reviewer_2": "The authors perform a computational study of quantum Ising models in longitudinal and transverse fields using the tensor renormalization group. The focus of the investigation is on comparative investigation of the magneto-thermodynamic behavior at finite temperature above the vicinity of the quantum critical point. One of the main findings is that a longitudinal field gives rise to a stronger magnetocaloric effect than tuning the transverse field through the quantum critical point. This is good work and constitutes a timely investigation with interesting results and potential applications. The manuscript is overall well written. I am therefore in principle in favor of publication in Nature Communications.\nHowever, I have a few points which I think the authors should address:\n1) The classical supercritical behavior and related complex phase diagrams have been thoroughly investigated since before 1970. If one consults the authors' preprint arXiv:2407.05455v3 [1], it becomes clear that they are aware of this history, and I think it would be fair to add a few short comments with appropriate references on this historical background also in the present manuscript.\n2) There are 5 Supplementary Figures. These are referenced exclusively from the main text. This is not very friendly to the reader and makes the impression that the authors would have liked to include these in the main text, but were not able to do so for lack of space. I recommend a bit of reorganization with an admission which elements are not sufficiently important to be discussed in the main text, shifting the related discussion the the Supplementary Material. For example, I would shift the discussion of the two-dimensional quantum Ising model on lines 183-185 and 354-363 to the Supplementary Material. Likewise, it would seem more consistent to me to move the discussion of the Supplementary Figure 2 from lines 407-413 to the Supplementary Material. Also the discussion of the Supplementary Figure 3 around line 226 may be a bit difficult to access in the main text and might be easier to digest when suitably expanded and moved to the Supplementary Material.\nThere are a few further relatively minor items:\n3) Line 14: claims of discovery are delicate, in particular in the abstract. I thus recommend to reformulate this statement and to remove the \"discover\".\n4) Insets of Figs. 2(b) and (d): it did not become entirely clear to me which values of h and T these refer to. Can the authors explain? Maybe a different notation would also be better (after all, both h and T vary on the horizontal axes of Fig. 2).\n5) The abbreviation \"HDM\" appears in Fig. 4 and its caption before it has been introduced.\n6) Line 354: I think that some justification (e.g., a reference) should be provided for the value of gc=1.52. Incidentally, this number changes to gc=1.45 in the Supplementary Figure 1.\n7) I believe that the first identity in the equation between lines 403 and 404 needs justification (e.g., via a reference).\n8) Line 446: I recommend to remove the \"a careful\" (it is not clear to me why care is needed: this is either evident from Table I, or did the authors mean to say something that is not clear from this discussion?).\n9) I believe that the numbers in Table I need justification, e.g., via references. In particular the exact values of the critical exponents in 3D might depend on the investigation. Hence, the source(s) should really be specified.\n10) Notwithstanding a globally well written manuscript, a spell checker might still find and eliminate two or three spelling errors.\n11) A final suggestion: according to the abstract, one might have the impression that this is experimental work; maybe the authors should specify its theoretical nature explicitly. In the same spirit, maybe the list of experimental references could be shortened, in particular if the authors need to make space for some further theoretical references.\nI tried, but did not manage to download the file iLTRG.zip. Well, maybe it is not that important, but then why provide a link?"