Frustrated mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice

Mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice with sublattices A, B, and C are studied for two spin-value distributions $(S_{\mathrm{A}},S_{\mathrm{B}},S_{\mathrm{C}})=(1/2,1/2,1)$ and $(1/2,1,1)$ by Monte Carlo simulations. The nonbipartite character of the lattice induces geometrical frustration in both systems, which leads to the critical behavior rather different from their ferromagnetic counterparts. We confirm second-order phase transitions belonging to the standard Ising universality class occurring at higher temperatures, however, in both models these change at tricritical points (TCP) to first-order transitions at lower temperatures. In the model $(1/2,1/2,1)$, TCP occurs on the boundary between paramagnetic and ferrimagnetic $(\pm 1/2,\pm 1/2,\mp 1)$ phases. The boundary between two ferrimagnetic phases $(\pm 1/2,\pm 1/2,\mp 1)$ and $(\pm 1/2,\mp 1/2,0)$ at lower temperatures is always first order and it is joined by a line of second-order phase transitions between the paramagnetic and the ferrimagnetic $(\pm 1/2,\mp 1/2,0)$ phases at a critical endpoint. The tricritical behavior is also confirmed in the model $(1/2,1,1)$ on the boundary between the paramagnetic and ferrimagnetic $(0,\pm 1,\mp 1)$ phases.

Figure 1

Mixed-spin $\mathbf{S}=(S_{\mathrm{A}},S_{\mathrm{B}},S_{\mathrm{C}})$ models on a triangular lattice consisting of sublattices A, B, and C, with (a) $\mathbf{S}=(1/2,1/2,1)$ mixing, model I; and (b) $\mathbf{S}=(1/2,1,1)$ mixing, model II. Small and large circles denote spin-1/2 and spin-1 sites, respectively.

Figure 2

Time evolutions of the internal energy per site $e/\left|J\right|$ and the staggered magnetizations $m_{s1},m_{s3}$ (see definitions below), starting from the disordered phase at the temperatures $k_{B}T/\left|J\right|=0.7$ in model I and 1.0 in model II, for $D/\left|J\right|=0$ and $L=120$.

Figure 3

Temperature dependencies of (a) the specific heat and (b) the order parameters, for selected values of $D/\left|J\right|$ and a fixed lattice size $L=48$. In (a) additionally the results for $L=72$ are shown (squares).

Figure 4

FSS analysis of the critical exponents ratios $1/{\nu }_O$ and ${\gamma }_O/{\nu }_O$ (a, b), the fourth-order cumulant $U_O$ temperature dependencies for different $L$ (c, d) and the $U_O$ data collapse analysis (e, f), for $O=M_{s1}$ at $D/\left|J\right|=0$ (left column) and for $O=M_{s2}$ at $D/\left|J\right|=-2$ (right column). The coefficients of determination $R^2$ for the respective fits (from top to bottom) are 0.9999, 0.9998, 0.9997 in (a) and 0.9999, 1.0000, 1.0000 in (b).

Figure 5

Order parameter $m_{s1}$ as a function of the increasing $(▹)$ and decreasing $(◃)$ single-ion anisotropy parameter $D/\left|J\right|$ at various temperatures and $L=48$. The double-headed arrows mark the hysteresis widths.

Figure 6

(a) Energy distributions for $D/\left|J\right|=-1.47$ and different $L$. The respective temperatures are tuned by the reweighing technique to achieve approximately equal peak heights. (b) FSS analysis of the susceptibility ${\chi }_{M_{s1}}$ and the specific heat $c$, for $D/\left|J\right|=-1.47,-1.48$, and $-1.4825$. For $D/\left|J\right|=-1.48$ and $-1.4825$ the log-log plots are, respectively, fitted to the exact values of the tricritical exponents ratios ${\gamma }_t/{\nu }_t=1.85,{\alpha }_t/{\nu }_t=1.60$ and the exponent 2, corresponding to the system volume. (c) FSS analysis of ${\chi }_{M_{s1}},D{1}_{M_{s1}},D{2}_{M_{s1}}$ for $D/\left|J\right|=-1.46$, with the coefficients of determination $R^2$ for the respective fits (from top to bottom) 0.9999, 1.0000, 0.9999.

