Revealing three-dimensional quantum criticality by Sr substitution in Han purple

Classical and quantum phase transitions (QPTs), with their accompanying concepts of criticality and universality, are a cornerstone of statistical thermodynamics. An excellent example of a controlled QPT is the field-induced ordering of a gapped quantum magnet. Although numerous “quasi-one-dimensional” coupled spin-chain and -ladder materials are known whose ordering transition is three-dimensional (3D), quasi-two-dimensional (2D) systems are special for multiple reasons. Motivated by the ancient pigment Han purple (${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$), a quasi-2D material displaying anomalous critical properties, we present a complete analysis of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$. We measure the zero-field magnetic excitations by neutron spectroscopy and deduce the spin Hamiltonian. We probe the field-induced transition by combining magnetization, specific-heat, torque, and magnetocalorimetric measurements with nuclear magnetic resonance studies near the QPT. With a Bayesian statistical analysis and large-scale Quantum Monte Carlo simulations, we demonstrate unambiguously that observable 3D quantum critical scaling is restored by the structural simplification arising from light Sr substitution in Han purple.

Figure 1

Structure and inelastic neutron spectrum of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$: (a) Tetragonal crystallographic structure of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ [55]. (b) Representation of the minimal magnetic unit cell, indicating the relevant Heisenberg interaction parameters between dimers on the two bilayers, which are structurally equivalent with body-centered geometry. $(\widehat{a},\widehat{b},\widehat{c})$ and $({\widehat{a}}^{\prime },{\widehat{b}}^{\prime },{\widehat{c}}^{\prime })$ denote respectively the axes of the crystallographic and minimal magnetic unit cells [54]. Dotted gray lines indicate the correspondence between ${\mathrm{Cu}}^{2+}$ ion positions in the two cells. (c, d) Neutron scattering spectra measured on CAMEA at 1.5 K for two high-symmetry $\vec{Q}$-space directions; $\vec{Q}$ is indexed in reciprocal lattice units (r.l.u.) of the crystallographic unit cell. (e, f) Spectra modeled using the interaction parameters and line widths extracted from the data collected on both CAMEA and TASP (Fig. 2).

Figure 2

Inelastic neutron scattering analysis: (a) Scattered intensity as a function of energy transfer, measured on TASP at $\vec{Q}=\left(004\right)$ r.l.u. for ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ at 1.5 K (blue symbols) and ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ at 1.6 K (red). Shaded regions represent Gaussian fits used to extract the triplon energy, linewidth, and intensity. (b–e) Multiple ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ energy scans, of the type shown in panel (a), were analyzed for $\vec{Q}$-vectors covering two high-symmetry directions in reciprocal space. (b, c) Triplon mode energies obtained from data gathered on TASP (blue symbols) and CAMEA (open). (d, e) Corresponding scattered intensities. Solid lines are fits obtained from the spin Hamiltonian represented in Fig. 1 with the optimal parameters given in the text.

Figure 3

Phase boundary of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$: Results from a wide range of thermodynamic measurements performed on single-crystalline ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ at high magnetic fields, all with $\widehat{H}\parallel \widehat{c}$. (a) Magnetization, $M\left(H\right)$, measured in pulsed fields up to 50 T. Curves are shown with successive vertical offsets of 0.3. Left and right labels are the calibrated temperatures obtained respectively for the lower and upper phase transitions after correction for the strong magnetocaloric effects (MCEs) acting during the field pulse. The inset shows the second derivative, ${\partial }^{2}M/{\partial H}^2$, highlighting the locations of the ordering transitions. (b) Specific heat, $C_p\left(T\right)$, measured in pulsed fields up to 37.8 T and showing a $\lambda$ anomaly at the phase boundary. The maximum transition temperature, $T_c^{\max }\simeq 3.8$ K at 35.2 T, demarcates the top of the phase-boundary dome. (c) Magnetic torque, $\tau \left(H\right)$, measured in static fields around the onset of magnetic order. Curves above 0.35 K are shown with vertical offsets of 0.2. The inset shows the second derivative, ${\partial }^2\tau /{\partial H}^2$. (d) Field-temperature phase diagram of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ obtained by collecting all of our experimental data from magnetization, specific heat, magnetic torque, MCE, and NMR. Solid lines are lines of constant entropy obtained from our MCE measurements. Color contours represent the magnetic Grüneisen parameter, ${\mathrm{\Gamma }}_H=-(\partial M/\partial T)/C_p=-{(\partial T/\partial H)}_S/T$, which shows a sharp change of sign at the phase boundary. The black dashed line shows for comparison the phase boundary of ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ [43].