}, "author_response": "Manuscript: NCOMMS-25-47321\nTitle: QuantumSupercriticalRegimewithUniversalMagnetocaloricScalinginIsingMagnets\nDearReviewers,\nWe sincerely appreciate your favorable assessment of our work and thank you for your insightful comments. Wearedelightedthatyoubothrecognizetheinnovativevalueofthisresearchandconsideritworthyofpublication. Yourvaluablecommentsandsuggestionsarealsohighappreciated.\nIn the following, we give a point-by-point and detailed response to your comments and questions.\nThe text from your reports are cited in blue, with our subsequent response in normal format. With comprehensive revisions in the resubmitted manuscript, we believe the study is further strengthened and nowsuitableforpublicationinNatureCommunications.\nBestregards,\nEnzeLv,NingXi,YuliangJinandWeiLi\nResponsetothefirstReviewer’sreport\nReviewer #1 : The authors define new quantum supercritical regime, where — as they show — small variations of the symmetry-breaking field induce large variations in temperature. They are demonstrated bytensornetworksimulationsandtheiruniversalfeaturesarederivedbasedonscalinganalysis, showinguniversality of the phenomenon. They could have potential applications in new cooling techniques. Large temperature sensitivity was observed in experiments on Ising magnets and this is where the manuscript findings could find potential applications. I find the paper suitable for Nature Communications after the followingpointsareaddressedinthemanuscript:\nReply: We sincerely appreciate your concise summary and positive assessment of our research. As highlighted in your report, our work introduces the concept of a quantum supercritical regime (QSR) and elucidates related universal scaling behaviors in the magnetocaloric effect. Specifically, we demonstrate a universallydivergingGru¨neisenratioastemperatureapproachesabsolutezero,characterizingtheenhancedmagneticcooling. Moreover,QSRisnotmerelyatheoreticalconstruct;itcanbeexperimentallyrealizedinIsingmagnetssuchasCoNb O andothermaterials. Wefurtherproposeafield-tiltingprotocoltoharness2 6\nquantumsupercriticalcooling,offeringpracticalpathwaysforitsapplication.\nThus, our findings provide fresh perspectives on the fascinating interplay between symmetry-breaking field and spin fluctuations (both quantum and thermal) near the QCP at the endpoint of a first-order line. We expect these results to inspire further research on quantum Ising magnets and related systems, while alsoadvancingultralow-temperaturerefrigeration. WesincerelyappreciateyourrecommendationforpublicationinNatureCommunicationsandtheveryconstructivefeedback,towhichwehaveprovideddetailedresponsesbelow.\nReviewer #1 : 1) The Ising magnets are Ising only in first approximation. There are extra terms in the Hamiltonian to which the near-critical system can be quite sensitive. For instance, the universality class canchangefromE toD in1Dmagnetsinthesymmetry-breakingfield. Dotheexperimentalpredictions8 8\nofthemanuscriptsurvivetheseperturbations?\nReply: Thank you for the insightful question. For Ising-chain compounds like CoNb O , there are2 6\nindeed interchain coupling that leads to a low-temperature 3D ordered phase, with massive excitations(1)\nemerging. Thesemassivequasi-particlescanbedescribedbytheexceptionalE Liealgebra,orevenD8 8\nwhen considering the second-order approximation of inter-chain coupling. In experiments, signatures of(1)\nboth D (see arXiv:2403.10785, PRB 102, 104431 and J. Phys. A: Math. Theor. 55 484005) and E8 8\n(Science327, 177)spectrumappearonlyatalowtemperatures(< 1.5K).Thistemperaturescaleismuchlowerthanthatofthepredictedquantumsupercriticalscaling,whichappearsbelowabout4K,asshowninFig.4ofthemaintext.