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arXiv:2206.11788 · Physical review. B./Physical review. B

Partial lifting of degeneracy in the J1−J2−J3 Ising antiferromagnet on the kagome lattice

Jeanne Colbois, Bram Vanhecke, Laurens Vanderstraeten, Andrew Smerald, Frank Verstraete, Frédéric Mila

Published Nov 2, 2022• 19 citations
arXivHTMLPDF

Abstract

Motivated by dipolar-coupled artificial spin systems, we present a theoretical study of the classical ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Ising antiferromagnet on the kagome lattice. We establish the ground-state phase diagram of this model for ${J}_{1}>|{J}_{2}|,|{J}_{3}|$ based on exact results for the ground-state energies. When all the couplings are antiferromagnetic, the model has three macroscopically degenerate ground-state phases, and using tensor networks, we can calculate the entropies of these phases and of their boundaries very accurately. In two cases, the entropy appears to be a fraction of that of the triangular lattice Ising antiferromagnet, and we provide analytical arguments to support this observation. We also notice that, surprisingly enough, the dipolar ground state is not a ground state of the truncated model, but of the model with smaller ${J}_{3}$ interactions, an indication of a very strong competition between low-energy states in this model.

DOI
10.1103/physrevb.106.174403

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Motivated by dipolar-coupled artificial spin systems, we present a theoretical study of the classical J1\ensuremath-J2\ensuremath-J3{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}J1​\ensuremath-J2​\ensuremath-J3​ Ising antiferromagnet on the kagome lattice. We establish the ground-state phase diagram of this model for J1>∣J2∣,∣J3∣{J}_{1}>|{J}_{2}|,|{J}_{3}|J1​>∣J2​∣,∣J3​∣ based on exact results for the ground-state energies. When all the couplings are antiferromagnetic, the model has three macroscopically degenerate ground-state phases, and using tensor networks, we can calculate the entropies of these phases and of their boundaries very accurately. In two cases, the entropy appears to be a fraction of that of the triangular lattice Ising antiferromagnet, and we provide analytical arguments to support this observation. We also notice that, surprisingly enough, the dipolar ground state is not a ground state of the truncated model, but of the model with smaller J3{J}_{3}J3​ interactions, an indication of a very strong competition between low-energy states in this model.
Despite their simple formulation, short-range classical antiferromagnetic Ising models on frustrated lattices give rise to exotic phases of matter, in particular, due to their macroscopic ground-state degeneracy. Recent experiments on artificial spin systems comprising arrays of chirally coupled nanomagnets provide a significant strengthening of the nearest-neighbor couplings compared to systems with dipolar-coupled nanomagnets. This opens the way to design artificial spin systems emulating Ising models with nearest-neighbor couplings. In this paper, we compare the results of an extensive investigation with tensor network and Monte Carlo simulations of the nearest- and farther-neighbor $({J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3||})$ kagome Ising antiferromagnet with the experimental spin-spin correlations of a kagome lattice of chirally coupled nanomagnets. Even though the ratios between the farther-neighbor couplings and the nearest-neighbor coupling estimated from micromagnetic simulations are much smaller than for dipolar-coupled nanomagnets, we show that they still play an essential role in the selection of the correlations.
We investigate a nearest-neighbor Ising model on the kagome lattice coupled to elastic lattice distortions. By integrating out the lattice degrees of freedom, we derive an approximate spin-only description in terms of a kagome J1J_1J1​-J2J_2J2​-J3J_{3}J3​ Ising model with interactions extending to third nearest-neighbors. Using tensor network methods complemented by classical Monte Carlo simulations, we revisit its zero-temperature phase diagram and characterize the ground states studied more than thirty years ago. We discover three phases with extensive zero-temperature entropy and algebraically decaying spin correlations. Importantly, these states are not confined to fine-tuned points or frontiers between phases, but occupy extended areas of parameter space. Their quasi long-range order is reflected in Bragg-like peaks in the structure factors; we relate this, and other properties, to diverse phases found in Ising and Heisenberg models. Finally, we trace the trajectories generated by the magnetoelastic coupling through the effective-model phase diagram, explaining the emergence of self-frustration for the unfrustrated magnetoelastic model with ferromagnetic interactions.
