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arXiv:2301.08891

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

Yun-Tong Yang, Chen, Fu-Zhou, Luo, Hong-Gang

Published Jan 21, 2023• 2 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2301.08891

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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=16L=16L=16 to L=128L=128L=128. Our results demonstrate that the ground state of the system comprises four phases: ferromagnetic, paramagnetic, floating, and ⟨2,2⟩\langle 2,2 \rangle⟨2,2⟩ 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.
In this work we analyze the ground-state properties of the s=1/2 one-dimensional axial next-nearest-neighbor Ising model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field B_{x} and the strength of the next-nearest-neighbor interaction J_{2}, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, floating, and 〈2,2〉 phases, and we predict an infinite number of modulated phases in the thermodynamic limit (L→∞). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second order, whereas the line between the floating and the 〈2,2〉 phases is possibly of first order.
We study the transverse-field axial next-nearest-neighbor Ising (ANNNI) model on the square lattice and on a bilayer. At the classical multiphase point κ=J2/J0=1/2κ=J_2/J_0=1/2κ=J2​/J0​=1/2, every configuration of straight domain walls that separates domains of at least two sites is a ground state. The walls are rigid, so perturbation theory in the transverse field g=Γ/J0g=Γ/J_0g=Γ/J0​ can be carried out for each configuration separately, and we take it to order g8g^8g8. Quantum fluctuations select ⟨3⟩\langle 3\rangle⟨3⟩ (↑↑↑↓↓↓\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow↑↑↑↓↓↓) at order g2g^2g2, because spins next to a wall gain the most energy from virtual flips. The structures ⟨23⟩\langle 23\rangle⟨23⟩, ⟨223⟩\langle 223\rangle⟨223⟩ and ⟨2223⟩\langle 2223\rangle⟨2223⟩ then open between ⟨3⟩\langle 3\rangle⟨3⟩ and the antiphase ⟨2⟩\langle 2\rangle⟨2⟩ at orders g4g^4g4, g6g^6g6 and g8g^8g8, in the order of the Fisher-Selke sequence of the classical three-dimensional model. A second layer suppresses each successive phase more strongly, beyond the stiffening of the local fields. We test the two widest phases with stochastic series expansion quantum Monte Carlo, comparing the free energies of the competing ordered states by thermodynamic integration, since annealing freezes in the modulation that forms at the ordering transition. The simulations reproduce the ⟨3⟩\langle 3\rangle⟨3⟩ wedge to about 1%, and in a pre-registered test ⟨23⟩\langle 23\rangle⟨23⟩ is stable at six of seven points, with the seventh resolved by post-hoc runs.
To study the ground state of an axial next-nearest-neighbor Ising chain under transverse field as a function of frustration parameter kappa and field strength Gamma, we present here two different perturbative analyses. In one, we consider the (known) ground state at kappa=0.5 and Gamma=0 as the unperturbed state and treat an increase of the field from 0 to Gamma coupled with an increase of kappa from 0.5 to 0.5+rGamma/J as perturbation. The first-order perturbation correction to eigenvalue can be calculated exactly and we could conclude that there are only two phase-transition lines emanating from the point kappa=0.5, Gamma=0. In the second perturbation scheme, we consider the number of domains of length 1 as the perturbation and obtain the zeroth-order eigenfunction for the perturbed ground state. From the longitudinal spin-spin correlation, we conclude that floating phase exists for small values of transverse field over the entire region intermediate between the ferromagnetic phase and antiphase.
The spin structure of an axial next-nearest-neighbor Ising (ANNNI) model in two dimensions (2D) is a renewed problem because different Monte Carlo (MC) simulation methods predicted different spin orderings. The usual equilibrium simulation predicts the occurrence of a floating incommensurate (IC) Kosterlitz-Thouless (KT) type phase, which never emerges in non-equilibrium relaxation (NER) simulations. In this paper, we first examine previously published results of both methods, and then investigate a higher transition temperature Tc1{T}_{c1}Tc1​ between the IC and paramagnetic phases. In the usual equilibrium simulation, we calculate the chain magnetization on larger lattices (up to $512\ifmmode\times\else\texttimes\fi{}512$ sites) and estimate ${T}_{c1}\ensuremath{\approx}1.16J$ with frustration ratio $\ensuremath{\kappa}(\ensuremath{\equiv}\ensuremath{-}{J}_{2}/{J}_{1})=0.6$. We examine the nature of the phase transition in terms of the Binder ratio gL{g}_{L}gL​ of spin overlap functions and the correlation-length ratio $\ensuremath{\xi}/L$. In the NER simulation, we observe the spin dynamics in equilibrium states by means of an autocorrelation function and also observe the chain magnetization relaxations from the ground and disordered states. These quantities exhibit an algebraic decay at $T\ensuremath{\lesssim}1.17J$. We conclude that the two-dimensional ANNNI model actually admits an IC phase transition of the KT type.
Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond (J1J_1J1​) and the next-nearest-neighbor one (J2J_2J2​). In the two limits of α=J2/J1α=J_2/J_1α=J2​/J1​, i.e. α≪1α\ll 1α≪1 and α≫1α\gg 1α≫1, the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size L=4×4L=4 \times4L=4×4 with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter κ=α/(1+α)κ=α/ (1+α)κ=α/(1+α). Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around αc=1.5α_c = 1.5αc​=1.5 is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around αc=1.277α_c = 1.277αc​=1.277, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.