Ordering phenomena in a heterostructure of frustrated and unfrustrated triangular-lattice Ising layers

We study critical and magnetic properties of a bilayer Ising system consisting of two triangular planes A and B, with the antiferromagnetic (AF) coupling $J_{\mathrm{A}}$ and the ferromagnetic (FM) one $J_{\mathrm{B}}$ for the respective layers, which are coupled by the interlayer interaction $J_{\mathrm{AB}}$ by using Monte Carlo simulations. When $J_{\mathrm{A}}$ and $J_{\mathrm{B}}$ are of the same order, the unfrustrated FM plane orders first at a high temperature $T_{c1}\sim J_{\mathrm{B}}$. The spontaneous FM order then exerts influence on the other frustrated AF plane as an effective magnetic field, which subsequently induces a ferrimagnetic order in this plane at low temperatures below $T_{c2}$. When short-range order is developed in the AF plane while the influence of the FM plane is still small, there appears a preemptive Berezinskii-Kosterlitz-Thouless–type pseudocritical crossover regime just above the ferrimagnetic phase transition point, where the short-distance behavior up to a rather large length scale exponentially diverging in $\propto J_{\mathrm{A}}/T$ is controlled by a line of Gaussian fixed points at $T=0$. In the crossover region, a continuous variation in the effective critical exponent $\frac{4}{9}\lesssim {\eta }^{\mathrm{eff}}\lesssim \frac{1}{2}$ is observed. The phase diagram by changing the ratio $J_{\mathrm{A}}/J_{\mathrm{B}}$ is also investigated.

Figure 1

(a) Structure of the bilayer triangular lattice. (b) The investigated regions of the parameter space. The solid red line and the dashed blue lines represent the case(s) with $-J_{\mathrm{A}}=J_{\mathrm{B}}$ and $-J_{\mathrm{A}}\ne J_{\mathrm{B}}$, respectively.

Figure 2

Integrated autocorrelation time ${\tau }_{\mathrm{int},m_o}$ near the low-temperature phase transition in the AF layer for ($J_{\mathrm{A}}$, $J_{\mathrm{B}}$, $J_{\mathrm{AB}}$) $=(-0.4,0.4,0.6)$ with different lattice sizes $L$. The inset shows the $L$ dependence of the peak height showing a power-law behavior with the estimated dynamic critical exponent $z\approx 2.3$.

Figure 3

Ground-state phase diagram.

Figure 4

Temperature variations of the sublattice magnetizations (a) $m_{\mathrm{A}}$ and (b) $m_{\mathrm{B}}$, (c) the ferrimagnetic order $m_o$ of the AF layer, (d) the total specific heat, and (e) inverse temperature variations of the entropy density, for various values of $J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}$ and $L=48$. The solid black curves in (d) and (e) show the exact solutions for the decoupled dimer limit ($J=0$). $T_{c1}$ and $T_{c2}$ in (d) represent two critical temperatures for $J=0.4$. (f) $T$ dependence of ${\chi }_{\mathrm{A}}$, ${\chi }_{\mathrm{B}}$, and ${\chi }_o$ for $J=0.4$.

Figure 5

Phase diagram in the $(T,J)$ parameter space where $J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}$. The transition temperatures $T_{c1}$ and $T_{c2}$ are determined in several ways: the maximum of the corresponding susceptibilities (i.e., ${\chi }_{\mathrm{B}}$ for $T_{c1}$ and ${\chi }_o$ for $T_{c2}$), the crossing of $U_{4,o}$ for $T_{c2}$, and the FSS analysis, as indicated within the parentheses of the legends, with the last one giving most accurate estimates. The pseudocritical crossover regime surrounded by the lines of $T^{*}$ at high $T$ and $T^{**}$ at low $T$, characterized by ${\eta }_o^{\mathrm{eff}}=\frac{1}{2}$ and ${\eta }_o^{\mathrm{eff}}=\frac{4}{9}$, respectively, is the part of the FM phase in this parameter space ($T^{**}$ is expected to coincide with $T_{c2}$ in the thermodynamic limit; see the text). The filled squares at $T=0$ indicate the exact interval of the stabilization of the FR-FM ground state, $\frac{1}{7}<J<1$. The ground state for $0\le J<\frac{1}{7}$ is the trivial disordered state comprising decoupled dimers (see the text).

Figure 6

(a) Critical exponent ratios and (b) $T_{\max }^{\mathcal{O}}\left(L\right)$ obtained in the FSS analysis at the high-temperature phase transition into the FM phase driven by spins in the plane B, for $J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}=0.2$. The arrow in (b) indicates the estimate of $T_{c1}$ in the thermodynamic limit.

