Puzzle of bicriticality in the XXZ antiferromagnet

Renormalization-group theory predicts that the XXZ antiferromagnet in a magnetic field along the easy $Z$ axis has asymptotically either a tetracritical phase diagram or a triple point in the field-temperature plane. Neither experiments nor Monte Carlo simulations procure such phase diagrams. Instead, they find a bicritical phase diagram. Here, this discrepancy is resolved: After generalizing a ubiquitous condition identifying the tetracritical point, we employ different renormalization-group recursion relations near the isotropic fixed point, exploiting group-theoretical considerations and using accurate exponents at three dimensions. These show that the results from experiments and simulations can only be understood if their trajectories flow towards the fluctuation-driven first-order transition (and the associated triple point), but reach this limit only for prohibitively large system sizes or correlation lengths. In the crossover region one expects a bicritical phase diagram, as indeed is observed. A similar scenario may explain puzzling discrepancies between simulations and renormalization-group predictions for a variety of other phase diagrams with competing order parameters.

Figure 1

Possible phase diagrams for the XXZ antiferromagnet in a longitudinal magnetic field. (a) Bicritical phase diagram. (b) Tetracritical phase diagram. (c) Diagram with a triple point. Thick lines: First-order transitions. Thin lines: Second-order transitions. The first-order transition lines between the ordered phases and the disordered paramagnetic phase end at tricritical points (small open circles). After Refs. [12, 14].

Figure 2

The function $p_4^{}(\ell -{\ell }_1^{})$ (blue) for $B=1$ and $p_4^{}\left({\ell }_1^{}\right)=0.3$ and $-0.1$. Below the horizontal (orange) line at $p_4^{}=-35u^I/8=-1.75$, the transition becomes first order and the bicritical point becomes a triple point.