I just submitted a new manuscript to PRB and posted it on arXiv (arXiv:1804.06045). This paper is a collaboration with Anders Sandvik and Kedar Damle. This is paper builds on our previous work on the field-driven saturation transition in the 1D J-Q model to study the saturation transition in 2D.

Fig 4. Magnetization density as a function of field for various values of s=Q with J+Q=1.

Metamagnetism: Using QMC, we find magnetization jumps to the saturated state (metamagnetism) above a critical coupling ratio (Q/J)_min. We then use an exact method based on a high magnetization expansion to determine (Q/J)_min. Above (Q/J)_min the saturation transition is discontinuous (featuring a magnetization jump) and below (Q/J)_min the transition is continuous.

Zero-scale-factor universality: When the saturation transition is continuous it is governed by zero-scale-factor universality. First proposed by Sachdev, Senthil and Shankar in 1994, zero-scale-factor universality is characterized by response functions that depend only on the bare parameters and no non-universal numbers. Remarkably, there had be no previous numerical or experimental verification of the scaling forms predicted in the 1994 Sachdev paper. Our QMC results confirm that the leading order scaling forms work and find logarithmic divergences at low temperature. These divergences are to be expected, since two spatial dimensions is the upper critical dimension of the zero-scale-factor universality, and the Sachdev paper even proposes a form for this divergence. When comparing to our QMC results, we find that the Sachdev form for the logarithmic divergences does not appear to be correct.

If you would like to know more all the details are on arxiv.