Liquid-gas analog multicriticality in a frustrated Ising bilayer

TL;DR

The paper studies a classical frustrated bilayer Ising model with tunable couplings that hosts three symmetry-breaking phases and reveals a multicritical line controlled by emergent $Z_2$ symmetry. Using large-scale Monte Carlo simulations and field-mixing analysis, the authors locate a tricritical line and a critical endpoint that merge into a single multicritical line at strong coupling. Along the multicritical line, the DF and DAF sectors share the same leading critical behavior in the $A$-stage $2$D tricritical Ising universality class, while the subleading exponent $y_g$ shifts from $0.80$ to $1.00$ due to the emergent symmetry, reorganizing scaling without changing the leading class. The results establish a general mechanism for symmetry-enforced multicriticality and point to broader relevance for quantum multicritical phenomena and programmable experimental platforms.

Abstract

We report the discovery of a multicritical point that extends the liquid-gas paradigm to systems with competing symmetry-breaking orders. Using large-scale Monte Carlo simulations of a frustrated bilayer Ising antiferromagnet with tunable couplings, we map out a rich finite-temperature phase diagram hosting three ordered phases separated by both continuous and first-order transitions. By tuning the couplings, a tricritical line and a critical end-point line converge into a single multicritical line. At all points along the multicritical line, symmetry-distinct phases exhibit identical leading critical behavior -- consistent with the tricritical Ising universality class -- while the subleading exponent exhibits a sharp shift from $y_g = 0.8$ to $y_g = 1$. This shift reflects an emergent $Z_2$ symmetry akin to that of the liquid-gas critical point, but realized here at a genuine multicritical point involving simultaneous microscopic symmetry breaking. Our results establish a universality scenario in which emergent symmetry preserves the leading class but reorganizes subleading scaling, providing a general mechanism for symmetry-enforced multicriticality.

Figures (9)

• Figure 1: (a) Ground-state phase diagram in the $(J_\parallel / J_\perp, J_\times / J_\perp)$ plane. (b) Finite-temperature phase diagram on the $J_\times = J_\parallel$ slice, showing finite temperature first-order (black) and second-order Ising (blue) transitions; their junction defines a tricritical Ising point (magenta). The cyan point at ($t=0$, $J_\parallel/J_\perp = 0.25$) marks the triple point in (a), which also serves as a critical endpoint where the DF transition line can terminate (e.g., on the $J_\times/J_\perp = 0.25$ slice). Inset: bilayer square-lattice geometry and interactions. (c) 3D phase diagram in $(t, J_\parallel / J_\perp, J_\times / J_\perp)$ space, shown for slices at $J_\times = J_\parallel$ and at fixed $J_\times/J_\perp = 0.4$, $0.6$, and $0.8$. Representative results for $J_\times/J_\perp = 0.4$ and $0.8$ are presented in the main text, while results for the $J_\times = J_\parallel$ slice are given in Appendix \ref{['app:slice1']}. Yellow and blue lines denote continuous Ising transitions into the DF and DAF phases; black lines indicate first-order transitions. Magenta and cyan dots mark tricritical points and critical endpoints within each slice, which evolve into tricritical and critical endpoint lines merging into a single multicritical line at strong coupling. The finite-temperature extension of the BAF phase, symmetric to DF, is omitted here for clarity.
• Figure 2: (a) Finite-temperature phase diagram at $J_\times / J_\perp = 0.4$. The black line denotes a first-order transition, with a thermal CEP (cyan) and a TCP (magenta) separating first- and second-order transitions. (b) DF order parameter versus $J_\parallel / J_\perp$ at various temperatures near the CEP. Inset: DAF order parameter across the transition line near the TCP. (c) Symmetric $P_L(\tilde{Q})$ at $t = t_c$ for various $L$. (d) Fourth-order cumulant $U_Q$ along the transition line, determining the tricritical temperature $t_c = 0.2119(2)$. Inset: finite-size extrapolation of the transition points yields $(J_\parallel / J_\perp)_c$=0.27751(3).
• Figure 3: Tricritical scaling at $J_\times / J_\perp = 0.4$. (a) Scaling collapse of the symmetric $P_L(\tilde{Q})$ at $t = t_c$, yielding the leading exponent $y_t = 1.80(2)$. (b) Scaling collapse of the DAF correlation length at $t = t_c$, giving $\nu = 0.555(6) = 1/y_t$. Inset: Correlation length crossing yields the critical coupling $(J_\parallel / J_\perp)_c = 0.27751(2)$, consistent with the extrapolated value from Fig. 2(d). (c) Scaling collapse of the DAF order parameter at $t = t_c$, giving $\beta = 0.042(8)$. (d) Scaling collapse of the fourth-order cumulant $U_Q$ along the DAF transition line, yielding the subleading exponent $y_g = 0.80(2)$.
• Figure 4: (a) Finite-temperature phase diagram at $J_\times / J_\perp = 0.8$. (b) Scaling collapse of the fourth-order cumulant $U_Q$ along the transition line yields the subleading exponent $y_g = 1.00(4)$. Inset: crossing of $U_Q$ at $t_c = 0.4309(2)$. (c) Correlation length crossings for DF and DAF at $t = t_c$ occur at the same coupling $(J_\parallel / J_\perp)_c$=0.25176(4), confirming the convergence of the CEP and TCP. (d) Scaling collapse of the symmetric $P_L(\tilde{Q})$ at $t = t_c$ gives the leading exponent $y_t = 1.80(2)$, with $Q = \epsilon$ under optimal field mixing $s = 0$.
• Figure 5: Finite-size scaling collapses at the multicritical point of the $J_\times/J_\perp = 0.8$ slice, where $t=t_c$: correlation lengths for DAF (a) and DF (c) yield a common exponent $\nu = 0.555(6)$, while the corresponding order parameters in (b) and (d) collapse with $\beta = 0.042(9)$.