Figure 4

NMR measurements in the critical regime: (a, b) ${}^{29}\mathrm{Si}$ NMR spectra measured for a range of field values around the corresponding value of $H_c\left(T\right)$ at 120 mK (a) and 412 mK (b). The spectra show a splitting from a single, sharp peak (blue) in the quantum disordered phase at $H<H_c\left(T\right)$ to two generally broader peaks (red and green) in the magnetically ordered phase at $H>H_c\left(T\right)$. The measured intensities were modeled as the sum (gray) of the single or double peak(s) and a weak, constant contribution. The double peaks were modeled by a simultaneous fitting procedure in all panels other than the highest four fields shown at 120 mK (a), where individual peak fits were used. (c) Splitting of peaks in the NMR spectra. Solid lines show splitting functions deduced from simultaneous fits to all spectra measured at the same temperature, and crosses mark the field values at which data were taken. (d) Low-field phase boundary of ${\mathrm{Ba}}_{0.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuSi}}_2{\mathrm{O}}_6$ extracted from the NMR peak-splitting data (blue and white circles). Orange and white squares show the phase boundary obtained from QMC simulations. Blue and orange lines show the optimal fit to the phase boundary based on each set of points, obtained from the mean of the posterior distribution determined by a Bayesian inference procedure based on Eq. (1). The blue and orange shaded areas represent the estimated uncertainties in the form of the 68% (dark) and 95% (light shading) credible intervals (CIs). NMR and QMC data points marked by white circles and squares are those excluded from the statistical analysis of the quantum critical properties. Inset: NMR peak-splitting measured up to 25 T at $T=120$ mK (blue circles) juxtaposed with the magnetic order parameter, $m_{\perp }\left(T\right)$, obtained from QMC simulations.

Figure 5

Analysis of the critical exponent: (a–c) Posterior probability distributions of the parameters in the model of Eq. (1) for the field-temperature phase boundary. The input for the analysis is the set, ${\mathcal{D}}_{\mathrm{NMR}}$, of six blue NMR data points up to $T_{\max }=412$ mK in Fig. 4. The marginal posterior distributions, $p\left({\theta }_i\right|\mathcal{D})$, for each of the model parameters ${\theta }_i\equiv \alpha$ (a), $\varphi$ (b), and $H_{c1}$ (c), are visualized as histograms of samples drawn from the posterior distribution together with their mean (blue lines), 68%, and 95% CI boundaries. (d) Posterior distribution shown by projection to $\varphi$ and $H_{c1}$ (i.e., integrating over $\alpha$), which illustrates the strong negative correlation between the two parameters. (e) Phase-boundary points of Fig. 4 shown on logarithmic axes. The gradients of both sets of points correspond to the critical exponent, $\varphi$. For comparison we indicate the cases $\varphi =1$ (dotted) and $\varphi =2/3$ (dashed lines). (f) Evolution of $\varphi$ as a function of the maximal temperature, $T_{\max }$, up to which phase-boundary data points are included. Horizontal lines show the mean and shaded boxes the 68% and 95% CIs of the marginal distribution of $\varphi$ deduced by including only points below $T_{\max }$; the 95% CIs for $T_{\max }=264$ mK lie at 0.465 and 0.923. The NMR symbols were obtained by including $5,6,\cdots ,10$ of the data points shown in Fig. 4 and panel (e), the QMC symbols by including between 8 and 30 data points.