Two-dimensional Triangular Lattice Research Articles

Realization of two-sublattice exchange physics in the triangular lattice compound Ba3Er(BO3)3

Geometric frustration commonly occurs in materials where magnetic rare-earth ions are arranged on a two-dimensional triangular lattice. These compounds have been gaining significant attention lately, as they hold the promise of revealing unique quantum states of matter. However, little attention has been devoted to cases where spin-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2}$$\\end{document} rare-earth ions are substituted with ions exhibiting higher spin multiplicities. Here, we successfully synthesize high-quality single crystal samples of Ba3Er(BO3)3, which is part of the family of triangular lattice compounds. In our experiments, conducted at extremely low temperatures (around 100 millikelvin), we observe two sublattice exchange interactions in Ba3Er(BO3)3, resulting in the hexagonal lattice spins exhibiting a mixture of ferromagnetic and antiferromagnetic tendencies. Our theoretical analysis suggest that this behavior may be attributed to the distinct positions of magnetic ions within the crystal lattice. However, the presence of quantum effects adds an extra layer of complexity to our findings, calling for further exploration.

Equivariant neural networks for spin dynamics simulations of itinerant magnets

I present a novel equivariant neural network architecture for the large-scale spin dynamics simulation of the Kondo lattice model. This neural network mainly consists of tensor-product-based convolution layers and ensures two equivariances: translations of the lattice and rotations of the spins. I implement equivariant neural networks for two Kondo lattice models on two-dimensional square and triangular lattices, and perform training and validation. In the equivariant model for the square lattice, the validation error (based on root mean squared error) is reduced to less than one-third compared to a model using invariant descriptors as inputs. Furthermore, I demonstrate the ability to simulate phase transitions of skyrmion crystals in the triangular lattice, by performing dynamics simulations using the trained model.

Competition between magnetic interactions and structural instabilities leading to itinerant frustration in the triangular lattice antiferromagnet LiCrSe2

LiCrSe2 constitutes a recent valuable addition to the ensemble of two-dimensional triangular lattice antiferromagnets. In this work, we present a comprehensive study of the low temperature nuclear and magnetic structure established in this material. Being subject to a strong magnetoelastic coupling, LiCrSe2 was found to undergo a first order structural transition from a trigonal crystal system (Pbar{3}m1) to a monoclinic one (C2/m) at Ts = 30 K. Such restructuring of the lattice is accompanied by a magnetic transition at TN = 30 K. Refinement of the magnetic structure with neutron diffraction data and complementary muon spin rotation analysis reveal the presence of a complex incommensurate magnetic structure with a up-up-down-down arrangement of the chromium moments with ferromagnetic double chains coupled antiferromagnetically. The spin axial vector is also modulated both in direction and modulus, resulting in a spin density wave-like order with periodic suppression of the chromium moment along the chains. This behavior is believed to appear as a result of strong competition between direct exchange antiferromagnetic and superexchange ferromagnetic couplings established between both nearest neighbor and next nearest neighbor Cr3+ ions. We finally conjecture that the resulting magnetic order is stabilized via subtle vacancy/charge order within the lithium layers, potentially causing a mix of two co-existing magnetic phases within the sample.

Weak trimerization in the frustrated two-dimensional triangular Heisenberg antiferromagnet LuyY1−yMnO3

