Thermal-contact capacity of one-dimensional attractive Gaudin–Yang model

Tan’s contact $\mathcal{C}$ is an important quantity measuring the two-body correlations at short distances in a dilute system. Here we make use of the technique of exactly solved models to study the thermal-contact capacity ${\mathcal{K}}_{\mathrm{T}}$, i.e., the derivative of $\mathcal{C}$ with respect to temperature in the attractive Gaudin–Yang model. It is found that ${\mathcal{K}}_{\mathrm{T}}$ is useful in identifying the low temperature phase diagram, and using the obtained analytical expression of ${\mathcal{K}}_{\mathrm{T}}$, we study its critical behavior and the scaling law. Especially, we show ${\mathcal{K}}_{\mathrm{T}}$ versus temperature and thus the non-monotonic tendency of $\mathcal{C}$ in a tiny interval, for both spin-balanced and imbalanced phases. Such a phenomenon is merely observed in multi-component systems such as SU(2) Fermi gases and spinor bosons, indicating the crossover from the Tomonaga–Luttinger liquid to the spin-coherent liquid.