Attractive Tomonaga-Luttinger Liquid in a Quantum Spin Ladder

We present NMR measurements of a strong-leg spin-$1/2$ Heisenberg antiferromagnetic ladder compound $({\mathrm{C}}_7{\mathrm{H}}_{10}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$ under magnetic fields up to 15 T in the temperature range from 1.2 K down to 50 mK. From the splitting of NMR lines, we determine the phase boundary and the order parameter of the low-temperature (three-dimensional) long-range-ordered phase. In the Tomonaga-Luttinger regime above the ordered phase, NMR relaxation reflects characteristic power-law decay of spin correlation functions as $1/T_1\propto T^{1/2K-1}$, which allows us to determine the interaction parameter $K$ as a function of field. We find that field-dependent $K$ varies within the $1<K<2$ range, which signifies attractive interaction between the spinless fermions in the Tomonaga-Luttinger liquid.

Figure 1

(a) Typical ${}^{14}\mathrm{N}$ NMR spectra above and below the transition $T_c=340\mathrm{mK}$ in 15 T. Four NMR lines from inequivalent ${}^{14}\mathrm{N}$ sites are doubled by quadrupolar splitting ($I=1$), which results in a mirror image about the center (dashed line). The splittings of the lines at low temperature are different due to different hyperfine coupling tensors. (b) The splitting of the lowest-frequency line [filled with gray color in the spectrum shown in (a)] as a function of temperature in four different magnetic fields between 3.5 and 15.0 T. The inset shows fits to the power law, where the solid line represents the best fit. (c) Magnetic phase diagram obtained by plotting the onset temperature ($T_c$) for the line splitting, where the low-field part ($H\le 3.5\mathrm{T}$) was completed by ${}^1\mathrm{H}$ NMR (see the text). The solid line represents the critical scaling for magnon BEC. Previous results from thermodynamic measurements, squares for specific heat [27] and triangles for specific heat and magnetocaloric effects [29], are also shown.

Figure 2

(a)–(d) ${}^1\mathrm{H}$ $1/T_1$ (open circles) as a function of temperature at 3.5, 5.0, 12.0, and 15.0 T, respectively. Solid lines are fits to the power-law behavior $1/T_1\propto T^{\alpha }$ and represent the fitting range (see the text). Dash-dotted lines in (b)–(d) represent a noninteracting case $1/T_1\propto T^{-0.5}$ for the same fitting range, which show clear contrast to the fitted lines. Vertical dotted lines represent $T_c$. The ${}^{14}\mathrm{N}$ $1/T_1$ data (crosses) are overlaid in (d) for comparison. (e) Scaled plots of normalized $1/T_1{T}^{\alpha }$ as a function of $T/T_c$.

Figure 3

The extracted $\alpha$ as a function of applied magnetic field. The solid line represents the DMRG calculations reported in Ref. 27. The corresponding $K$ values can be read from the scale on the right axis.