\nMoreover, the quantum supercritical scaling is fundamentally governed by the universality class of the QCP,ratherthanthespecificalgebraicstructureofmassiveexcitationsduetointer-chainterms. Thequantumcriticalregime(QCR)isknowntopersistinquantumIsingchainsandemergesabovelow-temperature3D ordered systems, as demonstrated in materials such as CoNb O (PRX 4, 031008) and BaCo V O2 6 2 2 8\n(PRL 127, 077201), etc. Similarly, the QSR can preserve its characteristic scaling behavior despite inter-chain couplings and extends beyond the 3D ordered phase boundary. Consequently, we believe that the experimental predictions regarding quantum supercritical scaling and its associated enhanced cooling effectsremainrobustagainstsuchperturbations. Wesincerelyappreciatethisconstructivecommentandhaveincorporatedadditionalremarksaddressingthispointinline274-286oftherevisedmanuscript.\nReviewer#1:2)Impressivetensornetworkcalculationsin 1Dandonafinite-widthcylinderwereperformed. Fig. 1 SM demonstrates convergence in D. How about convergence in the cylinder’s width W? Is W=10 enough for convergence? Near the critical point correlation range should be long ranged making anyresultssensitivetoW.\nReply: Thank you for your insightful comment. The thermal tensor network method used in our work — tanTRG (see PRL 130, 226502) — is a highly efficient and powerful approach for finite-temperature calculations on 2D cylinders. That said, we agree that demonstrating the robustness of our conclusions acrossdifferentcylinderwidthsW isessential.\nIn Fig. R1 below (also included in Fig. SM 4), we present results for 2D cylinders with system sizes of N = 12×24,N = 10×20andN = 8×16. Indeed,thecriticalfieldg∗obtainedfromDMRGcalculationsc\non finite-width cylinders shows a dependence on the cylinder width W, increasing from g∗ ≃ 1.45 for c\nW = 10tog∗ ≃ 1.465forW = 12. However,theGru¨neisenratioΓ remainremarkablyconsistentacrossc h\ndifferent system sizes, exhibiting the quantum supercritical scaling Γ ∝ T−∆/zν with ∆/zν ≃ 2.482h\nfor the (2+1)D Ising universality class. This robustness of QSR scaling behaviors, which confirms the reliability of our findings, can be understood as the finite field (h) and temperature (T) impose a cutoff on the correlation length. Consequently, even using finite-width cylinders, our state-of-the-art tanTRG approachachievesaccurateresultsincapturingthequantumsupercriticalscaling.\nResponsetothesecondReviewer’sreport\nReviewer#2: TheauthorsperformacomputationalstudyofquantumIsingmodelsinlongitudinalandtransversefieldsusingthetensorrenormalizationgroup. Thefocusoftheinvestigationisoncomparativeinvestigationofthemagneto-thermodynamicbehavioratfinitetemperatureabovethevicinityofthequantumcritical point. One of the main findings is that a longitudinal field gives rise to a stronger magnetocaloric effectthantuningthetransversefieldthroughthequantumcriticalpoint. Thisisgoodworkandconstitutesa timely investigation with interesting results and potential applications. The manuscript is overall well written. IamthereforeinprincipleinfavorofpublicationinNatureCommunications.\nReply: Wesincerelyappreciateyourhighassessmentofourworkandyourrecommendationinprinciple.\nThequantumcriticalregimeisawell-establishedandwidelyusedconceptinthestudiesofquantumphasetransition. QCP can thus be explored through their finite-temperature extensions — making them experimentally accessible. However, the effects of symmetry-breaking fields (such as the longitudinal field h in theIsingmagnet)neartheQCPremainanintriguingandunderexploredproblem. Whilepriorstudieshavefocused on low-energy massive excitations with exceptional algebraic structures (e.g., E quasiparticles),8\ntheuniversalscalingbehaviorsdrivenbythesymmetry-breakingfieldhavereceivedmuchlessattention.\nIn this work, we propose that the field h can give rise to a quantum supercritical regime, characterized by distinct universal scaling law (with exponent ∆ ≡ β + γ) and intriguing emergent phenomena. This regime can be viewed as an analog of the supercritical regime observed above the critical points of liquid-gas systems or ferromagnets — except here, it originates from a zero-temperature QCP. In particular, here we investigate the universally enhanced magnetocaloric effect (MCE) with quantum supercritical scaling. Our findings of quantum supercritical regime with enhanced MCE pave a viable pathway for advancing ultralow-temperaturerefrigeration. Notably,wehavealreadyengagedwithexperimentalcollaborators,andpreliminaryinvestigationsarecurrentlyinprogress. Weanticipatethatsupercriticalphenomenainquantummagnets will bring novel ideas and significant advances to the field of quantum magnetism and, more broadly, correlated quantum materials. This research benefits from the interdisciplinary synergy between statisticalphysicsandcondensedmatterphysics,whichwebelievecouldyieldnewresearchopportunities.