Ground state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field

TL;DR

This paper develops and applies the pattern language to the one-dimensional ANNNI model in a transverse field to resolve conflicting phase diagrams and understand the floating phase. The method uses two diagonalizations to extract collective patterns $A_n$ with eigenvalues $\lambda_n$ that decompose the Hamiltonian as $\sum_n \lambda_n A_n^\dagger A_n$, enabling projection of the ground state onto pattern subspaces and identifying phases via pattern occupancy. The results reveal four ground-state phases—ferromagnetic, paramagnetic, floating, and $\langle 2,2\rangle$ antiphase—with a continuous FM–PM transition and a first-order FP–AP transition; the floating phase shows relay-like pattern activations whose sequence grows dense with system size, suggesting a route to continuous variation in the thermodynamic limit. The approach provides a microscopic, wave-function-based visualization of frustration-induced phases and hints at Monte Carlo-friendly observables derived from pattern occupancy for future numerical explorations.

Abstract

Describing and understanding the consequences of competing interactions remains profoundly challenging in both classical and quantum systems, as it is difficult to identify suitable order parameters, thereby hindering the characterization of certain phases such as the floating phase found in the one-dimensional axial next-nearest-neighbor Ising (ANNNI) model in the presence of the frustration interactions between the nearest and next-nearest neighbor sites. In this work, we employ the pattern picture to explore the frustration physics in such a model. This picture has been comprehensively detailed in our previous article [Yang and Luo, Phys. Rev. E \textbf{112}, 044102 (2025)]. Here, we apply it to the ANNNI model with periodic boundary conditions, considering system sizes ranging from $L=16$ to $L=128$. Our results demonstrate that the ground state of the system comprises four phases: ferromagnetic, paramagnetic, floating, and $\langle 2,2 \rangle$ antiphase. The transition from the ferromagnetic to the paramagnetic phase is continuous, analogous to that in the transverse-field Ising model, while the transition from the floating phase to the antiphase is first order. Furthermore, the floating phase exhibits particularly intriguing characteristics: states with distinct domain structures emerge successively as the frustration parameter increases. As the system size grows, this succession becomes progressively denser, leading to the reasonable inference that it eventually approaches a continuous variation in the thermodynamic limit. To validate the effectiveness of our picture, we computed the second derivative of the ground-state energy, which exhibits multiple dips within the floating phase$-$consistent with the pattern language.

Figures (9)

• Figure 1: Patterns of the Hamiltonian in Eq. (\ref{['annni1']}). Spin configurations (up/down) correspond to the signs of the coefficients $u_{n,i}$ in $A_n$. Results are shown for a system with lattice size $L=8$ under PBC. These patterns are roubst and remain unchanged across different parameters of the model.
• Figure 2: The total energy of the system (not shown) and the energies of the sub-Hamiltonians (colored curves) as functions of $\kappa$. The red curve corresponds to $H_1$, while the blue curve represents $H_{4,5}$. Only the sub-Hamiltonian energies are displayed, since the energies of $H_1$ and $H_{4,5}$ coincide with the total energy. This indicates that the system is in the ferromagnetic phase at $\kappa<0.5$ and in the antiphase at $\kappa>0.5$. All other sub-Hamiltonians have zero energy.
• Figure 3: Patterns and their relative phases obtained from the first diagonalization. Each pattern is labeled by the single-body operators $\hat{A}_n$ where the notation $(\pm,\pm)$ denotes the signs of the coefficients $(u_{n,2i-1},u_{n,2i})$. The patterns are divided into two groups, marked by the red and blue dashed frames, corresponding to $\lambda_n < 0$ and $\lambda_n > 0$, respectively. Results are shown for a system size of $L=8$. These patterns remain unchanged for different parameters of the model.
• Figure 4: Ground state energies as functions of the frustration parameter $\kappa$, computed using Eq. (\ref{['annni6a']}) (black solid lines) and direct numerical ED (circles). The corresponding energy components $\langle H_n \rangle$ are shown as colored solid lines. The patterns displayed below are obtained from the signs of the coefficients $u_{n,2i}$. The model parameter is taken as $J=2.0$.
• Figure 5: Ground state energy $E_0$ (black ines) and the energy components $\langle H_n \rangle$ (colored lines) as functions of the frustration parameter $\kappa$ for system sizes ranging from $L=16$ to $128$. Four distinct phases are identified: ferromagnetic (FM), paramagnetic (PM), floating phase (FP) and antiphase (AP). Vertical dashed lines indicate phase boundaries with shaded region denoting a crossover. The model parameter is fixed at $J=2.0$. The step of $\kappa$ is set to 0.003 for this and subsequent analyses.