We analyse the antiferromagnetic spin-1/2 Heisenberg model on a depleted kagome lattice, where some bonds have been reduced to exchange integral J_2 << J_1. The fully depleted system consists of 1D chains, each with a doubly degenerate singlet-pair ground state and a spectral gap (like the Majumdar-Ghosh model). There are localised and itinerant low-energy excitations. The modes from the lowest branch of excitations are incapable of lifting the 2D system's ground state degeneracy at finite J_2/J_1 << 1. Low-energy excitations of the 2D system are dominated by coherently propagating bound states of the 1D excitations.
Phase transitions of the antiferromagnetic J1J_1J1​-J2J_2J2​-J3J_3J3​ Ising model on the cubic lattice
Motivated by the conflicting findings in prior studies regarding the nature of the phase transition for the three-dimensional antiferromagnetic J1J_1J1​-J2J_2J2​-J3J_3J3​ Ising model on the cubic lattice with J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0, in this study we conduct a detailed investigation of this model using both Monte Carlo simulations and neural networks (NN). Here J1J_1J1​ and J2J_2J2​ are the intralayer nearest-neighbor and next-nearest-neighbor couplings, respectively, and J3J_3J3​ is the nearest-neighbor interlayer interaction. By investigating the data obtained from much larger lattices than those used in previous studies, we find that the phase transition for J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0 is second order without any ambiguity. Hence we bring a resolution to this debate. Apart from J1=J2=J3=1.0J_1 = J_2 = J_3 = 1.0J1​=J2​=J3​=1.0, we study the critical phenomenon associated with J2/J1=4/7J_2/J_1 = 4/7J2​/J1​=4/7 and J3=J1=1.0J_3 = J_1 = 1.0J3​=J1​=1.0 as well. In addition, by employing a dedicated training and testing strategy, our NN computation not only is much more efficient than those utilizing the standard procedures commonly used in the literature, but also requires only a per-mille fraction of the storage capacity needed by a conventional NN calculation when a large linear system size LLL is considered. The method employed here provides an extremely resource-efficient route for applying NN techniques to study the critical behaviors of higher-dimensional systems.
Using exact numerical diagonalizations as well as the semiclassical spin-wave approach, we study the quantum phase diagram of the spin-12 Heisenberg kagome strip, which is a particular cut-out from the kagome lattice. The unit cell contains five spins placed on the central cite and on the four end sites building two spin-12 along the strip. The model can be characterized by three different constants for the nearest-neighbor exchange bonds between (i) the central and chain spins (J1), (ii) the end spins in a cluster (J2 > 0), and (ii) the spins of neighboring clusters (J′2>0) In the present work we extend the previous studies of the model, which were restricted to antiferromagnetic J1 bonds (J1 > 0), by including the region with ferromagnetic J1 couplings (J1<0). It is demonstrated that for dominating ferromagnetic J1 couplings there is another gapless phase, which can be related to the critical ground state of the spin-52 Heisenberg chain. The phase boundaries separating the discussed critical phases, as well as the behavior of their low-lying excitations close to these boundaries, are examined as well.
We study the ground-state phase diagram of the quantum spin-1/2\text{spin-}1/2spin-1/2 Heisenberg model on the kagome lattice with first- (J1<0)({J}_{1}<0)(J1​<0), second- (J2<0)({J}_{2}<0)(J2​<0), and third-neighbor interactions (Jd>0)({J}_{d}>0)(Jd​>0) by means of analytical low-energy field theory and numerical density-matrix renormalization group (DMRG) studies. The results offer a consistent picture of the Jd-dominant{J}_{d}\text{-dominant}Jd​-dominant regime in terms of three sets of spin chains weakly coupled by the ferromagnetic interchain interactions J1,2{J}_{1,2}J1,2​. When either J1{J}_{1}J1​ or J2{J}_{2}J2​ is much stronger than the other one, the model is found to support one of two cuboctohedral phases, cuboc1, and cuboc2. These cuboc states host noncoplanar long-ranged magnetic order and possess finite scalar spin chirality. However, in the compensated regime ${J}_{1}\ensuremath{\simeq}{J}_{2}$, a valence bond crystal phase emerges between the two cuboc phases. We find excellent agreement between an analytical theory based on coupled spin chains and unbiased DMRG calculations, including at a very detailed level of comparison of the structure of the valence bond crystal state. To our knowledge, this is the first such comprehensive understanding of a highly frustrated two-dimensional quantum antiferromagnet. We find no evidence of either the one-dimensional gapless spin liquid or the chiral spin liquids, which were previously suggested by parton mean-field theories.