Quantum-Critical Spin Dynamics in Quasi-One-Dimensional Antiferromagnets

By means of nuclear spin-lattice relaxation rate $T_1^{-1}$, we follow the spin dynamics as a function of the applied magnetic field in two gapped quasi-one-dimensional quantum antiferromagnets: the anisotropic spin-chain system ${\mathrm{NiCl}}_2\mathrm{\text{−}}4\mathrm{SC}({\mathrm{NH}}_2{)}_2$ and the spin-ladder system $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2\mathrm{Cu}{\mathrm{Br}}_4$. In both systems, spin excitations are confirmed to evolve from magnons in the gapped state to spinons in the gapless Tomonaga-Luttinger-liquid state. In between, $T_1^{-1}$ exhibits a pronounced, continuous variation, which is shown to scale in accordance with quantum criticality. We extract the critical exponent for $T_1^{-1}$, compare it to the theory, and show that this behavior is identical in both studied systems, thus demonstrating the universality of quantum-critical behavior.

Figure 1

Characteristic field-temperature phase diagram of weakly coupled, gapped antiferromagnetic chains or ladders around the critical field $B_{c1}$ that closes the gap. The slopes $-g{\mu }_B$ and $0.76g{\mu }_B$ [15] of the temperature crossovers to the gapped and TLL regions, respectively, are indicated by dashed lines. Dotted lines indicate characteristic temperature scales set by the dominant, 1D coupling $J$ and weak, 3D couplings $J_{3\mathrm{D}}$.

Figure 2

$T_1^{-1}$ as a function of temperature $T$ for several magnetic field values around the QCP. (a) In BPCB, ${}^{14}\mathrm{N}$ $T_1^{-1}$ data are taken around $B_{c1}=6.723\mathrm{T}$ on $\mathrm{N}(1{)}_{\mathrm{II}}$ NMR lines as defined in Ref. 12. The value of $B_{c1}$ is determined from the low-temperature magnetization as measured by ${}^{14}\mathrm{N}$ NMR hyperfine shift. (A more precise determination provides a slightly higher value than reported in Ref. 12.) (b) In DTN, proton $T_1^{-1}$ data are taken around $B_{c2}=12.325\mathrm{T}$ on the highest-frequency NMR line. Magnetic field is aligned with the $c$ axis to within 1°. The value of $B_{c2}$ is determined from the low-temperature boundary $T_c(B)$ of the 3D ordered state as measured by proton NMR [31]. Thick gray lines in (a) and (b) are $T_1^{-1}(T)$ predictions for the TLL behavior close to the critical field ($T_1^{-1}\propto T^{-1/2}$), for the tentative quantum-critical behavior exactly at the critical field ($T_1^{-1}\propto T^{1/2}$), and for the gapped behavior, from top to bottom, respectively.

Figure 3

Demonstration of the scale invariance in the quantum-critical region of DTN. The best collapse of different $T_1^{-1}(T)$ data sets [displayed in Fig. 2b] on the same curve in scaling variables $-(B-B_{c2})/T^{\beta }$ and $T_1^{-1}/T^{\alpha }$ is obtained for the critical exponents $\alpha =0.46\pm 0.12$ and $\beta =1.00\pm 0.24$, with mean values used in the plot. Inset shows the color plot of ${\chi }^2(\alpha ,\beta )$ measuring the goodness of this collapse. Obtained mean values are indicated by solid white lines, their uncertainties are given by a standard deviation $\sigma$ (dashed white contour line).

Figure 4

Demonstration of the universality of quantum-critical behavior. (a) $T_1^{-1}$ versus $\pm (B-B_{c1,2})/T$ data sets for BPCB at 0.18 K and DTN at 2 K [determined from the data in Figs. 2a, 2b] overlap perfectly in the quantum-critical region. (b) Corresponding overlap of the $\alpha$ data sets, where the power-law exponents $\alpha$ are evaluated as the slopes of tangents to the $T_1^{-1}(T)$ data sets in a log-log scale. Inset shows the ${\chi }^2(T_{\mathrm{BPCB}})$ plot measuring the goodness of this overlap at a fixed $T_{\mathrm{DTN}}=2\mathrm{K}$, defining the optimal $T_{\mathrm{BPCB}}=0.18\mathrm{K}$ (indicated by the dashed line). Dashed lines in (a) and (b) indicate the crossover temperatures to the gapped (blue) and TLL (red) regions, as in Fig. 1, and the green lines in (b) are $\alpha$ versus $\pm (B-B_{c1,2})/T$ predictions in these two regions. Different $T_1^{-1}$ scales in (a) are due to different gyromagnetic ratios and different NMR hyperfine couplings for ${}^{14}\mathrm{N}$ and protons.