Tricriticality in crossed Ising chains

We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$ and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.

Figure 1

The geometry of the interactions in the CICM for four sites is shown. This model is studied on an $L\times L$ square lattice with periodic boundary conditions (in the $\widehat{x}$ and $\widehat{y}$ directions) containing $N$ ($=L^2$) “$S$ spins” and $N$ “$T$ spins” taking values $\pm 1$. There are $L$ parallel 1D chains of $S$ spins in the $\widehat{x}$ direction and $L$ parallel 1D chains of $T$ spins in the $\widehat{y}$ direction, illustrated by the blue ($J_x$) and red ($J_y$) lines, respectively. There is an interaction between an $S$ and a $T$ spin on the same site in the $\widehat{z}$ direction ($J_z$) illustrated by the green lines. Finally, there is an interaction between nearest-neighbor $S$ and $T$ spins ($J_{z^{\prime }}$) that is illustrated by the gray lines.

Figure 2

The four ordered phases found in the CICM are defined. Each pair of spins represents an $S$ and a $T$ spin on a single site (i.e., coupled by $J_z$). In the ferromagnetic (FM) phase, all $S$ and $T$ spins are aligned ferromagnetically. In the ferromagnetic prime (${\mathrm{FM}}^{\prime }$) phase, the $S$ and $T$ spins are aligned ferromagnetically on each site and antiferromagnetically along the $S$ and $T$ chains. In the antiferromagnetic (AFM) phase, the $S$ and $T$ spins are aligned antiferromagnetically on each site and ferromagnetically along the $S$ and $T$ chains. Finally, in the antiferromagnetic-prime (${\mathrm{AFM}}^{\prime }$) phase, the $S$ and $T$ spins are aligned antiferromagnetically on each site and also along the $S$ and $T$ chains. There is spin inversion symmetry in this model, so flipping all of the spins in any of these phases does not change the phase.

Figure 3

The phase diagram for the CICM with $J_{x,y}=1$ and $T=0$ in the $J_{z^{\prime }}$ vs $J_z$ parameter space is shown. The line separating the FM and AFM phases (where the internal energies are equal) is given by $J_{z^{\prime }}=-\frac{1}{4}{J}_z$. The AFM and ${\mathrm{AFM}}^{\prime }$ phases are separated by $J_{z^{\prime }}=\frac{J_{x,y}}{2}=\frac{1}{2}$; the FM and ${\mathrm{FM}}^{\prime }$ phases by $J_{z^{\prime }}=\frac{-J_{x,y}}{2}=\frac{-1}{2}$; the FM and ${\mathrm{AFM}}^{\prime }$ phases by $J_z=-2J_{x,y}=-2$; and the AFM and ${\mathrm{FM}}^{\prime }$ phases by $J_z=2J_{x,y}=2$. This phase diagram is consistent with the observation that for $J_z$ and $J_{z^{\prime }}$ both positive or both negative, there is no competition between phases. Additionally, the symmetry between FM and AFM and ${\mathrm{FM}}^{\prime }$ and ${\mathrm{AFM}}^{\prime }$ when switching the signs of $J_z$ and $J_{z^{\prime }}$ is evident.

Figure 4

A plot of the FM and AFM free energies at $J_{z^{\prime }}=0.1$ and $T=1.0$ is shown. For $J_z=-0.39$, the minimum of the FM free energy (thin blue solid curve) is lower than the minimum of the AFM free energy (thin blue dashed curve) and therefore the system is in the FM phase. For $J_z=-0.40$ the minimum of the FM free energy (thick red solid curve) is greater than the minimum of the AFM free energy (thick red dashed curve) and therefore the system is in the AFM phase. If the global minima were to be at $m=0$, the system would be in the PM phase where the total magnetization is 0. This is the analysis used to determine all of the first-order phase boundaries in the mean-field theory phase diagrams.

Figure 5

The mean-field theory phase diagram for $J_{z^{\prime }}=0$ and $J_{x,y}=1$ is shown. For these parameters, there is no tricritical point. For $J_z>0$, there is a second-order phase transition between a low-temperature FM phase and a high-temperature PM phase. For $J_z<0$, there is a second-order phase transition between a low-temperature AFM phase and a high-temperature PM phase. Also, there is a vertical first-order phase boundary between the FM and AFM phases at $J_z=0$ that extends up to $T=2$, the MF critical temperature for the 1D Ising model. In the large, positive $J_z$ limit, this model becomes an effective 2D Ising model with $T_C=4(J_{x,y}+2J_{z^{\prime }})=4$. In the large, negative $J_z$ limit, this model also becomes an effective 2D Ising model with $T_C=4(J_{x,y}-2J_{z^{\prime }})=4$. The FM and AFM phase shapes are symmetric about $J_z=0$ only when $J_{z^{\prime }}=0$.