\nReviewer#2: However,IhaveafewpointswhichIthinktheauthorsshouldaddress:\n1)Theclassicalsupercriticalbehaviorandrelatedcomplexphasediagramshavebeenthoroughlyinvestigated since before 1970. If one consults the authors’ preprint arXiv:2407.05455v3 [1], it becomes clear thattheyareawareofthishistory,andIthinkitwouldbefairtoaddafewshortcommentswithappropriatereferencesonthishistoricalbackgroundalsointhepresentmanuscript.\nReply: We sincerely appreciate this important suggestion. The classical supercritical regime traces its origins to 19th century discoveries, when researchers first identified a supercritical state emerging above the critical point at the end of the first-order liquid-gas line (Ann. Chim. Phys. 21, 127). In this regime, supercritical fluids exhibit strong thermal fluctuations that are highly sensitive to external fields. Subsequentyearswitnessedintensiveinvestigationsofsupercriticalphenomenaandcrossoverlinesinliquid-gassystems(seePNAS10216558-16562andPRL111.145901).\nNotably,recentworkbyoneofourcoauthorshasproposedthat—basedontheprincipleofuniversality—crossoverlinesinliquid-gassystemsshouldfollowascalinglaw|δPˆ±| ∼ (T −T )∆≡β+γ (PNAS121c\ne2400313121). Drawingonthewell-knownliquid-gas/ferromagnetanalogy,equivalentcrossoverbehavior emergesinIsingferromagneticsystemsneartheCuriepoint,withh ∼ (T −T )∆ underexternalmagneticc\nfieldh. Ourworkextendsthesesupercriticalcrossoverphenomenatothequantumregime,andthecrossoverlineh ∼ T∆/zν fallsintoaquantumscalinglawneartheQCP.\nTherefore,wecompletelyagreethatestablishingthisclassicalfoundationisessentialbeforeintroducingquantum supercriticality. Accordingly, we have added a concise background description (now located at line81-87oftherevisedmanuscript)thatcoversthesekeyconceptsandreferences.\nReviewer #2 : 2) There are 5 Supplementary Figures. These are referenced exclusively from the main text. This is not very friendly to the reader and makes the impression that the authors would have liked to include these in the main text, but were not able to do so for lack of space. I recommend a bit of reorganization with an admission which elements are not sufficiently important to be discussed in the main text, shifting the related discussion the the Supplementary Material. For example, I would shift the discussion ofthetwo-dimensionalquantumIsingmodelonlines183-185and354-363totheSupplementaryMaterial. Likewise, it would seem more consistent to me to move the discussion of the Supplementary Figure 2 from lines 407-413 to the Supplementary Material. Also the discussion of the Supplementary Figure 3 around line226maybeabitdifficulttoaccessinthemaintextandmightbeeasiertodigestwhensuitablyexpandedandmovedtotheSupplementaryMaterial.\nReply: Thank you for the very constructive suggestion, which we have fully taken into consideration. The discussion and methods related to the 2D Ising model (originally in lines 183-185 and 354-363) have beenmovedtotheSupplementaryNotes1and4,withappropriatereferencesinthemaintextandMethods. Some detailed discussion on quantum supercritical scaling of Gru¨neisen ratio Γ (originally in lines 407-h\n413)hasbeenmovedfromthemaintexttoSupplementaryNote2. Regardingthediscussionofg-hscaling at low and zero temperatures (around the original line 226, relating to Supplementary Fig. 3), we have expandedthedescriptionandmovedthemtoSupplementaryNote3.\nReviewer#2: Thereareafewfurtherrelativelyminoritems:\n3)Line14: claimsofdiscoveryaredelicate,inparticularintheabstract. Ithusrecommendtoreformulate thisstatementandtoremovethe“discover”.