Quantum Supercritical Regime with Universal Magnetocaloric Scaling in Ising Magnets

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 $T \propto h^{{zν}/Δ}$, where $z$, $ν$ and $Δ\equiv β+γ$ 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 $Γ_h \propto T^{-Δ/{zν}}$, which indicates that a small symmetry-breaking field $h$ can generate dramatic temperature variation. We propose to observe the quantum supercritical regime in Ising-chain compound CoNb$_2$O$_6$ and related quantum materials, revealing a helium-3-free pathway to millikelvin cooling via the supercritical magnetocaloric effect.

Figures (5)

• Figure 1: Schematic diagram of the quantum supercritical regime and universal magnetocaloric effects.a Above the Curie point (CP) $T_c$, there exists a classical supercritical regime (SR), reflecting the profound analogy between FM transitions and liquid-gas phase transitions. The SR is extended down by the quantum fluctuation $g$ and eventually becomes the quantum supercritical regime (QSR) originating from the quantum critical point (QCP). The QSR is located within the $h$-$T$ plane orthogonal to the quantum critical regime (QCR) situated in the $g$-$T$ plane. The two blue thick lines represent the first-order phase transitions, and the blue curves are quantum supercritical crossovers in the $g$-$h$ plane at zero temperature wang2024qsc. b The QSR is situated between two different ordered regimes in the $h$-$T$ plane, and the QSR crossover lines follow a distinct supercritical scaling $T \propto h^{{z\nu}/\Delta}$, with $z$, $\nu$ and $\Delta\equiv \beta+\gamma$ the critical exponents. c The QCR separates the ordered and disordered spin states in the $g$-$T$ plane. The two QCR crossover lines follow the quantum critical scaling $T \propto \tilde{g}^{z\nu}$, where $\tilde{g} \equiv g-g_c$ measures the distance to the QCP at $g_c$. d Schematic diagram of quantum supercritical cooling. Starting from an ordered phase at initial temperature $T_i$ ($T_i'$), the quantum system reaches $T_f$ ($T_f'$) within the QSR by reducing or tilting the magnetic field ($\theta$ represents the field-tilting angle). e Crystal structure of CoNb_2O_6. The Co$^{2+}$ ions carry spin-1/2 moments and form Ising chains along the $c$ axis.
• Figure 2: Universal quantum supercritical cooling with enhanced Grüneisen ratio.a The simulated isentropic lines of 1D quantum Ising model in the $h$-$T$ plane (with $\tilde{g}=0$). Red dashed lines represent the crossover lines enclosing the QSR. bThe double-peak specific heat curves for different field $h$. The position of the lower peak reveals the crossover line, scaling as $T_p \propto h^{{z\nu}/\Delta}$ (see inset). Solid circles in the main panel indicate the peaks of $C/T$, with one-to-one correspondence to those in the inset. c The Grüneisen ratio $\Gamma_h$ for each fixed temperature. Circles ($h_{\rm cr}, {\color[rgb]{.0,.0,.0}{T}}$), marking the peaks (dips) of $\Gamma_h$, constitute the crossover lines with an exponent ${z\nu}/\Delta$ (see inset). Red dashed lines mark the crossover boundary, as in panel a.d The calculated spin-lattice relaxation rate $\mathcal{S}_1(\omega=0)$, which exhibits a power-law scaling $T^{-3/4}$ within the QSR, whose peak locations also reveal the QSR crossover line (see inset). Solid squares in the inset represent the peak locations $T_m$ in the main panel, which also exhbits the quantum supercritical scaling. Black dashed lines in b and d represent the $\tilde{g} = h = 0$ data.e The comparison between peak values $\Gamma_h({\color[rgb]{.0,.0,.0}{h_{\rm cr}, T}})$ of QSR and those of QCR (shown in Methods), where $\Gamma_{h}$ diverges much more rapidly as $\Delta =15/8 > 1$. The distinct scaling laws for both cases are illustrated by dashed lines.f The scaling function $\phi_{\Gamma_h}(x)$ is obtained through data collapse from c, with color bar the same as c. The $\pm x_0$ points correspond to the crossover lines shown in a, and the QSR is within the regime $-x_0 \leq x \leq x_0$. Black dashed line indicates the linear behavior near $x=0$.
• Figure 3: Hyperscaling function of Grüneisen ratio and crossover surfaces near the QCP.a The hyperscaling function $\psi_{\Gamma_h} (x,y)$ near the QCP of quantum Ising model, obtained through data collapsing. The red lines indicate the locations of peaks/dips in $\psi_{\Gamma_h}(x,y)$ landscape when scanning $x$ for various fixed $y$. Each point $(x_i, y_i)$ in a corresponds to a crossover line in b, and these lines form a crossover surface in the $g$-$h$-$T$ diagram. The gray plane represents an isothermal cut, where the dashed intersection lines connect the locations of the maxima in $\Gamma_h$ at fixed $T$. In the low-temperature limit, they satisfy $h \propto (g - g_c)^{\Delta}$, i.e., quantum supercritical crossovers in the $g$-$h$ plane proposed in Ref. wang2024qsc.
• Figure 4: Quantum supercritical cooling effect for the Ising magnet CoNb_2O_6.a Simulated isentropic lines of CoNb_2O_6 by varying the longitudinal field $B_z$, while the transverse field is fixed at $B_x^c \simeq 5.25$ T. The simulations are performed using realistic parameters for the compound, with results presented in experimental units. Inset shows the field-tilting protocol for observing quantum supercritical behaviors in CoNb_2O_6. A small tilting angle of about 2$^\circ$ generates sufficient $B_z$ component to enable quantum supercritical cooling into the ultralow-temperature regime. b Temperature scaling of magnetic Grüneisen peak values for various systems. Blue circles represent the calculated results of CoNb_2O_6. The supercritical Heisenberg-Dzyaloshinskii-Moriya (HDM) system is also plotted as a comparison, where the coupling strength is set as $J\simeq 18.2$ K as in the compound Cu benzoate. The hollow dots are adapted from experimental results of different materials, such as the triangular-lattice Heisenberg antiferromagnet Cs_2CuCl_4 Wolf2014 and the spin chain CuP Oliver2017.
• Figure 5: Quantum critical cooling and diverging Grüneisen ratio $\Gamma_g$.a The isentropic lines of 1D quantum Ising model near QCP are illustrated in the $g$-$T$ plane with $h=0$. Blue dashed lines represent the crossover lines enclosing the QCR. b The Grüneisen ratio $\Gamma_g$ for each fixed temperature. The inset collects the peak/dip locations $(\tilde{g}_{\rm cr}, {\color[rgb]{.0,.0,.0}{T}})$, which exhibit a power-law scaling $\tilde{g}_{\rm cr}\propto T^{1/z\nu}$.