Hidden zero-temperature bicritical point in the two-dimensional anisotropic Heisenberg model: Monte Carlo simulations and proper finite-size scaling

By considering the appropriate finite-size effect, we explain the connection between Monte Carlo simulations of two-dimensional anisotropic Heisenberg antiferromagnet in a field and the early renormalization group calculation for the bicritical point in $2+ϵ$ dimensions. We found that the long-length-scale physics of the Monte Carlo simulations is indeed captured by the anisotropic nonlinear $\sigma$ model. Our Monte Carlo data and analysis confirm that the bicritical point in two dimensions is Heisenberg-like and occurs at $T=0$; therefore the uncertainty in the phase diagram of this model is removed.

Figure 1

Three candidates for the phase diagram. Solid lines are second-order phase boundaries, and the dashed line in (a) is a first-order phase boundary.

Figure 2

(Color online) Staggered magnetization across the finite-lattice spin-flop line at $T=0.2$. Each group of nearby data points is from one simulation performed with a single fixed magnetic field. Histogram reweighting was used to calculate observables in slightly different magnetic fields. The insets show the data measured from a larger range of magnetic field.

Figure 3

(Color online) Binder cumulant for various sizes at $T=0.2$. The crossing points of these curves are near $H=2.4055$. $U_4$ at the crossing point is different from the universal value for an Ising-like transition.

Figure 4

(Color online) Finite-size scaling of $M_z^{†}$. (a) Exponents used belong to first-order phase transition; (b) Ising exponents are used. Error bars are not plotted since most of them are smaller than or equal to the size of the symbols, as shown in Fig. 2.

Figure 5

Probability distribution of the staggered magnetization across the spin-flop transition. $P(r,z)$ is proportional to the gray scale. The data are from simulations with $L=100$ at $T=0.265$.

Figure 6

(Color online) Probability distribution of the tilt angle $\theta$. Symbols are calculated from the histogram of $\theta$ in the simulations, and the solid lines are curve fitting with Eq. (22). The fitting parameter $q$ is plotted in the inset, where the straight line is a linear fit. The simulations were performed with $L=80$ at $T=0.265$.

Figure 7

(Color online) Finite-size scaling analysis corresponding to Eq. (21) at $T=0.2$. (a) The AF order parameter; (b) the $XY$ order parameter. We introduce $T^{*}$ as a fitting parameter which is slightly above the actual temperature at which the simulation is done. Error bars are smaller than or equal to the size of the symbols.

Figure 8

(Color online) The same as Fig. 7, but for simulations at $T=0.1$. Unlike Fig. 7, we have used $T^{*}=T$ here.

Figure 9

(Color online) Finite-size scaling plot of the effective $\sigma$ spin component from simulations at $T=\left(\mathrm{a}\right)$ 0.2 and (b) 0.1. Instead of adjusting $T^{*}$, we have simply set $T^{*}=T$. (a) and (b) are expected to display the same curve; see text for details. Error bars are smaller than or equal to the size of the symbols.

Figure 10

(Color online) Scaling plot for susceptibility near the spin-flop transition according to Eq. (23), at $T=\left(\mathrm{a}\right)$ 0.1 and (b) at 0.2.

Figure 11

(Color online) Scaling plot for specific heat near the spin-flop transition according to Eq. (29), at $T=\left(\mathrm{a}\right)$ 0.1 and (b) 0.2.