Figure 7

Time evolution of the sublattice magnetizations at the points (a) $(D/|J|,k_{B}T/|J\left|\right)=(-1.487,0.33)$ and (b) $(-1.47,0.35)$, for $L=48$.

Figure 8

Phase diagram of the model I in $(k_{B}T/|J|-D/|J\left|\right)$ parameter space. The empty circles represent the phase transition temperatures $k_B{T}_c/\left|J\right|$ between the paramagnetic state P and the ferrimagnetic states ${\mathrm{FR}}_1^{\mathrm{I}}(\pm 1/2,\pm 1/2,\mp 1)$ and ${\mathrm{FR}}_2^{\mathrm{I}}(\pm 1/2,\mp 1/2,0)$, estimated from the specific heat peaks for $L=48$, the empty triangles mark the hysteresis widths at first-order transitions between the phases ${\mathrm{FR}}_1^{\mathrm{I}}$ and ${\mathrm{FR}}_2^{\mathrm{I}}$ with the expected phase transition boundary marked by the dash-dot line. The filled symbols at finite temperatures show more precise values obtained from the FSS analysis and the Binder cumulant crossing, where the diamond is the tricritical point (TCP), the hexagon is the critical endpoint (CE), and the square at $(D_c/|J|,k_B{T}_c/|J\left|\right)=(-3/2,0)$ represents the exact value of the GS transition point.

Figure 9

(a) FSS analysis of the critical exponent ratios $1/{\nu }_{M_{s3}}$ and ${\gamma }_{M_{s3}}/{\nu }_{M_{s3}}$, (b) the fourth-order cumulant $U_{M_{s3}}$ temperature dependencies for different $L$, and (c) the $U_{M_{s3}}$ data collapse analysis at $D/\left|J\right|=0$. In (a) the coefficients of determination $R^2$ for the respective fits (from top to bottom) are 0.9998, 0.9999, and 0.9999. Inset in (b) shows an alternative way of the critical temperature estimation from the FSS analysis.

Figure 10

Order parameter $m_{s3}$ as a function of the increasing $(▹)$ and decreasing $(◃)$ single-ion anisotropy parameter $D/\left|J\right|$ at various temperatures and $L=48$.

Figure 11

(a) Energy distributions for $D/\left|J\right|=-1.48$ and different $L$. The respective temperatures are tuned by the reweighing technique to achieve approximately equal peak heights. Time evolution of the internal energy shown in the inset demonstrates tunneling between the coexisting phases for $L=48$. (b) FSS analysis of the susceptibility ${\chi }_{M_{s3}}$ and the specific heat $c$, for $D/\left|J\right|=-1.47,-1.48$ and $-1.4825$. For $D/\left|J\right|=-1.4825$ the log-log plots are fitted to the exponent 2, corresponding to the system volume. (c) FSS analysis of ${\chi }_{M_{s3}},D{1}_{M_{s3}},D{2}_{M_{s3}}$ for $D/\left|J\right|=-1.46$ with the fitted Ising values of the critical exponents ratios ${\gamma }_I/{\nu }_I=1.75,1/{\nu }_I=1.00$.

Figure 12

Phase diagram of the model II in $(k_{B}T/|J|-D/|J\left|\right)$ parameter space. The empty circles represent the phase transition temperatures $k_B{T}_c/\left|J\right|$ between the paramagnetic P and the ferrimagnetic phase ${\mathrm{FR}}^{\mathrm{II}}(0,\pm 1,\mp 1)$ estimated from the specific heat peaks for $L=48$, the filled circle shows a more precise value obtained from the FSS analysis and Binder cumulant crossing, the filled diamond is the tricritical point and the filled squares represent a first-order transition point determined from FSS analysis at $D/\left|J\right|=-1.48$ and the exact value of the GS transition point at $D_c/\left|J\right|=-3/2$. The empty triangles mark the first-order transition related discontinuities in the order-parameter $m_{s3}$ in the $D/\left|J\right|$ increasing $(▹)$ and decreasing $(◃)$ processes.