Short-time Monte Carlo simulation of a frustrated Ising model in two dimensions

The frustrated Ising antiferromagnet with first- and
second-nearest-neighbor interactions is studied within the
framework of short-time non-equilibrium behavior on square lattices.
The Hamiltonian reads
\begin{equation}
{\cal H} = -J_1 \sum_{\langle i,j \rangle} s_i s_j -
J_2 \sum_{\langle i,k \rangle} s_i s_k,
\label{hamiltonian}
\end{equation}
where $s_i = \pm 1$, $J_1 < 0$, $J_2 < 0$, and the first summation
is carried out only over nearest-neighbor pairs of spins whereas
the second one is over next-nearest-neighbor pairs of spins.
Notice that the Hamiltonian given by Eq. ($1$) is frustrated
on square lattices.

Recently, the phase diagram of this model system was obtained through
an effective-field theory with correlations [Phys. Rev. E {\bf 91}, 032145 (2015)] and approximations based
on different cluster sizes in the $R = J_1 / |J_2|$ {\it versus} temperature plane.
It was found that the phase transitions between the antiferromagnetic
and paramagnetic phases $(R > -0.5)$ close to $R = -0.5$ are of first order
and turn into second order for $R > R_{\mbox{tcp}}$,
with $(T_{\mbox{tcp}} , R_{\mbox{tcp}})$ the coordinates of the tricritical point.
Obviously, the effective-field theory does not give any information
on the true critical exponents.

In this work, the critical properties of the frustrated Ising antiferromagnet, Eq.
($1$), is investigated through short-time Monte Carlo relaxation on
square lattices. The main goal is to locate the line separating the paramagnetic
phase from the antiferromagnetic phase $(-0.5 \le R \le 0)$ and obtain estimates
to the critical exponents along this line.
The values of the critical exponents to $R$ in the range $[-0.4,0]$ are
compatible with the Ising universality class in two dimensions. Therefore,
no signal of a tricritical point was found in the range of $R$ investigated.