Evidence for spinon localization in the heat transport of the spin-$\frac{1}{2}$ ladder compound ${\left({\text{C}}_5{\text{H}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$

We present experiments on the magnetic-field-dependent thermal transport in the spin-$\frac{1}{2}$ ladder system ${\left({\text{C}}_5{\text{H}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$. The thermal conductivity $\kappa \left(B\right)$ is only weakly affected by the field-induced transitions between the gapless Luttinger-liquid state realized for $B_{c1}<B<B_{c2}$ and the gapped states, suggesting the absence of a direct contribution of the spin excitations to the heat transport. We observe, however, that the thermal conductivity is strongly suppressed by the magnetic field deeply within the Luttinger-liquid state. These surprising observations are discussed in terms of localization of spinons within finite ladder segments and spinon-phonon umklapp scattering of the predominantly phononic heat transport.

Figure 1

(Color online) (a) Zero-field thermal conductivity of ${\left({\text{C}}_5{\text{H}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$ parallel and perpendicular to the ladder direction as a function of temperature. (b) Relative change in the thermal conductivity parallel to the ladders as a function of magnetic field at constant temperatures. Inset: data for $T=0.77\text{K}$ parallel and perpendicular to the ladders.

Figure 2

(Color online) (a) $B$ dependence of the magnetic specific heat of ${\left({\text{C}}_5{\text{H}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$. The symbols are experimental data (Ref. 12) and the lines are calculated within the MFT model (see text). (b) Comparison of the expected spin thermal conductivity ${\kappa }_{s,\text{min}}$ (line) and the experimental total thermal conductivity ${\kappa }^{\parallel }$ along the ladders (symbols).

Figure 3

(Color online) [(a) and (b)] Energy of the spin excitations normalized by $k_{B}T$ as a function of the wave vector. The dispersion curves are calculated via Eq. (3) for $T$ and $B$ corresponding to the minima of $\kappa \left(B\right)$. The arrows indicate increasing $T$ for $ka>\pi /2$. (c) Spin excitation energy at $ka=\pi /2$ for $B=B_{\text{min}}$, calculated via Eq. (3), as a function of temperature for ${\left({\text{C}}_5{\text{H}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$ (BPCB) and copper pyrazine dinitrate (CuPzN). Both scales are normalized by the respective exchange couplings $J$ (see text).