\nReply: Thankyouforthissuggestion. Wehaverevisedthewording,changing“discover”to“propose”tomoreaccuratelyreflectthetheoreticalnatureofourfindings.\nReviewer #2 : 4) Insets of Figs. 2(b) and (d): it did not become entirely clear to me which values of h and T these refer to. Can the authors explain? Maybe a different notation would also be better (after all, bothhandTvaryonthehorizontalaxesofFig. 2).\nReply: Thankyoufortheverycarefulreviewingofourmanuscript. ThehandT valuesrefertothepeak locations of specific heat C/T (panel b) and relaxation rate S(ω = 0) (panel d), respectively. These are indicated by solid circles and squares in both the main panel and inset, with a one-to-one correspondence betweenthem. Wehaverevisedthefigurecaptiontomakeitmoreexplicitandavoidpotentialambiguity.\nReviewer #2 : 5) The abbreviation ”HDM” appears in Fig. 4 and its caption before it has been introduced.\nReply: The abbreviation “HDM” refers to the Heisenberg-Dzyaloshinskii-Moriya chain, and we have nowexplicitlyincludedthisfullterminologyinthefigurecaptionforclarity.\nReviewer#2: 6)Line354: Ithinkthatsomejustification(e.g., areference)shouldbeprovidedforthevalueofgc=1.52. Incidentally,thisnumberchangestogc=1.45intheSupplementaryFigure1.\nReply: Thank you for the careful reviewing, g ≃ 1.522 is the critical transverse field in the 2D limit,c\nwhich can be obtained by quantum Monte Carlo (PRE 66, 066110). We have included related references in the revised manuscript. In our tanTRG (T > 0) and DMRG (T = 0) simulations, we can address long cylinders with finite width W. For W = 10 (W = 12) cylinder, the critical field g∗ ≃ 1.45 (g∗ ≃ 1.465)c c\nis determined, at which we conduct finite-temperature calculations. These details have been expanded in SupplementaryNotes1and4.\nReviewer #2 : 7) I believe that the first identity in the equation between lines 403 and 404 needs justification(e.g.,viaareference).\nReply: Thank you for bringing this point to our attention. We have now significantly expanded our analysis of the Gru¨neisen ratio (Γ ) in Methods to provide a more detailed derivation, ensuring all keyh\nstepsarenowclearlypresented(seelines420-429oftherevisedversion). Moreoverandassuggested, we haveaddedappropriatereferencefortheuniversalform(M.Continentino,Quantumscalinginmany-bodysystems,2nd)discussedinourmanuscript.\nReviewer #2 : 8) Line 446: I recommend to remove the “a careful” (it is not clear to me why care is needed: thisiseitherevidentfromTableI,ordidtheauthorsmeantosaysomethingthatisnotclearfromthisdiscussion?).\nReply: Thank you for bringing this to our attention. We have revised the wording, replacing “careful” with“thorough”tobetterconveythatwehavesystematicallyinvestigatedthecommonuniversalityclasses. We have also revised the section “Critical exponents of typical universality classes” to include additional analysisbasedonthescalingrelationsandcriticalexponentspresentedinTableI.\nReviewer#2: 9)IbelievethatthenumbersinTableIneedjustification,e.g.,viareferences. Inparticular the exact values of the critical exponents in 3D might depend on the investigation. Hence, the source(s) shouldreallybespecified.\nReply: Weappreciatethissuggestion, andhaveincludedthereferencesfor(2+1)Duniversalityclasses in Tab. I through a new column. In addition, we also put ∆/zν explicitly in the table to facilitate the comparisonsbetweendifferentuniversalityclasses.\nReviewer #2 : 10) Notwithstanding a globally well written manuscript, a spell checker might still find andeliminatetwoorthreespellingerrors.\nReply: We appreciate your careful reading and helpful reminder. We have now conducted a thorough checkofourmanuscriptandidentifiedseveraltyposthatarenowcorrected.\nReviewer#2: 11)Afinalsuggestion: accordingtotheabstract,onemighthavetheimpressionthatthis isexperimentalwork;maybetheauthorsshouldspecifyitstheoreticalnatureexplicitly. Inthesamespirit, maybethelistofexperimentalreferencescouldbeshortened,inparticulariftheauthorsneedtomakespaceforsomefurthertheoreticalreferences.