Magnetic field-induced phases in a model S=1 Haldane chain system

Abstract

An $S=1$ Haldane chain is a one-dimensional (1D) quantum magnet where strong fluctuations result in quantum disordered singlet ground state with a gapped excitation spectrum. The gap magnitude is primarily set by the dominant intrachain interaction ($J_\text{1D}$). An applied magnetic field closes the gap at $B_\text{c1}$ and drives the system into a gapless Tomonaga-Luttinger liquid (TLL) regime, followed by, at lower temperatures, a Bose-Einstein condensate (BEC) ground state, persisting up to $B_\text{c2} \propto 4 J_\text{1D}/gμ_B$. Almost all previously studied experimental realizations of such systems were based on transition-metal complexes which typically suffer from intrinsic anisotropies or large $J_\text{1D}$ values, limiting the access to the full theoretical phase diagram. We report a comprehensive study of TLL and BEC phases in the organic Haldane chain system 3,5-bis(N-tert-butylaminoxyl)-3'-nitrobiphenyl (BoNO). The absence of anisotropy and a moderate $J_\text{1D}$ enable exploration of the complete $B-T$ phase diagram. Through $^1$H nuclear magnetic resonance, combined with theoretical analysis, we characterize the TLL properties, map the BEC phase boundary $T_c (B)$, determine the associated critical exponent $ν\approx 0.66$ at $B_\text{c2}$, and demonstrate universal quasiparticle scaling in the quantum-critical regime. These results provide full experimental validation of theoretical predictions for field-induced phases in an $S=1$ Haldane chain, made over two decades ago.

Figures (3)

• Figure 1: The low-temperature phase diagram of a Haldane $S=1$ chain in a magnetic field. In absence of the external magnetic field, a gap separates the singlet quantum-disordered (QD) ground state from the first excited triplet. The energy of the triplet is lowered via the Zeeman effect in a finite magnetic field, eventually closing the gap at $B_\text{c1}$. In the range $B_\text{c1}$$< B <$$B_\text{c2}$, the system is gapless and can be described by the TLL framework below energy scale set by $J_\text{1D}$. At lower temperatures, $J_\text{3D}$ becomes relevant, and the system enters a long-range coherent 3D XY-AFM state, corresponding to the BEC in the bosonic mapping. Crossovers are indicated by dotted lines, while the solid line marks the phase transition. Universal quasiparticle behavior is observed in the quantum-critical (QC) region. At higher temperatures, system is an isotropic 3D $S=1$ paramagnet.
• Figure 2: Complete phase diagram of BoNO.a Temperature and b magnetic-field dependence of the $^1$H relaxation rate $T_{1}^{-1}$ across the phase transition at various external magnetic fields and temperatures, respectively. Solid lines are cubic-spline fits, and arrows mark the estimated transition temperatures $T_c(B)$. c Temperature evolution of the ordered transverse magnetic moment $m_\perp \propto \Delta f$ across $T_c$ at $7.5$ T (red) and $12$ T (blue), determined from spectral splitting at various $^1$H NMR lines (see Fig. S4). d Complete phase diagram derived from $T_{1}^{-1}$ and spectral data (green circles). At low magnetic fields, additional heat-capacity measurements (magenta diamonds) corroborate the NMR-determined phase boundary. QMC simulations (black line) reproduce the experimental phase diagram perfectly up to 15 T, above which a small discrepancy develops, which can be reconciled by allowing for a field-dependent interchain coupling $J_\text{3D}$. e Probability density map of the upper critical field $B_\text{c2}$ and critical exponent $\nu$, calculated for $T_c \propto (B_\text{c2}-B)^{\nu}$ with the mean values of $B_\text{c2}=33.645~$T and $\nu=0.663$. Dashed line marks the 68% highest density interval (HDI). f Measured magnetostriction along the chains $\lambda_a=(\Delta L_a/L_a)$ vs. $B$ at 1.4 K shows a parabolic contraction up to $B_\text{c2}$. Red line is a guide to the eye.
• Figure 3: Spin dynamics in Tomonaga–Luttinger and quantum-critical regimes. a Qualitative determination of the interaction parameter $K$ from constant-temperature cuts of $T_1^{-1}$ in the TLL regime (red circles), compared with DIMPY (blue squares) and BPCB (green triangles) from Ref. Jeong2016. The data predict attractive quasiparticle interactions ($K>1$), as expected for the Haldane system. b, c Experimental $T_1^{-1}(T)$ data (red circles) overlaid with Bayesian probability distributions (green colormap) at $19$ T and $27$ T. Bayesian point estimates (mean and 95% HDI, black lines) are compared to the least-squares fits (LS, orange lines) and the theoretical $T_1^{-1}(T)$ dependence using DMRG data (blue lines). d, e Probability density maps (green) and point estimates (mean and 68% HDI, black lines) for $K$ and $T_c$, compared with theory (blue lines) and LS fits (red circles). f Universal quasiparticle scaling of $T_1^{-1}~(B)$ data measured across the quantum-critical region in BoNO (red circles), consistent with the three-magnon model (grey line), also observedMukhopadhyay2012 in DTN (blue squares) and BPCB (green triangles).