Unambiguous Experimental Verification of Linear-in-Temperature Spinon Thermal Conductivity in an Antiferromagnetic Heisenberg Chain

The everlasting interest in spin chains is mostly rooted in the fact that they generally allow for comparisons between theory and experiment with remarkable accuracy, especially for exactly solvable models. A notable example is the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain (AFHC), which can be well described by the Tomonaga-Luttinger liquid theory and exhibits fractionalized spinon excitations with distinct thermodynamic and spectroscopic experimental signatures consistent with theoretical predictions. A missing piece, however, is the lack of a comprehensive understanding of the spinon heat transport in AFHC systems, due to difficulties in its experimental evaluation against the backdrop of other heat carriers and complex scattering processes. Here we address this situation by performing ultralow-temperature thermal conductivity measurements on a nearly ideal spin-$\frac{1}{2}$ AFHC system copper benzoate $\mathrm{Cu}({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{COO}{)}_2·3{\mathrm{H}}_2\mathrm{O}$, whose field-dependent spin excitation gap enables a reliable extraction of the spinon thermal conductivity ${\kappa }_s$ at zero field. ${\kappa }_s$ was found to exhibit a linear temperature dependence ${\kappa }_s\sim T$ at low temperatures, with ${\kappa }_s/T$ as large as $1.70\mathrm{mW}{\mathrm{cm}}^{-1}{\mathrm{K}}^{-2}$, followed by a precipitate decline below $\sim 0.3\mathrm{K}$. The observed ${\kappa }_s\sim T$ clarifies the discrepancies between various spin chain systems and serves as a benchmark for one-dimensional spinon heat transport in the low-temperature limit. The abrupt loss of ${\kappa }_s$ with no corresponding anomaly in the specific heat is discussed in the context of many-body localization.

Figure 1

(a) Schematics showing the generation of spinons in a spin-$\frac{1}{2}$ antiferromagnetic chain. An initial spin flip results in two unsatisfied bonds that can freely propagate, acting as domain walls and dubbed spinons. (b) Typical spectrum for spinons, featuring a continuum with clear boundaries. (c) The crystal structure of copper benzoate $\mathrm{Cu}({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{COO}{)}_2·3{\mathrm{H}}_2\mathrm{O}$. The ${\mathrm{Cu}}^{2+}$ ions carry spin-$\frac{1}{2}$ and form chains along the $c$ axis. Each spin site locates in the center of a distorted octahedron formed by oxygen ions. The intrachain coupling $J\sim 18\mathrm{K}$ and the extremely weak interchain coupling $J^{\prime }$ ($T_N\sim 0.8\mathrm{mK}$) make copper benzoate a model compound of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain. For all measurements, the external magnetic field was applied along $b$.

Figure 2

The specific heat of copper benzoate. (a) The total specific heat $C$ at zero magnetic field. The black solid line is a fit to $C=C_s+C_{\mathrm{ph}}=\gamma T+\beta T^3$, where $C_s$ and $C_{\mathrm{ph}}$ are the contributions from gapless spinons and phonons, respectively. The linear behavior of $C_s$ is better illustrated as a temperature-independent $C_s/T$ in the inset. (b) Magnetic specific heat $C_{\mathrm{mag}}$ at ${\mu }_{0}H=0$ and 7 T, obtained by subtracting the small $C_{\mathrm{ph}}$ determined from the fitting in (a). In the temperature range of our measurement, $C_{\mathrm{mag}}(0\mathrm{T})=C_s$ is linear (black solid line), attributed to spinons, while $C_{\mathrm{mag}}(7\mathrm{T})$ comes from gapped solitonlike excitations. The blue solid line is a fit to extract the spin excitation gap (see text).

Figure 3

The thermal conductivity of copper benzoate. (a) The thermal conductivity of $S4$ ($J_Q\perp c$) plotted as $\kappa /T$ against $T^{1.55}$. A straight line fits well the data under different fields. (b) The thermal conductivity of $S1$ ($J_Q\parallel c$) plotted as $\kappa /T$ against $T$. At 0 T, the data above 0.3 K can be fitted by $\kappa /T=({\kappa }_s+{\kappa }_{\mathrm{ph}})/T=a+b{T}^{\alpha -1}$, where ${\kappa }_s$ and ${\kappa }_{\mathrm{ph}}$ are the contributions from gapless spinons and phonons, respectively. At 14.5 T, ${\kappa }_{\mathrm{ph}}/T$ obtained from fitting the zero-field data accounts for the data below 0.3 K. (c) The zero-field ${\kappa }_s/T$ of various samples $S1$, $S2$, and $S3$, plotted as ${\kappa }_s/T$ vs $T$. The arrows indicate the abrupt drop.