Finite-temperature properties of frustrated classical spins coupled to the lattice

We present extensive Monte Carlo simulations for a classical antiferromagnetic Heisenberg model with both nearest $\left(J_1\right)$ and next-nearest $\left(J_2\right)$ exchange couplings on the square lattice coupled to the lattice degrees of freedom. The Ising-like phase transition, which appears for $J_2∕J_1>1∕2$ in the pure spin model, is strengthened by the spin-lattice coupling and is accompanied by a lattice deformation from a tetragonal symmetry to an orthorhombic one. Evidence that the universality class of the transition does not change with the inclusion of the spin-lattice coupling is reported. Implications for ${\mathrm{Li}}_2\mathrm{V}\mathrm{O}\mathrm{Si}{\mathrm{O}}_4$, the prototype for a layered $J_1-J_2$ model in the collinear regime, are also discussed.

Figure 1

Zero-temperature phase diagram of the Hamiltonian of Eq. (1): the Néel state, with a tetragonal lattice, is stabilized for $J_2∕J_1<0.5$ and below a critical value of the spin-lattice coupling ${\theta }_c$, otherwise the ground state has collinear spins, with antiferromagnetic correlations along one spatial direction and ferromagnetic along the other, and the lattice has an orthorhombic structure. Lines are guides to the eye.

Figure 2

$V_{\min }\left(L\right)$ as a function of $1∕L^2$ for $\theta =5$, 10, and 15. The horizontal line, located at $2∕3$, indicates the value for a continuous phase transition.

Figure 3

Upper panel: Specific heat as a function of the temperature for $\theta =5$ for three different clusters $L=20$, 40, and 80. Lower panel: The same for the spin susceptibility.

Figure 4

The same as in Fig. 3 for $\theta =10$.

Figure 5

Upper panel: Size scaling of the maximum of the specific heat as a function of the size of the cluster for $\theta =15$. Lower panel: The same for the maximum of the spin susceptibility. In both cases, the dashed line is a three-parameter fit (see text).

Figure 6

Entropy difference $S\left(T\right)-S\left(\overline{T}\right)$ as a function of the temperature for $L=80$ and $\theta =5$ (full triangles), 10 (full squares), and 15 (full circles). The reference temperature is $\overline{T}∕J_1=0.2$. Inset: The maximum of the specific heat for $L=80$ as a function of $\theta$.

Figure 7

The critical temperature $T_c$ as a function of the frustrating ratio $J_2∕J_1$ for three different values of the spin-lattice couplings, $\theta =5$, 10, and 15. The case for the pure spin model, corresponding to $\theta =0$, is also reported for comparison.

Figure 8

Upper panel: lattice parameters $l_x$ and $l_y$ as a function of the temperature for $\theta =5$ for $L=20$ and 80. Lower panel: The same for the Ising magnetization $M_{\sigma }$.

Figure 9

The same as in Fig. 8 for $\theta =10$.