Field-Induced Quantum Criticality and Universal Temperature Dependence of the Magnetization of a Spin-$1/2$ Heisenberg Chain

High-precision dc magnetization measurements have been made on $\mathrm{Cu}({\mathrm{C}}_4{\mathrm{H}}_4{\mathrm{N}}_2){({\mathrm{NO}}_3)}_2$ in magnetic fields up to 14.7 T, slightly above the saturation field $H_s=13.97\mathrm{T}$, in the temperature range from 0.08 to 15 K. The magnetization curve and differential susceptibility at the lowest temperature show excellent agreement with exact theoretical results for the spin-$1/2$ Heisenberg antiferromagnet in one dimension. A broad peak is observed in magnetization measured as a function of temperature, signaling a crossover to a low-temperature Tomonaga-Luttinger-liquid regime. With an increasing field, the peak moves gradually to lower temperatures, compressing the regime, and, at $H_s$, the magnetization exhibits a strong upturn. This quantum critical behavior of the magnetization and that of the specific heat withstand quantitative tests against theory, demonstrating that the material is a practically perfect one-dimensional spin-$1/2$ Heisenberg antiferromagnet.

Figure 1

(a) Field dependence of the magnetization $M$ (solid circles) and the differential susceptibility $dM/dH$ (solid squares) at 0.08 K, along with the result of exact QTM calculations for the 1D spin-$1/2$ HAF at 0.08 K (open symbols). (b) Enlarged plot near $H_s=13.97\mathrm{T}$. The dashed line is a Bethe-anzatz result for $T=0$. In both panels, thin solid lines are guides to the eye.

Figure 2

(a) Temperature dependence of $M/H$ at various fields. The black arrows indicate the peak position $T_p$. (b) Low-temperature part ($T\le 1.5\mathrm{K}$) of the $M/H$ plot for magnetic fields slightly below (open symbols) and slightly above (solid symbols) $H_s$. Thin lines are guides to the eye.

Figure 3

(a) Log-log plot of $(M_s-M)/H$ near $H_s$ as a function of temperature below 2 K. The saturation magnetization $M_s=1.15{\mu }_{\text{B}}$ has been taken from the magnetization curve at 0.08 K (see Fig. 3). The best fit of a power law, $(M_s-M)/H\propto T^{\beta }$, to the 14 T data yields $\beta =0.48(1)$ (solid line). (b) Comparison of $M$ at 14 T with the result of a QTM calculation for a 1D spin-$1/2$ HAF at $H_s$ (crosses). The solid line is the best fit with $\beta =1/2$ described in the text. Inset: $C/T$ as a function of temperature at 14 T (open circles). Nuclear and phonon contributions have been subtracted. Crosses are QTM results. The best fit of the power law $C/T\propto T^{-\alpha }$ below 2 K yields $\alpha =0.49(1)$ (dotted line).

Figure 4

$T$ vs $H$ phase diagram of CuPzN based on the temperature derivative of $M/H$. Open circles denote the positions of $T_p$, the temperature of the broad peak in $M$. The dotted line is the universal crossover line for the free-fermion limit, Eq. (6).