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arXiv:1406.6464 · Physical Review B

Kosterlitz-Thouless phase transition of the axial next-nearest-neighbor Ising model in two dimensions

Takayuki Shirakura, Fumitaka Matsubara, Naoki Suzuki

Published Oct 8, 2014• 8 citations
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Abstract

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 ${T}_{c1}$ 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 ${g}_{L}$ 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.

DOI
10.1103/physrevb.90.144410

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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×\else\texttimes\fi512512\ifmmode\times\else\texttimes\fi{}512512\ifmmode×\else\texttimes\fi512 sites) and estimate Tc1\ensuremath≈1.16J{T}_{c1}\ensuremath{\approx}1.16JTc1​\ensuremath≈1.16J with frustration ratio \ensuremathκ(\ensuremath≡\ensuremath−J2/J1)=0.6\ensuremath{\kappa}(\ensuremath{\equiv}\ensuremath{-}{J}_{2}/{J}_{1})=0.6\ensuremathκ(\ensuremath≡\ensuremath−J2​/J1​)=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 \ensuremathξ/L\ensuremath{\xi}/L\ensuremathξ/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≲1.17JT\ensuremath{\lesssim}1.17JT\ensuremath≲1.17J. We conclude that the two-dimensional ANNNI model actually admits an IC phase transition of the KT type.
We examine the ground state of a dilute two-dimensional axial-next-nearest-neighbor Ising model with the interaction ratio κ (≡ −J2/J1) > 1/2. We calculate the spin overlap function Q together with the antiphase (〈2〉 phase) magnetization M〈2〉 on medium lattices of L × L (L ≤ 24) using a cluster heat bath method. The 〈2〉 phase is destroyed by a small quantity of non-magnetic atoms due to a random impurity effect, and a critical phase is realized, which is composed of segments of four types of 〈2〉 domains arranged in order. As the spin concentration c decreases, a critical phase with different ferromagnetic clusters appears near above the percolation threshold cp of the lattice. A phase diagram is proposed in the (κ,c)(\kappa ,c)(κ,c) plain at low temperatures.
Some features of the phase diagram of the two-dimensional axial next-nearest-neighbor Ising model have long been debated. The extended structural correlations and long relaxation times associated with its Kosterlitz-Thouless phase indeed result in analytical and numerical treatments making contradictory predictions. Here, we introduce a numerical transfer matrix approach that bypasses these problems and thus clears up various ambiguities. In particular, we confirm the transition temperatures and the order of the transition to the floating incommensurate phase. Our approach motivates considering transfer matrices for solving long-standing problems in related models.
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.
We investigate the dynamics of a two-dimensional biaxial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=-J_{0}∑_{i,j=1}^{L}[(S_{i,j}S_{i+1,j}+S_{i,j}S_{i,j+1})-κ(S_{i,j}S_{i+2,j}+S_{i,j}S_{i,j+2})]. For κ<1, the system does not reach the equilibrium ground state and keeps evolving in active states forever. For κ≥1, though, the system reaches a final state, but it does not reach the ground state always and freezes to a striped state with a finite probability like a two-dimensional ferromagnetic Ising model and an axial next-nearest-neighbor Ising (ANNNI) model. The overall dynamical behavior for κ>1 and κ=1 is quite different. The residual energy decays in a power law for both κ>1 and κ=1 from which the dynamical exponent z has been estimated. The persistence probability shows algebraic decay for κ>1 with an exponent θ=0.22±0.002 while the dynamical exponent for ordering z=2.33±0.01. For κ=1, the system belongs to a completely different dynamical class with θ=0.332±0.002 and z=2.47±0.04. We have computed the freezing probability for different values of κ. We have also studied the decay of autocorrelation function with time for a different regime of κ values. The results have been compared with those of the two-dimensional ANNNI model.
We use the finite-entanglement scaling of infinite matrix product states (iMPS) to explore supposedly infinite order transitions. This universal method may have lower computational costs than finite-size scaling. To this end, we study possible MPS-based algorithms to find the ground states of the transverse axial next-nearest-neighbor Ising (ANNNI) model in a spin chain with first and second neighbor interactions and frustration. The ground state has four distinct phases with transitions of second order and one of supposedly infinite order, the Kosterlitz–Thouless transition. To explore phase transitions in the model, we study general quantities such as the correlation length, entanglement entropy and the second derivative of the energy with respect to the external field, and test the finite-entanglement scaling. We propose a scaling ansatz for the correlation length of a non-critical system in order to explore infinite order transitions. This method provides considerably less computational costs compared to the finite-size scaling method in [8], and quantities obtained by applying fixed boundary conditions (such as domain wall energy in [8]) are omitted. The results show good agreement with previous studies of finite-size scaling using DMRG.
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.