Critical behaviours of contact near phase transitions

A central quantity of importance for ultracold atoms is contact, which measures two-body correlations at short distances in dilute systems. It appears in universal relations among thermodynamic quantities, such as large momentum tails, energy and dynamic structure factors, through the renowned Tan relations. However, a conceptual question remains open as to whether or not contact can signify phase transitions that are insensitive to short-range physics. Here we show that, near a continuous classical or quantum phase transition, contact exhibits a variety of critical behaviours, including scaling laws and critical exponents that are uniquely determined by the universality class of the phase transition, and a constant contact per particle. We also use a prototypical exactly solvable model to demonstrate these critical behaviours in one-dimensional strongly interacting fermions. Our work establishes an intrinsic connection between the universality of dilute many-body systems and universal critical phenomena near a phase transition.

: Scaling behaviour of the derivative of contact ∂ hc √ t vs external field h. The left (right) panel shows the intersection of the derivatives of contact at different temperatures near the phase transition F-PP (P-PP). Here the critical field hc = 1.1 and hc = 0.9, respectively. This plot read off the critical dynamics exponent z = 2 and correlation length exponent ν = 1/2 respectively.

Supplementary Note 1 Tan's contact
By definition of Tan's contact c Here we denote the dimensionless contactc = c/ 2 b and f x n = Li n (−e x/T ). The above equation of contact looks very complex. Nevertheless, the universal scaling form of contact is hidden in such complexity of this kind. In the Supplementary Figure 2 shows a 3D contour plotc/ñ against dimensionless temperature t and chemical potentialμ at h = 0.8. Near the lower critical pointμ c = −0.5, the flatness ofc/ñ is the consequence of the criticality of the model as discussed in the main paper. The values ofc/ñ drops very faster for the chemical potential excesses the upper critical pointμ c = −0.335 due to the increase of the polarization. We will present further discussions on the critical behaviour of contact in the following part.
The derivatives of contact connect various thermal and magnetic properties such as density, magnetization and entropy We can analytically calculate these derivatives, namely Again, we can work out the scaling functions of these derivatives directly from the above equations. In Supplementary  Figure 1, we plot the derivative of contact ∂c/∂μ against chemical potentialμ. It is clearly see that the derivative of contact evolve into a sharp peak at the critical point.

Universal Scaling Forms
Quantum phase transitions occur at absolute zero temperature as the driving parameters µ and H are varied across the phase boundaries. The phase transitions are driven by quantum fluctuations with quantum critical points governed by divergent correlation lengths. Near a quantum critical point, the many-body system is expected to show universal scaling behaviour in the thermodynamic quantities. In the critical regime, a universal and scale-invariant description of the system is expected through the power-law scaling of the thermodynamic properties [3,4]. Quantum phase transitions are uniquely characterized by the critical exponents depending only on the dimensionality and symmetry of the system. In order to work out the connection of Tan's contact to the criticality of the model, we first present the dimensionless functionsÃ From equations(8) and (9), we could expand contact (2) in the critical regime, i.e. |μ −μ c | 1 and |μ −μ c | > t near different quantum phase transitions.
V-P: From vacuum V to the fully-paired phase P, the critical point isμ c = −1/2, h < 1. Taking low temperature limit near the critical point, we can obtaiñ Substituting Eq.(10) into Eq. (2), we can obtain the scaling forms of contact and its derivative with respect to µ.