To understand the 2D triangular Heisenberg antiferromagnetic system, we investigated the magnetic structures and the dynamics of ${\mathrm{Lu}}_{y}{\mathrm{Y}}_{1\ensuremath{-}y}{\mathrm{MnO}}_{3}$ in detail. The substitutions are adjusted to the Mn atomic position close to ${x}_{\text{Mn}}=\frac{1}{3}$. The neutron powder diffraction data claims that the magnetic structure of ${\mathrm{Lu}}_{y}{\mathrm{Y}}_{1\ensuremath{-}y}{\mathrm{MnO}}_{3}$ is described as a mixture of ${\mathrm{\ensuremath{\Gamma}}}_{3}$ ($P{6}_{3}^{\ensuremath{'}}c{m}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\Gamma}}}_{4}$ ($P{6}_{3}^{\ensuremath{'}}{c}^{\ensuremath{'}}m$) at the ${x}_{\text{Mn}}$ position for $y=0.15$, 0.30, and 0.45. The ratio of ${\mathrm{\ensuremath{\Gamma}}}_{3}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}$ depends on temperature and composition and the fraction of ${\mathrm{\ensuremath{\Gamma}}}_{3}$ increases upon cooling, while no clear trimerization was observed at the ${x}_{\text{Mn}}$ position. We estimated exchange parameters from the analysis of the low-energy part of the spin waves. The results showed a weak trimerization effect on cooling because the nearest-neighbor exchange interaction is slightly enhanced. The temperature dependence of the spin-wave dispersion around the $\mathrm{\ensuremath{\Gamma}}$ point shows that the spin gap closes with increasing temperature because the exchange interactions in the nearest Mn-Mn neighbor become smaller. Gapless diffusive magnetic excitation from a Mn triangular lattice has been observed in a wide range in $Q$ and $E$ space of ${\mathrm{Lu}}_{y}{\mathrm{Y}}_{1\ensuremath{-}y}{\mathrm{MnO}}_{3}$. We found that ${\mathrm{Lu}}_{0.7}{\mathrm{Y}}_{0.3}{\mathrm{MnO}}_{3}$ could be an ideal case to investigate the trimerization, frustrated magnetism, and magnetoelastic coupling often observed in two-dimensional triangular lattice Heisenberg antiferromagnet systems.

Signatures of the order parameter of a superconducting adatom layer in magnetic field dependent quasiparticle interference

Experiments have observed superconductivity in atomically-thin metallic layers deposited on semiconducting substrates. As in any superconductor, it is important to determine the structure of the superconducting pairing function in order to reveal the mechanism responsible for superconductivity. To that end, we study the possible superconducting states of two-dimensional triangular lattices. We calculate the quasiparticle interference (QPI) patterns which would result from various nearest-neighbor pairing order parameters, and show how the QPI can be used to distinguish between those order parameters. The QPI patterns are the momentum-space representations of real-space local density-of-states fluctuations: the QPI signal at momentum $q$ reveals the strength of scattering processes at that momentum transfer. We show how characteristic differences between scattering from charge disorder (i.e. impurities) and from order-parameter disorder (i.e. vortices) can be used to identify the angular momentum of the superconducting pairs.

Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence

We predict and examine various topological states on a two-dimensional (2D) triangular kagome lattice (TKL) using the tight-binding (TB) models and theory of topological quantum chemistry (TQC). Firstly, on the basis of TQC, we diagnose band structures with fragile topology and calculate Wilson-loop spectra and Hofstadter butterfly spectra to confirm their non-trivial nature. Secondly, we examine the bulk-boundary correspondence and find that an obstructed-atomic-limit (OAL) insulator hosts fractional corner states without being accompanied by fragile topological band structures, which implies that the presence of OALs and corner states is not a sufficient condition to fragile topology. Last but not least, we predict a topological phase transition from a second-order topological phase to a first-order topological phase that can be realized in the TKL under the action of a magnetic field.

Magnetic phase diagrams and large magnetocaloric effects of the two-dimensional antiferromagnetic triangular lattice of Gd3+ ions in KBaGd(BO3)2

We report a detailed study of the magnetic properties of KBaGd(BO$_3$)$_2$, in which magnetic Gd$^{3+}$ ($S=7/2$) ions form into two-dimensional triangular layers. Magnetization, specific heat and magnetocaloric effect (MCE) measurements have been performed on KBaGd(BO$_3$)$_2$ single crystals. The results show that a long-range antiferromagnetic state is established below $T_{\rm N}=0.24$ K. In zero fields, only about half of the full entropy is released at $T_{\rm N}$, indicating that not all the magnetic moments are frozen below the ordering temperature, as expected from the geometrical frustration of the triangular spin lattice. Further studies under external fields were performed down to 50 mK, and the magnetic phase diagrams are established with magnetic fields applied both within and perpendicular to the triangular plane. KBaGd(BO$_3$)$_2$ serves as an example of a two-dimensional triangular lattice with large spin values ($S=7/2$) and can be directly compared with the iso-structure KBaR(BO$_3$)$_2$ (R = Dy-Yb) family of doublet ground states, which exhibit effective spins of $S=1/2$.