Figure 7

(a) Critical exponent ratios and (b) $T_{\max }^{\mathcal{O}}\left(L\right)$ for $J=0.4$, obtained in the FSS analysis at the low-temperature phase transition into the FR phase ($J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}$). (c) Critical exponent ratios and (d) $T_{\max }^{\mathcal{O}}\left(L\right)$ for $J=0.6$. The arrows in (b) and (d) indicate the estimates of $T_{c2}$ in the thermodynamic limit for $J=0.4$ and $0.6$, respectively.

Figure 8

(a) Critical exponent ratios obtained for various $J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}$ at the FR transition at $T=T_{c2}$, based on FSS using the lattice sizes $L\ge L_{\min }=72$. (b) Evolution of the values for $J=0.6$ when the data for $L<L_{\min }$ are gradually dropped from FSS. The dashed lines indicate the universal values of the three-state ferromagnetic Potts model.

Figure 9

Temperature variation of (a) the order parameter $m_o$ for various $L$ and (b) the exponent ${\eta }_o^{\mathrm{eff}}$ obtained from the scaling relation (14), for $J=-J_{\mathrm{A}}=J_{\mathrm{B}}=1-J_{\mathrm{AB}}=0.9$. In (b), the value $\frac{4}{9}\le {\eta }_o^{\mathrm{eff}}\le \frac{1}{2}$ is expected for the BKT-type spin-correlation in the plane A induced by a coupling to the FM ordered layer B (see the text in Sec. 5). The insets show the FSS log-log plots of $m_o\left(L\right)$ for some representative values of $T$ (right) and the coefficient of determination of the linear fit $R^2$ as a function of $T$ (left); the region with $R^2\approx 1$ at low temperatures is indicated by the shaded area here and also in the main plot of (b).

Figure 10

Temperature dependencies of the quantities $m_{\mathrm{A}}$, $m_{\mathrm{B}}$, and $m_o$ (left column), ${\chi }_{\mathrm{A}}$, ${\chi }_{\mathrm{B}}$, and ${\chi }_o$ (central column), and ${\eta }_{\mathrm{A}}$, ${\eta }_{\mathrm{B}}$, and ${\eta }_o$ (right column) for $J_{\mathrm{AB}}=0.6$, $J_{\mathrm{B}}=0.4$, varying values of $J_{\mathrm{A}}$, and $L=24\text{–}120$. The panel (h) is the reproduction of Fig. 4 to make a comparison. In the panel (o), ${\eta }_o^{\mathrm{eff}}$ for $J_{\mathrm{A}}=-10$ is also included.

Figure 11

Phase diagram as a function of $J_{\mathrm{A}}$ for $(J_{\mathrm{B}},J_{\mathrm{AB}})=(0.4,0.6)$. $T_{c1}$ and $T_{c2}$ are determined by either the maximum of ${\chi }_{\mathrm{B}}$ for $T_{c1}$ (${\chi }_o$ for $T_{c2}$) or the crossing of $U_{4,o}$ for $T_{c2}$, as indicated. The filled square at $J_{\mathrm{A}}=-\left(\frac{1}{6}\right)J_{\mathrm{AB}}=-0.1$ indicates the transition point in the ground state.

Figure 12

(a) Temperature dependence of ${\eta }_o^{\mathrm{eff}}$ for $J_{\mathrm{A}}=-1.6$, $-4$, and $-10$ with $(J_{\mathrm{B}},J_{\mathrm{AB}})=(0.4,0.6)$. (b)–(d) Temperature dependence of the coefficient of determination of the linear fit $R^2$ for each case.

Figure 13

(a) The order parameter $m_o$ for $L=24$ and (b) the effective critical exponent ${\eta }_o^{\mathrm{eff}}$, for the bilayer model (blue circles) and the single-layer TLIA model in the field $\tilde{H}=0.1$ (red squares).

Figure 14

(a) The height rule (see the text). (b) A plaquette of three parallel spins, which can be three up or three down, is a vortex (left) or an antivortex (right) in the height description.

Figure 15

Normalized correlation length $\xi /L$, as functions of the temperature, for several system sizes $L$ for ${\mathcal{H}}_{\mathrm{TLIA}}$. The inset shows the behavior in a broader temperature range.

Figure 16

(a) Schematic projected RG flow diagram including three isolated fixed points (FPs) and a line of Gaussian FPs at $T=0$. The dashed line indicates a typical trajectory of the system as $T$ is lowered. (b) Schematic picture illustrating the situation where the pseudocritical behavior is observed. The system parameters are in proximity of the line of Gaussian FPs and the average vortex-antivortex separation $\approx \xi$ well exceeds the size $L$ of the system (the area represented by the filled square).