Phase diagrams of S=1/2 bilayer Models of SU(2) symmetric antiferromagnets

TL;DR

This work maps the zero-temperature phase diagrams of bilayer S=1/2 SU(2) antiferromagnets with two interlayer coupling schemes—spin-spin (S-S) and energy-energy (E-E)—including two-, four-, and six-spin exchanges. Using sign-problem-free stochastic series expansion QMC, the authors identify Néel, VBS, and dimer phases and characterize transitions across multiple cuts: a 3D O(3) transition in the S-S A cut, first-order Néel–VBS transitions in B and C, and a continuous VBS–dimer transition in D with persistent Z4 anisotropy. In the E-E class, Néel and VBS phases coexist with locked VBS order across layers, while Néel order remains independent, and the Néel–VBS transition appears first-order but weaker, with signs of proximity to an emergent-like symmetry near criticality. The study advances understanding of Landau versus deconfined criticality scenarios in bilayer quantum magnets and highlights intriguing, unresolved questions about symmetry and critical behavior in these systems.

Abstract

We study the $T=0$ phase diagrams of models of bilayers of $S=1/2$ square lattices antiferromagnets with SU(2) Heisenberg symmetry that have 2, 4, and 6 spin exchanges. We study two families of bilayer models with distinct internal symmetries and, hence, different phase diagram topologies. A traditional bilayer model in which the interlayer interaction is Heisenberg so that the two layers can exchange spin (and energy) with each other, making it possible to achieve a simple dimerized valence bond liquid-like state. The resulting phase diagram is rich with Néel, valence bond solid and simple dimer phases, and both first-order and continuous transitions, which we demonstrate are consistent with the conventional Landau theory of order parameters. In the second family of models in which the layers can exchange only energy but no spin (reminiscent of the Ashkin-Teller coupling), the simple dimer state cannot occur. The phase diagrams reveal a number of phase transitions that are accessed for the first time. We find that the phase transition between Néel and VBS is first order in both the spin-spin and energy-energy coupled models, although they have strikingly distinct finite-size scaling behavior and that the transition from VBS to dimer in the spin-spin coupling model deviates from the expected scenario of an XY model with dangerously irrelevant four-fold anisotropy.

Figures (14)

• Figure 1: The bilayer lattice and its interactions. Yellow spheres represent the spins in the first layer, while blue spheres denote the spins in the second layer. The thin black lines in (a) indicate the intralayer exchange interactions denoted as $J$, whereas the thin red lines in (a) represent the interlayer S-S exchange interactions denoted as $J_\perp$. The ellipses depict singlet projection interactions. (b) depicts the intralayer six-spin interaction, denoted as $Q_3$. (c) depicts the intralayer four-spin interactiondenoted as $Q_2$. (d) depicts the interlayer four-spin interaction denoted as $Q_\perp$.
• Figure 2: The cartoons of phases of this model. a) Néel phase. b) Dimer phase. c) VBS phase. Arrows in (a) represent the direction of spins. However, it is important to note that the Néel state in a quantum model is not identical to the antiferromagnetic phase in a classical model. This difference arises because, in the quantum model, spins are not arranged regularly due to quantum fluctuations in the ground state. Thick bonds in b) and c) represent singlet states, which are $\frac{1}{\sqrt{2}} (| \uparrow \downarrow \rangle -| \downarrow \uparrow \rangle)$. The VBS state breaks the $Z_4$ translational symmetry of the lattice. It has four distinct ground states. Figure (c) illustrates one of these ground states.
• Figure 3: Phase diagram of bilayer S-S coupling model, Eq. \ref{['eq:SS3']}. We have constructed the phase boundaries by extensive simulations of the model on moderate system sizes. In addition to probing the nature of the phase transitions, we have studied four cuts $\mathcal{A}$ ($J_\perp/J=14$), $\mathcal{B}$ ($J_\perp/J=5$), $\mathcal{C}$ ($J=0, J_\perp<0$), $\mathcal{D}$ ($J=0, J_\perp>0$) in detail on large lattices. Briefly, we find that the phase transition on the $\mathcal{A}$ cut belongs to 3D O(3) universality, whereas $\mathcal{B}$ and $\mathcal{C}$ cuts are first-order phase transitions. All of these are consistent with the expectations of a straightforward Landau theory. Most interesting is $\mathcal{D}$ cut, where we observe behavior consistent with a continuous phase transition. One natural scenario for a continuous transition is the three-dimensional XY transition with a dangerously irrelevant four fold magnetic field anistropy (for recent work see e.g. carmona6loopShao_2020). However, from our numerical simulations we find in our model a persistent $Z_4$ anisotropy that contradicts the expected emergent O(2) symmetry of the dangerously irrelevant scenario. We are unable to offer a consistent theoretical scenario for this numerical observation. In our phase diagrams, we represent first-order phase transitions with dashed lines and continuous phase transitions with solid lines. The blue line indicates the O(3) phase transition, whereas the universality class of the red line is not clear yet. Additionally, two stars are marked: the blue star denotes the single-layer Néel-VBS transition, and the yellow star represents a multi-critical point where the three phases meet.
• Figure 4: The finite-size behavior of $U_m$ and $\rho_sL$ at cut $\mathcal{A}$. (a) shows $U_m$ varies with $g_\mathcal{A} = Q_3/J$ near the phase transition point for different system sizes $L$. (b) illustrates how $L\rho_s$ changes with $g_\mathcal{A}$ near the phase transition for various system sizes $L$. Both tend to 0 in the disordered phase and approach finite values in the N'eel ordered phase, with their crossings converging to the phase transition point as the system size $L$ increases.
• Figure 5: The finite size scaling study of $1/\nu$ by the crossing points analysis of $(L,2L)$ on cut $\mathcal{A}$ of Fig. \ref{['jq3jpd']} where $J_\perp/J=14$. The red color refers to the crossing points from $U_{m}$. Blue colors refer to the results from spin stiffness. Filled circles refer to the QMC data, and the solid lines are fits using Eq. (\ref{['eq:nuc']}). We find $1/\nu=1.40(1)$ from $U_m$ and $1.403(2)$from $L\rho_s$. The fitting windows is $L=12 \sim 32$ for $U_m$, $L=16 \sim 32$ for $L\rho_s$.