XXZ Heisenberg antiferromagnet

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 effects¹. 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 phenomenology³, 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.