Figure 4: Contact per particle at finite temperatures.
(a) Density plot of the entropy obtained from the numerical solution of thermodynamic Bethe ansatz (TBA) equations for highlighting different regions for the phase transition V–P on the 
plane. We denote by SC the semiclassical region with very low density. QC stands for the critical regime with non-relativistic dispersion, TLLp is the Tomonaga–Luttinger liquid (TLL) of pairs with linear relativistic dispersion and HT stands for high temperature region where universal behaviour of thermodynamics are absent. Dashed white lines represent the crossover temperatures from QC to SC and TLL regions. (b) Contact per particle 
versus the temperature at fixed values of low densities, where 
is the dimensionless density. The flatness of 
confirms the constant contact per particle as shown in equation (15) in the QC region. (c) 
at high densities. The solid lines show the numerical result derived from TBA equations. The deviations of the TLL result (dotted lines) from TBA results indicate the breakdown of the TLLp phase at the crossover temperature T* from TLLp to QC. T* here is consistent with the result obtained from the deviation of entropy from the linear temperature dependence of TLL. The inset shows 
versus temperature in which the deviation is more visible. A maximum of 
demonstrates the enhancement of contact when quantum effects become important in the quantum degenerate region where 
.