\nReply: Thanks for the kind suggestion. Now we have new revised the abstract to emphasize that the QSRscalingbehaviorsareobtained“throughthermaltensornetworksimulations”,tospecifythetheoretical progress of our study explicitly. Regarding the references, we have removed several experimental studies (original Refs. 13, 22, 24, 25, 26, 30, 34 and 37) that, while valuable, were not directly pertinent to CoNb O —theprimaryIsing-chainmaterialconsideredinthiswork. Additionally,weaddsomerelevant2 6\ntheoreticalreferencestoourmanuscript,asindicatedintherevisionlist.\nReviewer #2 : I tried, but did not manage to download the file iLTRG.zip. Well, maybe it is not that important,butthenwhyprovidealink?\nReply: Thank you for your kind reminder. Our code of thermal tensor network is available on GitHub at https://github.com/ThermoTN/QSpinTN.ml. To facilitate easier access, we have also uploaded the code tohttps://doi.org/10.5281/zenodo.16919271."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am entirely satisfied with the thorough response to my comments and the amendments in the manuscript/SM. Therefore, I would like to recommend publication in the present form.", "reviewer_2": "With their revised version, the authors have improved their manuscript. In particular, the reorganization between main text and the Supplementary Information significantly improves readability. I thus believe that the manuscript is almost ready for publication in Nature Communications.\nNevertheless, I still stumbled across a few minor details while rereading the manuscript and the Supplementary Information:\n1) Lines 74-75, \"the exceptional Lie algebra\": There are five, see, e.g., Wikipedia, https://en.wikipedia.org/wiki/Exceptional_Lie_algebra. The authors should thus either specify which one (I guess E ) or 8 change the formulation.\n2) Inset of Fig. 2b: Even if this is now clear to me, I still suggest to adapt notation and write, e.g., T on the vertical axis \\max rather than just T.\n3) Line 184: \"larger\" might be better than \"greater\".\n4) Line 398: It is not clear to me what \"There exhibits\" is supposed to mean. In my opinion, this is grammatically incorrect and might merit being revisited.\n5) New text on lines 420-427: I was not able to follow the logic. Maybe this is due to the argument being interrupted by a full stop in the middle of the sentence (line 426). Can the authors simplify the discussion and make it clearer? See also the next point.\n6) Eq. (2) (between lines 429 and 430): It seems that here I did not get across. The first identity is not the definition of Γ , but h uses a Maxwell relation. Therefore, I still think that justification via a reference would be appropriate.\n7) I appreciate the addition of references for the (2+1)D universality classes in Table I. However, this leaves one wanting for a reference also for the (1+1)D and mean-field universality classes in Table I. I thus recommend to add a source also for these values of the exponents. If the authors can find one textbook or review article that covers them all, this would be fine for me.\n8) Supplementary Information, inset of Supplementary Figure 3b: I recommend to use a notation that is different from $h$ on the vertical axis (see also point 2 above).\n9) Supplementary Information, line below Eq. (1): remove \"we have\" just before Γ .\nh\n10) Supplementary Information, 4 lines below Supplementary Figure 4: \"thorough\" -> \"through\".\nI read the comment in the resubmission letter. Beyond this, I trust the authors that they know what they are doing. In any case, I believe that the manuscript no longer refers to a code."}, "author_response": "ResponsetothefirstReviewer’sreport\ninthemanuscript/SM.Therefore,Iwouldliketorecommendpublicationinthepresentform.\nReply: Thank you for your positive assessment and recommendation of our work! We are delighted\nto hear that you are satisfied with our responses and revisions, which have helped us to improve our work\nsignificantly. Thankyouforyourtimeandeffortsreviewingourwork.