Ground-state phase diagram of S = 1/2 Heisenberg model on 2D square-hexagon-octagon lattice

TL;DR

The paper addresses how a $S=\tfrac{1}{2}$ Heisenberg model on the square-hexagon-octagon lattice organizes its ground state under competing nearest and selective next-nearest couplings. It combines SSE-QMC and DMRG with finite-size scaling, Binder cumulants, spin stiffness, and entanglement analysis to map a detailed phase diagram containing AFM, hexagon, OSD, LSD, and a Haldane-like SPT phase, and shows that nonmagnetic-to-AFM transitions fall into the 3D $O(3)$ universality class. A key finding is the identification of a Haldane-like SPT phase in a 2D spin system, evidenced by degenerate ground states under open boundaries and a doubly degenerate entanglement spectrum, with $c=1$ at the relevant transitions. The results provide a foundational framework for understanding 2D magnetism on the SHO lattice and offer benchmarks for potential experimental realizations in SHO-structured materials, cold-atom systems, or MOFs.

Abstract

Using stochastic series expansion quantum Monte Carlo method and density matrix renormalization group, we study the ground-state phase diagram of $S=1/2$ Heisenberg model on 2D square-hexagon-octagon (SHO) lattice. In this model, we consider two kinds of nearest-neighbor interaction (intra-hexagon interaction $J_1$ and inter-hexagon $J_2$) and the selected third nearest-neighbor interaction $J_3$ along $x$ direction. From our calculations, there are five phases in the parameters regime $0<λ_1=J_2/J_1<4, 0<λ_2=J_3/J_1<4$, including a Néel antiferromagentic phase, a Haldane-like symmetry protected topological phase, a hexagon phase and two dimer phases. In the Haldane-like SPT phase, we characterized its topological nature using the degeneracy of ground-state energy under open boundary condition and the entanglement spectrum. To characterize the phase boundaries, we use spin stiffness and Binder cumulant to do the comprehensive finite-size scalings. From data collapse, the critical behaviors of all the nonmagnetic phases to the antiferromagnetic phase belong to the 3D $O(3)$ Heisenberg universality class. As a theoretical exploration, our work establishes a foundational framework for understanding 2D magnetism on the SHO lattice.

Figures (8)

• Figure 1: (a)The sketch of SHO structure, constituted with hexagon unit cells. There are three types of bonds with different interactions from our definition: $J_1$ indicated with black color, $J_2$ is red and $J_3$ is blue, respectively. We also define the ratios between them: $\lambda_1=J_2/J_1$ and $\lambda_2=J_3/J_1$. A unit cell of SHO lattice consists of six sites, which is shown in the black dashed square. (b) The lattice in square form which is topologically equivalent to the SHO lattice.
• Figure 2: The ground-state phase diagram of $S = 1/2$ Heisenberg model on 2D SHO lattice. In total there are five phases in the phase diagram, namely the hexagon phase, the orthogonal staggered dimer (OSD) phase, the ladder staggered dimer (LSD) phase, the Haldane-like symmetry protected topological (SPT) phase, and the Néel antiferromagnetic (AFM) phase.
• Figure 3: (a)Variation of Binder cumulant $U_2$ with $\lambda_1$ at different sizes when $\lambda_2 = 0$. Inset in (a) is the extrapolation of the two intersection positions of the Binder cumulant as a function of the size, using the extrapolation function in the form of $\lambda_{1,c}(L)=\lambda_{1,c}(\infty)+aL^{-\omega}$10.1063/1.3518900. (b) Second-order polynomial extrapolation of $M_s^2$ as a function of $L$ when $\lambda_1 = 1$. The nonzero extrapolated value indicates that there is a nonzero staggered magnetic order in the thermodynamic limit: $M^2_s(\infty) = 0.0609(2)$, and $SU(2)$ continuous symmetry breaking occurs at the intermediate phase.
• Figure 4: (a) When $\lambda_1 =0$, the Binder cumulant which is defined on a single ladder varies with $\lambda_2$. As the intermediate phase is nonmagnetic, the Binder cumulant fails to reveal phase transitions between nonmagnetic phases, and no intersection points are visible. (b) When $\lambda_1 = 0$, the spin stiffness in the $x$-direction changes with $\lambda_2$. Unlike the Binder cumulant, spin stiffness can detect phase transitions between nonmagnetic phases. However, there are four intersections, so extrapolation is needed to pinpoint the exact phase transition points. (c) and (d) show the extrapolation results for the spin stiffness intersections at the two phase transition points, with the following form of extrapolation function: $\lambda_{2,c}(L)=\lambda_{2,c}(\infty)+aL^{-\omega}$10.1063/1.3518900. Near $\lambda_{2,c} = 1.040(2)$ and $\lambda_{2,c} = 1.634(2)$, the extrapolation curves approach and intersect each other.
• Figure 5: (a) and (b) display the extrapolation of the excitation energy gaps for singlet ($S=0$), triplet ($S=1$) and quintuplet ($S=2$) states at $\lambda_2=1.4$, under periodic and open boundary condition, respectively. The fitting function used is: $\Delta(L)=a+e^{-L/\xi}(b/L + c/L^2)$Hohenadler2012JKFang2021. Panel (c) shows the entanglement spectrum for a linear system size $L=32$ at $\lambda_2=1.4$ under open boundary conditions. Pannel (d) presents the entanglement entropy near the phase transition points at $\lambda_1=0$, with $L=32$ under periodic boundary condition. The inset includes liner extrapolations of the central charges $c$ at the these transition points.