The XXZ Heisenberg antiferromagnet features an anisotropic magnetic coupling and is given by

\[ \mathcal{H}_\mathrm{XXZ} = J \sum\limits_{\langle i,j\rangle} \Big[ \Delta(\phi_i^x\phi_j^x+\phi_i^y\phi_j^y)+\phi_i^z\phi_j^z \Big] - H\sum_i\phi_i^z \]

where \(\Delta<1\) encodes the so-called uniaxial exchange anisotropy. Already in one dimension this Hamiltonian shows a rich variety of physical effects^{1}. Depending on the interplay of temperature and external field \(H\), antiferromagnetic (AF) and paramagnetic (P) phases, as well as so-called spin flop phases arise (SF)^{2}.

Here we consider the model equipped with an *additional single-ion anisotropy* term of the form

\[ \mathcal{H} = \mathcal{H}_\mathrm{XXZ} + D\sum_i\big(\phi_i^z\big)^2 \]

This model presents an even richer phenomenology^{3}, in the ground states as well as in the finite-temperature phase diagram. The latter is shown in Figure 1 for the case of a 2D square lattice. Particularly interesting is the very narrow *biconical* (BC) phase, separating the AF and SF regimes and ending in a *tetra-critical point*.

**Figure 1**. Phase diagram of the XXZ antiferromagnet on a square lattice with a competing
single-ion anisotropy, \(\Delta=0.8\) and \(D/J = 0.2\), from Holtschneider & Selke (2008).

Using MARQOV, we study the AF to SF transition shown in Figure 1. Therefore, we fix the temperature at \(T=0.3\), and sweep the external magnetic field through the critical line. Measured observables include the Binder ratios of the *staggered longitudinal and transversal magnetizations* since these quantities should be able to detect a second-order transition. Moreover, we average all curves over a number of 50 independent replicas. The results are shown in Figure 2, where we observe a clear transition in both quantities. At a closer look one can even guess the existence of the intermediate narrow BC phase, as the corresponding transition points are separated by a small amount in the external field.

**Figure 2**. Longitudinal and transversal Binder ratios indicate the AF to SF transition and the intermediate BC phase.

- Mikeska, H. J., & Kolezhuk, A. K. (2004). One-dimensional magnetism. In Quantum magnetism (pp. 1-83). Springer, Berlin, Heidelberg. ↩
- Holtschneider, M., Wessel, S., & Selke, W. (2007). Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets. Physical Review B, 75(22), 224417. ↩
- Holtschneider, M., & Selke, W. (2008). Uniaxially anisotropic antiferromagnets in a field on a square lattice. The European Physical Journal B, 62(2), 147-154. ↩