Evidence for chiral superconductivity on a silicon surface

Tin adatoms on a Si(111) substrate with a one-third monolayer coverage form a two-dimensional triangular lattice with one unpaired electron per site. These electrons order into an antiferromagnetic Mott-insulating state, but doping the Sn layer with holes creates a two-dimensional conductor that becomes superconducting at low temperatures. Although the pairing symmetry of the superconducting state is currently unknown, the combination of repulsive interactions and frustration inherent in the triangular adatom lattice opens up the possibility of a chiral order parameter. Here we study the superconducting state of Sn/Si(111) using scanning tunnelling microscopy, scanning tunnelling spectroscopy and quasiparticle interference imaging. We find evidence for a doping-dependent superconducting critical temperature with a fully gapped order parameter, the presence of time-reversal symmetry breaking and a strong enhancement in zero-bias conductance near the edges of the superconducting domains. Although each individual piece of evidence could have a more mundane interpretation, our combined results suggest the possibility that Sn/Si(111) is an unconventional chiral d-wave superconductor.

Nuclear and magnetic spin structure of the antiferromagnetic triangular lattice compound LiCrTe2 investigated by mu ^+SR, neutron and X-ray diffraction

Two-dimensional (2D) triangular lattice antiferromagnets (2D-TLA) often manifest intriguing physical and technological properties, due to the strong interplay between lattice geometry and electronic properties. The recently synthesized 2-dimensional transition metal dichalcogenide LiCrTe_2, being a 2D-TLA, enriched the range of materials which can present such properties. In this work, muon spin rotation (mu ^+SR) and neutron powder diffraction (NPD) have been utilized to reveal the true magnetic nature and ground state of LiCrTe_2. From high-resolution NPD the magnetic spin order at base-temperature is not, as previously suggested, helical, but rather collinear antiferromagnetic (AFM) with ferromagnetic (FM) spin coupling within the ab-plane and AFM coupling along the c-axis. The value if the ordered magnetic Cr moment is established as mu _{textrm{Cr}}= 2.36~mu _{textrm{B}}. From detailed mu ^+SR measurements we observe an AFM ordering temperature T_{textrm{N}}approx 125 K. This value is remarkably higher than the one previously reported by magnetic bulk measurements. From mu ^+SR we are able to extract the magnetic order parameter, whose critical exponent allows us to categorize LiCrTe_2 in the 3D Heisenberg AFM universality class. Finally, by combining our magnetic studies with high-resolution synchrotron X-ray diffraction (XRD), we find a clear coupling between the nuclear and magnetic spin lattices. This suggests the possibility for a strong magnon–phonon coupling, similar to what has been previously observed in the closely related compound LiCrO_2.

Synthesis, structure and magnetism of RTa7O19 (R = Pr, Sm, Eu, Gd, Dy, Ho) with perfect triangular lattice

We present the synthesis, structure and magnetic properties of a hitherto unstudied family of rare-earth based RTa7O19 (R = Pr, Sm, Eu, Gd, Dy, Ho) compounds with space groupP6¯2c. The structural analysis reveals that magnetic R ions form layers of ideal two-dimensional triangular lattices in the ab-plane separated by several nonmagnetic polyhedra layers along the c-axis. The magnetic susceptibility and magnetization measurements indicate the absence of long-range magnetic order and field-induced magnetic transitions in all the studied compounds. The details about the magnetism strongly depend on the type of R ions. The presence of geometrical frustration is discussed. Thus, RTa7O19 provides a new platform for exploring the frustrated magnetism.

Spin direction tunable topological transition in two-dimensional frustrate antiferromagnetic triangular lattice T-FeO2 monolayer

Recently, numerous two-dimensional (2D) magnetic materials are predicted with various promising properties, whereas noncollinear antiferromagnetic 2D materials are rarely reported. In this paper, we predicted a stable 2D noncollinear antiferromagnetic triangular lattice T-FeO2. The ground state is 120° antiferromagnetic with stronger next nearest neighbor exchange coupling than that of nearest neighbor exchange coupling because of the RKKY interaction. Our Monte Carlo simulations reveal that the magnetic phase transition is from a 120° antiferromagnetic state to a stripy state and then to a paramagnetic state with increasing temperature. In addition, by tuning the spin direction from an in plane antiferromagnetic state to a canted weak ferromagnetic state, a nontrivial topological phase transition could be induced. Our investigation about magnetic property and nontrivial topological phase transition is very promising for antiferromagnetic spintronics.

