Quantum criticality of spinons

The free fermion nature of interacting spins in one dimensional (1D) spin chains still lacks a rigorous study. In this letter we show that the length-$1$ spin strings significantly dominate critical properties of spinons, magnons and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution we analytically calculate exact scaling functions of thermal and magnetic properties of the model, providing a rigorous understanding of the quantum criticality of spinons. It turns out that the double peaks in specific heat elegantly mark two crossover temperatures fanning out from the critical point, indicating three quantum phases: the Tomonaga-Luttinger liquid (TLL), quantum critical and fully polarized ferromagnetic phases. For the TLL phase, the Wilson ratio $R_W=4K_s$ remains almost temperature-independent, here $K_s$ is the Luttinger parameter. Furthermore, applying our results we precisely determine the quantum scalings and critical exponents of all magnetic properties in the ideal 1D spin-1/2 antiferromagnet Cu(C${}_4$H${}_4$N${}_2$)(NO${}_3$)${}_2$ recently studied in Phys. Rev. Lett. {\bf 114}, 037202 (2015)]. We further find that the magnetization peak used in experiments is not a good quantity to map out the finite temperature TLL phase boundary.