Near a quantum critical point (QCP), the low-temperature thermodynamics follows universal scaling laws. Using copper sulfate pentahydrate-a canonical spin-1/2 antiferromagnetic Heisenberg chain compound-we report the observation of a universal magnetocaloric effect (MCE) near a field-driven QCP. Remarkably, in the 1D quantum-critical regime, we obtain the universal magnetocaloric scaling function via adiabatic demagnetization measurements, which agrees with the analytical solutionof the critical 1D Fermi gas. This establishes the copper sulfate crystal as an ideal platform for studying quantum criticality and universal phenomena. Upon further cooling, our MCE and nuclear magnetic resonance measurements reveal a dimensional crossover to a 3D quantum-critical regime of the Bose-Einstein condensation (BEC) universality class, characterized by the scaling lawand a clear data collapse of the magnetic Grüneisen ratio with the 3D Bose-gas scaling function. Practically, this quantum-critical MCE enables cooling to 68.7 mK near the QCP and achieves a lowest temperature of 12.8 mK at zero field without the need for helium-3. Our work identifies a universal MCE in a common compound, establishing this magnon BEC system as a prototype quantum-critical coolant and a platform for next-generation millikelvin refrigeration.
Near a quantum critical point (QCP), the low-temperature thermodynamics follows universal scaling laws. Using copper sulfate pentahydrate-a canonical spin-1/2 antiferromagnetic Heisenberg chain compound-we report the observation of a universal magnetocaloric effect (MCE) near a field-driven QCP. Remarkably, in the 1D quantum-critical regime, we obtain the universal magnetocaloric scaling function via adiabatic demagnetization measurements, which agrees with the analytical solutionof the critical 1D Fermi gas. This establishes the copper sulfate crystal as an ideal platform for studying quantum criticality and universal phenomena. Upon further cooling, our MCE and nuclear magnetic resonance measurements reveal a dimensional crossover to a 3D quantum-critical regime of the Bose-Einstein condensation (BEC) universality class, characterized by the scaling lawand a clear data collapse of the magnetic Grüneisen ratio with the 3D Bose-gas scaling function. Practically, this quantum-critical MCE enables cooling to 68.7 mK near the QCP and achieves a lowest temperature of 12.8 mK at zero field without the need for helium-3. Our work identifies a universal MCE in a common compound, establishing this magnon BEC system as a prototype quantum-critical coolant and a platform for next-generation millikelvin refrigeration.
Bose-Einstein condensation (BEC), a macroscopic quantum phenomenon arising from phase coherence and bosonic statistics, has been realized in quantum magnets. Here, we report the observation of a universal magnetocaloric effect (MCE) near a BEC quantum critical point (QCP) in copper sulfate crystal (CuSO4⋅5H2O). By conducting magnetocaloric and nuclear magnetic resonance measurements, we uncover a field-driven BEC QCP, evidenced by the universal scaling law Tc∝(Bc−B)2/3 and the perfect data collapse of the magnetic Grüneisen ratio. Thermal excitation triggers a dimensional crossover to a 1D quantum-critical regime, where the MCE scaling strictly matches the universality class of 1D Fermi gases. Notably, the quantum-critical MCE enables cooling down to 12.8 mK without helium-3, with very fast thermal relaxation rate that is critical for high cooling power. This work demonstrates the universal MCE in magnon BEC systems, using a common copper sulfate compound as a paradigmatic example, and paves the way for next-generation sub-Kelvin cooling.
The magnetic Grüneisen ratio, defined as Γ g ≡ (1/ T )(∂ T /∂ g ) S , characterizes the magnetocaloric effect (MCE) driven by an external field g and serves as a highly sensitive probe for field-induced quantum critical points (QCPs). Near a QCP, Γ g displays diverging behaviors described by a universal scaling function depending on the universality class. Here, we systematically investigate the universal scaling functions of Grüneisen ratio Γ g in typical one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, XY model, q -state quantum Potts model ( q = 3, 4), and the J 1 - J 2 columnar dimer model. Our approach employs the linearized tensor renormalization group (LTRG) method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Γ g near QCPs. We extract universal scaling functions via data collapse analysis. This universal magnetocaloric framework offers a quantitative tool for analyzing measurements in quantum materials and for guiding the search for quantum critical cooling applications.
The Grüneisen ratio, defined as Γg≡(1/T)(∂T/∂g)S, serves as a highly sensitive probe for detecting quantum critical points (QCPs) driven by an external feild g and for characterizing the magnetocaloric effect (MCE). Near a QCP, the Grüneisen ratio displays a universal divergence which is governed by a universality-class-dependent scaling function stemming from the scale invariance. In this work, we systematically investigate the universal scaling functions of Grüneisen ratio in both one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, the spin-1/2 Heisenberg model, the quantum q-state Potts model (q=3,4) and the J1-J2 columnar dimer model. Our approach employs the thermal tensor-network method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Grüneisen ratio near QCPs. Through data collapse analysis, we extract the corresponding scaling functions, which establish quantitative frameworks to interpret magnetocaloric experiments and guide the development of ultralow-temperature refrigeration.
Abstract Quantum critical points (QCPs) ubiquitously emerge in strongly correlated systems, with their influence extending beyond zero temperature. There exist extended quantum critical regimes (QCRs) at finite temperatures and external fields. A paradigmatic example exhibiting QCP is quantum Ising magnets like CoNb 2 O 6 , where critical quantum fluctuations and robust universal scaling behavior provide crucial insights into the quantum material. In this work, we discover a fundamentally distinct quantum supercritical regime (QSR) emerging also from the QCP but controlled by symmetry-breaking fields (h) coupled to the order parameter. The QSR is enclosed by the crossover boundaries T~h zν/Δ , where z, ν and Δ≡β+γ are critical exponents. Amongst other intriguing phenomena, there exists an enhanced magnetocaloric effect (MCE) in QSR, characterized by a universally diverging magnetic Gr\"uneisen ratio Γ h ≡1/T(∂T/∂h) S ~T -Δ/zν . Even small symmetry-breaking fields generate dramatic temperature variations. The supercritical scaling can be comprehended via thermal data collapse based on the derived scaling form. We propose to observe the QSR and related supercritical MCE in CoNb 2 O 6 and other quantum Ising magnets, which may enable cryogenic technologies capable of reaching the millikelvin regime without relying on helium-3.
Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (MCE). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the form of Ising and Berezinskii-Kosterlitz-Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling T∝h2/3 and a Grüneisen ratio scaling Γh∝T−3/2 for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function ϕΓ(x) of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression ϕΓ(x)≈x/(1+x2). Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.
Abstract Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (Lv et al 2025 Nat. Commun. 16 10646). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the forms of Ising and Berezinskii–Kosterlitz–Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling T ∝ h 2 / 3 and a Grüneisen ratio scaling Γ h ∝ T − 3 / 2 for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function ϕ Γ ( x ) of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression ϕ Γ ( x ) ≈ x / ( 1 + x 2 ) . Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.