Static and dynamic magnetic properties of the spin- triangle lattice antiferromagnet Na3Fe(PO4)2 studied by 31P NMR

31P nuclear magnetic resonance (NMR) measurements have been carried out to investigate the magnetic properties and spin dynamics of Fe3+ (S = 5/2) spins in the two-dimensional triangular lattice (TL) compound Na3Fe(PO4)2. The temperature (T) dependence of nuclear spin-lattice relaxation rates () shows a clear peak around Néel temperature, K, corresponding to an antiferromagnetic (AFM) transition. From the temperature dependence of NMR shift (K) above , an exchange coupling between Fe3+ spins was estimated to be K using the spin-5/2 Heisenberg isotropic-TL model. The temperature dependence of divided by the magnetic susceptibility (χ), , above proves the AFM nature of spin fluctuations below in the paramagnetic state. In the magnetically ordered state below , the characteristic rectangular shape of the NMR spectra is observed, indicative of a commensurate AFM state in its ground state. The strong temperature dependence of 1/T 1 in the AFM state is well explained by the two-magnon (Raman) process of the spin waves in a 3D antiferromagnet with a spin-anisotropy energy gap of 5.7 K. The temperature dependence of sublattice magnetization is also well reproduced by the spin waves. Those results indicate that the magnetically ordered state of Na3Fe(PO4)2 is a conventional 3D AFM state, and no obvious spin frustration effects were detected in its ground state.

Gapless triangular-lattice spin-liquid candidate PrZnAl11O19

A quantum spin liquid (QSL) is an exotic state in which electron spins are highly entangled, yet keep fluctuating even at zero temperature. Experimental realization of model QSLs has been challenging due to imperfections, such as antisite disorder, strain, and extra or a lack of interactions in real materials compared to the model Hamiltonian. Here we report the magnetic susceptibility, thermodynamic, inelastic neutron scattering (INS), and muon-spin relaxation studies on a polycrystalline sample of PrZnAl$_{11}$O$_{19}$, where the Pr$^{3+}$ ions form an ideal two-dimensional triangular lattice. Our results demonstrate that this system does not order nor freeze, but keep fluctuating down to 50 mK despite large antiferromagnetic couplings ($\sim$ -10 K). Furthermore, the INS and specific-heat data suggest that PrZnAl$_{11}$O$_{19}$ is best described as a gapless QSL.

Zero-field skyrmion, meron, and vortex crystals in centrosymmetric hexagonal magnets

We investigate the interaction conditions to stabilize topological spin textures in centrosymmetric magnets without an external magnetic field. By performing the simulated annealing for an effective spin model with the momentum-resolved interaction on a two-dimensional triangular lattice, we find that the competing interactions between the finite-q channel and the uniform (q = 0) channel lead to a zero-field skyrmion crystal with the skyrmion number of ±1. Moreover, we obtain a variety of topological spin textures, such as a meron crystal, vortex crystal, and other multiple-Q states with and without a scalar chirality. We especially show that the bond-dependent anisotropic exchange interaction originating from the relativistic spin–orbit coupling plays an important role in inducing such topological spin textures, while the single-ion anisotropy does not.

Field-induced magnetic states in the metallic rare-earth layered triangular antiferromagnet TbAuAl4Ge2

Magnetic frustration in metallic rare earth lanthanides ($Ln$) with $4f$-electrons is crucial for producing interesting magnetic phases with high magnetic anisotropy where intertwined charge and spin degrees of freedom lead to novel phenomena. Here we report on the magnetic, thermodynamic, and electrical transport properties of TbAuAl$_4$Ge$_2$. Tb ions form 2-dimensional triangular lattice layers which stack along the crystalline $c$-axis. The magnetic phase diagram reveals multiple nearly degenerate ordered states upon applying field along the magnetically easy $ab$-plane before saturation. The magnetoresistance in this configuration exhibits intricate field dependence that closely follows that of the magnetization while the specific heat reveals a region of highly enhanced entropy, suggesting the possibility of a non-trivial spin textured phase. For fields applied along the $c$-axis (hard axis), we find linear magnetoresistance over a wide range of fields. We compare the magnetic properties and magnetoresistance with an isostructral GdAuAl$_4$Ge$_2$ single crystals. These results identify TbAuAl$_4$Ge$_2$ as an environment for complex quantum spin states and pave the way for further investigations of the broader $Ln$AuAl$_4$Ge$_2$ family of materials.

