Controlling Luttinger Liquid Physics in Spin Ladders under a Magnetic Field

We present a ${}^{14}\mathrm{N}$ nuclear magnetic resonance study of a single crystal of ${\mathrm{CuBr}}_4({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2$ (BPCB) consisting of weakly coupled spin-$1/2$ Heisenberg antiferromagnetic ladders. Treating ladders in the gapless phase as Luttinger liquids, we are able to fully account for (i) the magnetic field dependence of the nuclear spin-lattice relaxation rate $T_1^{-1}$ at 250 mK and for (ii) the phase transition to a 3D ordered phase occurring below 110 mK due to weak interladder exchange coupling. BPCB is thus an excellent model system where the possibility to control Luttinger liquid parameters in a continuous manner is demonstrated and the Luttinger liquid model tested in detail over the whole fermion band.

Figure 1

(a) A ladder formed by the supposed exchange interactions, $J_{ij}{\mathbf{S}}_i·{\mathbf{S}}_j$, between $S=1/2$ spins of ${\mathrm{Cu}}^{2+}$ ions in the crystal structure of BPCB. Solid thick blue lines and dashed thick blue lines stand for $J_{\perp }$ and $J_{\parallel }$, respectively. The 10 protons attached to the C atoms are not shown. (b) ${}^{14}\mathrm{N}$ NMR spectra at 120 and 40 mK recorded at $H=9.0\mathrm{T}$.

Figure 2

(a) Magnetic field dependence of the longitudinal uniform magnetization $m_z$ per ${\mathrm{Cu}}^{2+}$ ion at 40 mK (○). Inset shows linear $m_z(H)$ dependence very close to $H_{c2}$ (solid line) and the determination of $H_{c2}^{1\mathrm{D}}$ (dashed line). The data are compared to the result of the DMRG calculation for $J_{\perp }/J_{\parallel }=3.6$ (solid line) and to the prediction of the $XXZ$ chain model (dashed line), both at $T=0$. (b) Difference between each prediction and the experimental data. (c) Variation of the LL parameters $K(H)$ and $u(H)$ (in kelvin units) over the fermion band as calculated for $J_{\perp }/J_{\parallel }=3.6$ (solid line) and for the $XXZ$ chain model (dashed line).

Figure 3

Magnetic field dependence of (a) ${}^{14}\mathrm{N}$ $T_1^{-1}$ at 250 mK (○), in the 1D LL phase, (b) the temperature of the transition between the 3D ordered phase and the 1D LL phase (○), (c) the transverse staggered magnetization $m_x$ per ${\mathrm{Cu}}^{2+}$ ion at 40 mK (○) and its extrapolation to zero temperature (●), all measured on $\mathrm{N}(1{)}_1$ site as described in the text. Each data set is compared to the corresponding prediction of the LL model based on $K(H)$ and $u(H)$ [from Fig. 2c] for $J_{\perp }/J_{\parallel }=3.6$ (solid red line) and to the prediction of the simplified $XXZ$ chain model (dashed green line). The only adjustable parameter in each case is an overall scaling factor. Dotted blue line in (c) is the result of the DMRG calculation.