Berezinskii-Kosterlitz-Thouless phase transitions of the antiferromagnetic Ising model with ferromagnetic next-nearest-neighbor interactions on the kagome lattice

TL;DR

This work analyzes the antiferromagnetic Ising model on the kagome lattice with ferromagnetic next-nearest-neighbor interactions to reveal two Berezinskii-Kosterlitz-Thouless transitions and six-state clock universality. It combines level-spectroscopy via transfer-matrix methods, large-scale Monte Carlo with parallel tempering, and a neural-network classifier trained on clock-model data to map the phase diagram and extract transition temperatures. The study finds a three-phase structure (ordered, critical BKT, disordered) with a $c\simeq 1$ critical line and demonstrates consistent $T_{1,2}$ across methods, validating the effective Gaussian/d sine-Gordon description of the transitions. These results advance understanding of frustrated kagome systems and illustrate the applicability of hybrid numerical approaches to verify universality classes in complex spin models.

Abstract

We investigate the six-state clock universality of the Ising model on the kagome lattice, considering antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interactions. Our comprehensive study employs three approaches: the level-spectroscopy method, Monte Carlo simulations, and a machine-learning phase classification technique. In this system, we observe two Berezinskii-Kosterlitz-Thouless (BKT) transitions. We present a phase diagram consisting of three phases: the low-temperature ordered phase with sublattice magnetizations, the intermediate BKT phase, and the high-temperature disordered phase, as a function of the ratio of the NNN interaction to the NN interaction. We verify the six-state clock universality through the machine-learning study, which uses data from the six-state clock model on the kagome lattice for training.

Figures (7)

• Figure 1: The schematic representation of Hamiltonian (\ref{['eq_H']}) on the kagome lattice. Open, double, and cross circles denote sites in the three sublattices, $\Lambda_0$, $\Lambda_1$, and $\Lambda_2$, respectively. The NN couplings $J_1$ are denoted by solid lines, and the NNN couplings $J_2$ are denoted by dashed lines. The shaded area indicates the unit region of the $L=12$ system with a periodic boundary condition. The row-to-row transfer matrix ${\bf T}_{\rm even}(s,s')$ [${\bf T}_{\rm odd}(s,s')$] accounts for the Boltzmann weights denoted by red (blue) lines; see the text.
• Figure 2: The global phase diagram of the kagome-lattice AF Ising model with NNN F couplings on the $(u,v)$ plane which consists of the ordered phase with sublattice magnetizations, the critical phase, and the disordered phase. The blue squares (red circles) with a curve plot the BKT transition points $T_1$ ($T_2$) for the finite ratios $r=1/8$, 1/4, 1/3, 1/2, 1.0, 2.0, 3.0, 4.0, 8.0 and for the limiting case of $u=0$. The cross mark, $\times$, on $v$-biaxis denotes the exact phase transition point of the kagome-lattice F Ising model $v_c=(3+2\sqrt3)^{\frac{1}{4}}$. The dotted line satisfies the condition $r=1/3$; the double circle in the critical phase denotes the point $T=0.85$ on the line.
• Figure 3: The system-size dependence of $T_1(L)$ and $T_2(L)$ for $r=1/3$. We draw the straight lines by fitting the $L=18$ and 24 data. Insets (a) and (b) display, respectively, the level-cross conditions Eq. (\ref{['eq_LS1']}) and Eq. (\ref{['eq_LS2']}) for the scaled gaps of $L=24$ as functions of $T$.
• Figure 4: The system-size dependence of $v_1(L)$ and $v_2(L)$ for $u=e^{-J_1/T}=0$. We draw straight lines by fitting the $L=18$ and 24 data. Insets (a) and (b) respectively display the level-crossing conditions Eq. (\ref{['eq_LS1']}) and Eq. (\ref{['eq_LS2']}) for the scaled gaps of $L=24$ as functions of $v=e^{+J_2/T}$.
• Figure 5: The finite-size dependence of the largest eigenvalue of the TM for $r=1/3$ and $T=0.85$; see Fig. \ref{['fig_phase']}. We fit the two largest size data, $l=L/2=9$ and 12, with an inverse geometric factor $v=\sqrt{3}$. The slope of the fitting line estimates $c\simeq1.03$.