Dielectric catastrophe at the Wigner-Mott transition in a moiré superlattice

The bandwidth-tuned Wigner-Mott transition is an interaction-driven phase transition from a generalized Wigner crystal to a Fermi liquid. Because the transition is generally accompanied by both magnetic and charge-order instabilities, it remains unclear if a continuous Wigner-Mott transition exists. Here, we demonstrate bandwidth-tuned metal-insulator transitions at fixed fractional fillings of a MoSe2/WS2 moiré superlattice. The bandwidth is controlled by an out-of-plane electric field. The dielectric response is probed optically with the 2s exciton in a remote WSe2 sensor layer. The exciton spectral weight is negligible for the metallic state with a large negative dielectric constant. It continuously vanishes when the transition is approached from the insulating side, corresponding to a diverging dielectric constant or a ‘dielectric catastrophe’ driven by the critical charge dynamics near the transition. Our results support the scenario of continuous Wigner-Mott transitions in two-dimensional triangular lattices and stimulate future explorations of exotic quantum phases in their vicinities.

Exploiting weak magnetic anisotropy on the stability of spin-spirals and skyrmions lattice in VX2 (X=S, Se) monolayers

The effects of competing magnetic interactions in stabilizing different spin configurations attracts renewed attention to simultaneously reveal emerging topological spin textures and highlight microscopic mechanisms leading to their stabilization. By performing Density Functional Theory (DFT) and Monte-Carlo (MC) simulations, we studied the magnetic properties of VS2 and VSe2 monolayer. We found a profusion of different spin configurations going from spin-spirals to skyrmions lattice. In this work, we demonstrate that magnetic anisotropic interaction deduced from DFT calculation, plays a crucial role in the formation of spin-spirals. We have extracted energy barriers between 0.039 meV and 0.067 meV responsible for the transition between the spin-spiral and skyrmion phases. This transition phase occurs at the same value of the external magnetic field (B = 0.051 T) for the different values of the ratio exchange (J)/Dzyaloshinskii-Moriya interaction (DMI) noted J/DMI. Finally, we have established the phase diagram of a frustrated localized magnet on a two-dimensional centrosymmetric triangular lattice, focusing on the interplay between the DMI and B applied perpendicularly to the magnetic layer.

Small-world disordered lattices: spectral gaps and diffusive transport

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts–Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation.

Optical Conductivity Spectra of Charge-Crystal and Charge-Glass States in a Series of θ-Type BEDT-TTF Compounds

In the 3/4-filled band system θ-(BEDT-TTF)2X with a two-dimensional triangular lattice, charge ordering (CO) often occurs due to strong inter-site Coulomb repulsion. However, the strong geometrical frustration of the triangular lattice can prohibit long-range CO, resulting in a charge-glass state in which the charge configurations are randomly distributed. Here, we investigate the charge-glass states of orthorhombic and monoclinic θ-type BEDT-TTF salts by measuring the electrical resistivity and optical conductivity spectra. We find a substantial difference between the charge-glass states of the orthorhombic and monoclinic systems. The charge-glass state in the orthorhombic system with an isotropic triangular lattice exhibits larger low-energy excitations than that in the monoclinic one with an anisotropic triangular lattice and becomes more metallic as the isotropy of the triangular lattice increases. These results can be understood by the different charge-glass formation mechanisms in the two systems: in the orthorhombic system, the charge-glass state originates from geometric frustration due to the equilateral triangular lattice, leading to metallic 3-fold COs, whereas in the monoclinic system, the charge-glass formation originates from geometric frustration of the isosceles triangular lattice, in which the charge-glass state is described by the superposition of insulating 2-fold stripe COs.

Magnon valley thermal Hall effect in triangular-lattice antiferromagnets

We study the magnetic excitations of the antiferromagnetic XXZ model on two-dimensional triangular lattice. We find there is a magnon valley thermal Hall effect. As far as we know, this effect has been studied on only the ferromagnetic honeycomb lattice. We reveal that the valley thermal Hall conductivity also conforms to the universal behavior of thermal Hall conductivity proposed by Yang et al. [Phys. Rev. Lett. 124, 186602 (2020)]. We derive a low-energy effective theory near the valleys and find that although the low-energy effective dynamic matrix is non-Hermitian, the Berry curvatures are mainly determined by its Hermitian part. We show that our results are general to some extent in the Y phase of the triangular lattice. By an explicit calculation with experimental parameters, we demonstrate that ${\mathrm{Rb}}_{4}\mathrm{Mn}{({\mathrm{MoO}}_{4})}_{3}$ is a suitable material to realize the magnon valley thermal Hall effect.