Field-Tunable Berezinskii-Kosterlitz-Thouless Correlations in a Heisenberg Magnet

We report the manifestation of field-induced Berezinskii-Kosterlitz-Thouless (BKT) correlations in the weakly coupled spin-$1/2$ Heisenberg layers of the molecular-based bulk material $[\mathrm{Cu}(\mathrm{pz}{)}_2(2\text{−}\mathrm{HOpy}{)}_2]({\mathrm{PF}}_6{)}_2$. At zero field, a transition to long-range order occurs at 1.38 K, caused by a weak intrinsic easy-plane anisotropy and an interlayer exchange of $J^{\prime }/k_B\approx 1\mathrm{mK}$. Because of the moderate intralayer exchange coupling of $J/k_B=6.8\mathrm{K}$, the application of laboratory magnetic fields induces a substantial $XY$ anisotropy of the spin correlations. Crucially, this provides a significant BKT regime, as the tiny interlayer exchange $J^{\prime }$ only induces 3D correlations upon close approach to the BKT transition with its exponential growth in the spin-correlation length. We employ nuclear magnetic resonance measurements to probe the spin correlations that determine the critical temperatures of the BKT transition as well as that of the onset of long-range order. Further, we perform stochastic series expansion quantum Monte Carlo simulations based on the experimentally determined model parameters. Finite-size scaling of the in-plane spin stiffness yields excellent agreement of critical temperatures between theory and experiment, providing clear evidence that the nonmonotonic magnetic phase diagram of $[\mathrm{Cu}(\mathrm{pz}{)}_2(2\text{−}\mathrm{HOpy}{)}_2]({\mathrm{PF}}_6{)}_2$ is determined by the field-tuned $XY$ anisotropy and the concomitant BKT physics.

Figure 1

Phase diagram of CuPOF for out-of-plane magnetic fields from experiment and numerics. The pentagons denote the spin-anisotropy crossover temperature $T_{\mathrm{co}}$ from Ref. [25]. White diamonds indicate the transition temperature $T_{\mathrm{LRO}}$ to long-range order, and squares show the BKT transition temperature $T_{\mathrm{BKT}}$, as obtained from the analysis of the ${}^{31}\mathrm{P}$ $1/T_1$ rate (Fig. 2). $T_{\mathrm{LRO}}$ at zero field is determined by ${\mu }^{+}\mathrm{SR}$ measurements [25]. The green pluses and red crosses denote $T_{\mathrm{LRO}}$ and $T_{\mathrm{BKT}}$, respectively, as obtained from QMC calculations (Fig. 3). The diamond at 17.5 T denotes the saturation field, which was determined from magnetization experiments [25], and is in agreement with QMC results. All lines are guides to the eye.

Figure 2

(a)–(c) Temperature-dependent ${}^{31}\mathrm{P}$ nuclear spin-lattice relaxation rate $1/T_1$ of CuPOF, recorded at out-of-plane fields of 2, 7, and 16 T. The solid lines are best fits according to $1/T_1\propto {\xi }^{z-\eta }$ for the temperature dependent correlation lengths ${\xi }_{3\mathrm{DHeis}}$ and ${\xi }_{2\mathrm{DXY}}$ of the 3D Heisenberg and the 2D $XY$ cases (see main text). The transition temperature $T_{\mathrm{LRO}}$, marked with a downward triangle, is inferred from the $1/T_1$ peak position, and $T_{\mathrm{BKT}}$, marked with a dotted line, is determined from fits according to $1/T_1\propto {\xi }_{2\mathrm{DXY}}^{z-\eta }$. At all fields, but most noticeably at 7 T, $1/T_1$ is described best by ${\xi }_{2\mathrm{DXY}}$ at $T\gtrsim T_{\mathrm{LRO}}$.

Figure 3

Finite-size scaling analysis performed to obtain the critical temperatures $T_{\mathrm{BKT}}$ and $T_{\mathrm{LRO}}$ from the QMC simulations at 2 T. (a) Data collapse of the finite-size in-plane spin stiffness $\rho$ fit closely above $T_{\mathrm{BKT}}$, for the $J^{\prime }=0$ model, which should collapse to a single curve if the fit is perfect, reaffirming the calculated $T_{\mathrm{BKT}}$. The different curves correspond to different linear sizes $L$. (b) Crossings of the $L\rho$ curves for the $J^{\prime }=1\mathrm{mK}$ model; the inset shows the $L\to \infty$ scaling of the crossing temperature $T^{*}$. The red line denotes a second-order polynomial fit, which is extrapolated to $1/L\to 0$ to estimate $T_{\mathrm{LRO}}$.