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Physical review. B, Condensed matter

Critical properties of a one-dimensional frustrated quantum magnetic model

Parongama Sen, Bikas K. Chakrabarti

Published Jun 1, 1991• 10 citations
Publisher

Abstract

The phase diagram and ground-state energy along the phase boundary are estimated for a one-dimensional spin-1/2 Ising model with competing nearest- and next-nearest-neighbor interactions in presence of a transverse field. We have used a self-consistent Hartree-Fock approximation and a real-space renormalization-group treatment. The phase diagram (of this one-dimensional quantum model) is compared to that of a two-dimensional classical axial next-nearest-neighbor model and the Lifshitz point is located. The estimated ground-state energy, along the critical line, is seen to reduce considerably (like the critical fields), with increasing competition, showing the effect of frustration.

DOI
10.1103/physrevb.43.13559

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arXiv:cond-mat/0511658•Physical Review B•Feb 7, 2006•50 citations
The phase diagram and ground-state energy along the phase boundary are estimated for a one-dimensional spin-1/2 Ising model with competing nearest- and next-nearest-neighbor interactions in presence of a transverse field. We have used a self-consistent Hartree-Fock approximation and a real-space renormalization-group treatment. The phase diagram (of this one-dimensional quantum model) is compared to that of a two-dimensional classical axial next-nearest-neighbor model and the Lifshitz point is located. The estimated ground-state energy, along the critical line, is seen to reduce considerably (like the critical fields), with increasing competition, showing the effect of frustration.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fiuid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.
Abstract This paper focuses on the ground state phase diagram of a 1D spin‐1/2 quantum Ising model with competing first and second nearest neighbour interactions known as the axial next nearest neighbour Ising model in the presence of a transverse magnetic field. Here, using quantum correlations, both numerically and analytically, some evidence is provided to clarify the identification of the ground state phase diagram. Local quantum correlations play a crucial role in detecting the critical lines either revealed or hidden by symmetry‐breaking. A non‐symmetry‐breaking disorder transition line can be identified by the first derivative of both entanglement of formation and quantum discord between nearest neighbour spins. In addition, the quantum correlations between the second neighbour spins can also be used to reveal Kosterlitz–Thouless phase transition when their interaction strength grows and becomes closer to the first nearest neighbour one. The results obtained using the Jordan–Wigner transformation confirm the accuracy of the numerical case.
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.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
We apply density-matrix renormalization-group methods to study the phase diagram of the quantum axial next-nearest-neighbor Ising model in the region of low frustration where the ferromagnetic coupling is larger than the next-nearest-neighbor antiferromagnetic one. By finite-size scaling on lattices with up to 80 sites we locate precisely the transition line from the ferromagnetic phase to a paramagnetic phase without spatial modulation. We then measure and analyze the spin-spin correlation function in order to determine the disorder transition line where a modulation appears. We give strong numerical support to the conjecture that the Peschel-Emery one-dimensional line actually coincides with the disorder line. We also show that the critical exponent governing the vanishing of the modulation parameter at the disorder transition is ${\ensuremath{\beta}}_{q}=1∕2$.