Figure 6

The mean-field theory phase diagram for $J_{z^{\prime }}=0.1$ and $J_{x,y}=1$ is shown. For these parameters, there is no tricritical point. For $J_z>-4J_{z^{\prime }}=-0.4$, there is a second-order phase transition between a low-temperature FM phase and a high-temperature PM phase. For $J_z<-4J_{z^{\prime }}=-0.4$, there is a second-order phase transition between a low-temperature AFM phase and a high-temperature PM phase. Also, there is a vertical first-order phase boundary between the FM and AFM phases at $J_z=-0.4$. In the large, positive $J_z$ limit, this model becomes an effective 2D Ising model with $T_C=4(J_{x,y}+2J_{z^{\prime }})=4.8$. In the large, negative $J_z$ limit, this model also becomes an effective 2D Ising model with $T_C=4(J_{x,y}-2J_{z^{\prime }})=3.2$. A positive $J_{z^{\prime }}$ shrinks the AFM phase and grows the FM phase as it is increased in magnitude, as evidenced by comparing with the phase diagram for $J_{z^{\prime }}=0$.

Figure 7

The mean-field theory phase diagram for $J_{z^{\prime }}=0.3$ and $J_{x,y}=1$ is shown. For $J_z\gtrsim -0.7$, there is a second-order phase transition between a low-temperature FM phase and a high-temperature PM phase. For $-1.1\lesssim J_z\lesssim -0.7$, there is a first-order phase transition between a low-temperature FM phase and a high-temperature PM phase. This section of the phase diagram is separated from the previous section by the green square tricritical point (located at $T=\frac{32}{15}\approx 2.13,J_z=\frac{-16ln\left(2\right)}{15}\approx -0.74$). For $-1.2\lesssim J_z\lesssim -1.1$, there is a low-temperature FM phase followed by a small higher-temperature AFM phase and then, for higher temperatures, a disordered PM phase. For $J_z\lesssim -1.2$, there is a second-order phase boundary between a low-temperature AFM phase and a high-temperature PM phase. Additionally, there is an approximately vertical first-order phase boundary between the FM and AFM phases at $J_z=-1.2$. This phase diagram is zoomed in relative to the other phase diagrams in order to show the details of the tricritical point and first-order phase boundary.

Figure 8

The Binder fourth-order cumulants for various lattice sizes and for a second-order phase transition (solid lines) at $J_z=-0.9$ and a first-order phase transition (dashed lines) at $J_z=-1.1$. A minimum below $U_{L,\mathrm{FM}}=0$, which gets deeper as the lattice size increases, is a signature of a first-order phase transition. The curves intersect at a single critical temperature in both cases. In both cases, at lower temperatures than the crossing, the order of the curves from top to bottom is $L=20$, 14, and 12, and finally $L=10$. $U_L$ was used to determine all of the critical points in the MC phase diagrams.

Figure 9

The MC-derived phase diagram for $J_{z^{\prime }}=0$ and $J_{x,y}=1$ is shown. For $J_z>0$, there is a second-order phase boundary separating a low-temperature FM phase from a high-temperature PM phase. The critical temperature increases as $J_z$ increases and saturates at $T_C\approx 2.269(J_{x,y}+2J_{z^{\prime }})=2.269$. For $J_z<0$, there is a second-order phase boundary separating a low-temperature AFM phase from a high-temperature PM phase. Similarly, the critical temperature increases as $J_z$ decreases until it saturates at $T_C\approx 2.269(J_{x,y}-2J_{z^{\prime }})=2.269$. This phase diagram is qualitatively similar to the MFT phase diagram with the same parameters, particularly in its lack of a tricritical point. As expected, the critical temperatures were overestimated in MFT.

Figure 10

Same as Fig. 9 except $J_{z^{\prime }}=0.1$. The phase diagram is qualitatively similar to the result of MFT, particularly in its lack of a tricritical point.

Figure 11

Same as Fig. 9 except $J_{z^{\prime }}=0.3.$ The phase diagram is qualitatively similar to the result of MFT in most regards. However, MC finds that a tricritical point, which is present only on the FM side in MFT, is also present on the AFM phase boundary.

Figure 12

Same as Fig. 9 except $J_{z^{\prime }}=1.0.$ As for $J_{z^{\prime }}=0.0$ and 0.1, there is no tricritical point, thus they appear to be restricted to intermediate $J_{z^{\prime }}\approx 0.3$.

Figure 13

The canonical distributions for three temperatures around a first-order phase boundary are shown (with $J_z=-1.1$ and $J_{z^{\prime }}=0.3$). The characteristic double peak behavior due to phase coexistence is clear. At the transition temperature ($T_T$) (black data), the two peaks are of equal height. The canonical distributions were normalized by the constant $k\left(T\right)$ such that the maximum height of the distribution is equal to 1. This simulation was performed on a size $L=30$ lattice.

Figure 14

The data collapse of $\left[L^{\frac{2\beta }{\nu }}\langle m_{\text{FM}}^2\left(t\right)\rangle \right]$ vs ($L^{\frac{1}{\nu }}t$) for $L=40$, 60, 80, and 100 using the two-dimensional Ising universality class exponents is shown.

Figure 15

The data collapse of $\left[L^{\frac{-\gamma }{\nu }}\langle {\chi }_L\left(t\right)\rangle \right]$ vs ($L^{\frac{1}{\nu }}t$) for $L=40$, 60, 80, and 100 using the two-dimensional Ising universality class exponents is shown.