Critical BKT dynamics in the archetypal 2D spin system Ba$_2$CuSi$_2$O$_6$Cl$_2$

TL;DR

The paper combines NMR measurements and large-scale QMC simulations to study BKT-type criticality in the quasi-2D spin-dimer compound Ba$_2$CuSi$_2$O$_6$Cl$_2$, revealing a robust 2D fluctuation regime above the Néel temperature and a field-independent BKT transition temperature ratio $T_{ m BKT}/T_N \approx 0.74$. By mapping the material to an effective XXZ model of hard-core bosons, the authors reproduce the BEC phase boundary and extract the 2D BKT scaling of the correlation length, $\xi_{2D}(T)=\xi_0\exp[b/\sqrt{T/T_{\rm BKT}-1}]$, with $b\approx2.30$, using finite-size scaling of the superfluid stiffness. The NMR relaxation rate $T_1^{-1}$ reflects the critical fluctuations via Moriya theory and shows a pronounced, extended peak that provides a clear signature of 2D BKT dynamics, while QMC supports a consistent scaling picture but exhibits quantitative discrepancies in the dynamical peak magnitude due to analytic continuation and finite-size limitations. Collectively, the work positions Ba$_2$CuSi$_2$O$_6$Cl$_2$ as a model system for exploring BKT dynamics in quantum magnets and highlights the challenges of connecting dynamical QMC results with experimental spin-lattice relaxation data in systems with dimensional crossover.

Abstract

We study the spin dynamics in the quasi-2D spin-$1/2$ dimer compound Ba$_2$CuSi$_2$O$_6$Cl$_2$, which exhibits a magnetic field-induced Bose-Einstein condensate (BEC) of triplons. Using nuclear magnetic resonance (NMR) spin-lattice relaxation rate ($T_1^{-1}$) measurements combined with large-scale quantum Monte Carlo (QMC) simulations, we investigate critical fluctuations across the field-temperature phase diagram. Bridging the behavior observed in 1D and 3D systems, the $T_1^{-1}$ relaxation rate shows a pronounced peak extending well above the Néel temperature $T_N$, indicating strong two-dimensional Berezinskii-Kosterlitz-Thouless (BKT)-type fluctuations. A quantitative match between experimental and theoretical BEC phase boundaries validates an effective XXZ model. The study determines the intrinsic BKT transition temperature $T_{\mathrm{BKT}}$ from QMC, revealing a nearly field-independent $T_{\mathrm{BKT}}/T_N \approx 0.74$. Scaling analysis of the relaxation rate shows critical exponents consistent with 2D universality, and a narrow temperature window is identified where 2D physics dominates. These findings establish Ba$_2$CuSi$_2$O$_6$Cl$_2$ as a model system for exploring BKT dynamics in quantum magnets.

Figures (9)

• Figure 1: The experimental phase diagram of Ba$_2$CuSi$_2$O$_6$Cl$_2$, obtained by NMR (red circles) from the maximum of critical fluctuations at the phase transition--measured by the peak of the $T_1^{-1}(T)$ (see Fig. \ref{['Fig_T1vsT']}) or $T_1^{-1}(H)$ data, is compared to the theoretical fit by the QMC simulations (blue diamonds). The latter fit allows for the QMC determination of the $T_{BKT}$ (green squares and brown dots for the $T_{BKT}/T_N$ ratio). Lines connecting the data points are a guide to the eye.
• Figure 2: Part of the Ba$_2$CuSi$_2$O$_6$Cl$_2$ crystal structure that contains two neighboring Cu$^{++}$ spin dimers and provide the main antiferromagnetic exchange couplings $J$, $J_d$ and $J_p$, respectively denoted by the solid, dashed and dotted thick black lines. Numbers on the Cu and Si atoms present different crystallographic sites Nawa2019. The exchange coupling paths between the dimers along the $a$ axis passes alternatively either through Si(2)-Si(2) and Si(4)-Si(4) atoms (as shown in the figure) or through Si(1)-Si(1) and Si(3)-Si(3) atoms, while all the couplings along the $b$ axis are identical, passing through Si(1)-Si(2) and Si(4)-Si(3) atoms. This provides a network of slightly different $J_d$ and $J_p$ values presented in Ref. Nawa2019. In our theoretical modelling we have neglected these details and considered a simplified average tetragonal (direction-independent) structure.
• Figure 3: Magnetic field dependence of nuclear spin-lattice relaxation rate $T_1^{-1}$ measured by $^{63}$Cu (red circles) and $^{29}$Si (dark cyan pentagons) NMR at 1.6 K. Solid line is a two-parameter (prefactors) two-exponential fit that confirms the opening of the Zeeman gap $\Delta$ below the first critical field $\mu_0 H_{c1}=13.42~T$, and the presence of the critical fluctuations near $H_{c1}$, characterized by tri-magnon processes Ranjith2022. Inset shows the field dependence of the $^{29}$Si NMR spectra, where arrows denote the low-frequency peak used for the $T_1^{-1}$ measurements. At low field and temperature values, where the spin polarization is nearly zero and the $^{29}$Si NMR spectrum is thus unresolved, $T_1^{-1}$ was measured by $^{63}$Cu NMR. The two vertical scales are related by $^{63}T_1^{-1}/^{29}T_1^{-1}=135$.
• Figure 4: Temperature dependence of the $^{29}$Si (circles) and $^{63}$Cu (stars) $T_1^{-1}$ NMR data for fixed magnetic field values covering the whole phase diagram. At the phase transition into the low-temperature BEC phase, the data sets present very strong peak that reflects the critical spin fluctuations. At both critical fields, $H_{c1}$ and $\mu_0 H_{c2}=28.385~T$, the relaxation rate is nearly constant, while below $H_{c1}$ it reflect the gap opening. The two vertical scales are related by $^{63}T_1^{-1}/^{29}T_1^{-1}=135$. Lines connecting the data points are a guide to the eye.
• Figure 5: The wave-vector dependance $T_1^{-1}(\mathbf{q})$ of the QMC simulations for 17 T presents a peak at the antiferromagnetic wave vector $\mathbf{Q}_{AF}$. For the two characteristic temperatures (2.4 and 7.6 K), the blue dotted line and the red dashed line present the $T_1^{-1}(\mathbf{q})$ dependence along the axes ($q_x$, $q_y$) of the unit cell and along its diagonal, respectively. The solid black line is the "flat-top exponential" fit of the $T_1^{-1}(\mathbf{q})$ peak, used to define the corresponding correlation length, as explained in the text. The two "3D plot" insets display the complete $\mathbf{q}$-dependence over a conveniently defined unit cell, shown by the red dashed line in the contour plot (for 7.6 K) given in the top-right inset.