\nResponsetothesecondReviewer’sreport\nReviewer#2: Withtheirrevisedversion,theauthorshaveimprovedtheirmanuscript. Inparticular,the\nreorganization between main text and the Supplementary Information significantly improves readability. I\nthusbelievethatthemanuscriptisalmostreadyforpublicationinNatureCommunications.\nNevertheless,IstillstumbledacrossafewminordetailswhilerereadingthemanuscriptandtheSupple-\nmentaryInformation:\nReply: Thank you for your positive assessment of our revised manuscript. We are delighted that you\nfound the revisions significantly improved the readability and are grateful for the additional suggestions.\nWehaveincorporatedthisfeedbackintothemanuscript,andbelieveitisnowreadyforpublication.\nReviewer #2 : (1) Lines 74-75, “the exceptional Lie algebra”: There are five, see, e.g., Wikipedia,\nhttps://en.wikipedia.org/wiki/Exceptional Lie algebra. Theauthorsshouldthuseitherspecifywhichone(I\nguessE )orchangetheformulation.\n8\n8 8\ninLine74inthemaintext.\nReviewer#2: (2)InsetofFig. 2b: Evenifthisisnowcleartome,Istillsuggesttoadaptnotationand\nwrite,e.g.,T ontheverticalaxisratherthanjustT.\nmax\nReply: Thankyouforthereminder. WehaveupdatedthenotationfromlnT tolnT andlnT (subscript\np m\n“p”forpeakand“m”formax)intheinsetofFig. 2banddassuggested.\nReviewer#2: (3)Line184: “larger”mightbebetterthan“greater”.\nReply: Assuggested,wehaveadjustedtheexpressionfrom“greater”to“larger”inLine184.\nReviewer #2 : (4) Line 398: It is not clear to me what ”There exhibits” is supposed to mean. In my\nopinion,thisisgrammaticallyincorrectandmightmeritbeingrevisited.\nReply: Thankyouforpointingoutthistypo,andwehaverevisedthissentence(nowinLines392-394):\n“we propose that the peak/dip values of Γ exhibits the characteristic T−1/zν scaling, even near such a\ng\nself-dualQCP”.\nReviewer#2: (5)Newtextonlines420-427: Iwasnotabletofollowthelogic. Maybethisisduetothe\nargumentbeinginterruptedbyafullstopinthemiddleofthesentence(line426). Cantheauthorssimplify\nthediscussionandmakeitclearer? Seealsothenextpoint.\nReply: Thank you for your question. We apologize for the omission of the Gru¨neisen ratio definition\nhere. Wehaveclarifiedthisrelationshipinline419andreorganizedtheexpressiontomakethecorrespond-\ningparagraphmorelogical(Lines422-425).\nReviewer#2: (6)Eq. (2)(betweenlines429and430): ItseemsthathereIdidnotgetacross. Thefirst\nidentityisnotthedefinitionofΓ ,butusesaMaxwellrelation. Therefore,Istillthinkthatjustificationvia\nh\nareferencewouldbeappropriate.\nReply: Thank you for the question. We have revised the description in Lines 422-425 and Eq. (2) for\nclarity,andcitedarelevantreferencePhys. Rev. B72,205129(2005)ontheGru¨neisenratio.\nReviewer#2: (7)Iappreciatethieadditionofreferencesforthe(2+1)DuniversalityclassesinTableI.\nHowever,thisleavesonewantingforareferencealsoforthe(1+1)Dandmean-fielduniversalityclassesin\nTableI.Ithusrecommendtoaddasourcealsoforthesevaluesoftheexponents. Iftheauthorscanfindone\ntextbookorreviewarticlethatcoversthemall,thiswouldbefineforme.\nReply: Thankyouforthesuggestion. Wenowprovidereferencesonthe(1+1)Dandmean-fielduniver-\nsalityclassesinTable.I.Inparticular,atextbook,“ConformalInvarianceandCriticalPhenomena”byM.\nHenkel,isnowincludedinthereferencelist.\nReviewer#2: (8)SupplementaryInformation,insetofSupplementaryFigure3b: Irecommendtousea\nnotationthatisdifferentfromhontheverticalaxis(seealsopoint2above).\nmax\nReviewer#2: (9)SupplementaryInformation,linebelowEq. (1): remove“wehave”justbeforeΓ .\nh\nReply: Wehaveremoved“wehave”belowEq. (1).\nReviewer#2: (10)SupplementaryInformation,4linesbelowSupplementaryFigure4: “thorough”→\n“through”.\nReply: WehaverevisedthetypobelowSupplementaryFigure4.\nReviewer #2 : I read the comment in the resubmission letter. Beyond this, I trust the authors that they\nknowwhattheyaredoing. Inanycase,Ibelievethatthemanuscriptnolongerreferstoacode.\nReply: Thank you very much for you trust. Meanwhile, we have updated our “Data Availability” and\n“CodeAvailability”sectionsandbelievethenewversionmeetsyourrequirements."}]

Decision letter