Triangular Lattice Antiferromagnet Research Articles

Magnetic phases of the frustrated ferromagnetic spin-trimer system Gd3Ru4Al12 with a distorted kagome lattice structure

The magnetization and specific heat measurements have been performed on single-crystalline Gd_3_Ru_4_Al_12_ with a distorted Kagome lattice structure. This spin system is regarded as an antiferromagnetic triangular lattice of XY like Heisenberg model at low temperatures. The magnetic phase diagrams indicate the existence of frustration and Z_2_ degeneracy. The magnetization and specific heat imply the successive phase transitions with partial disorder and a T-shaped spin structure in the ground state.

Stacking-induced magnetic frustration and spiral spin liquid

Like the twisting control in magic-angle twisted bilayer graphene, the stacking control is another mechanical approach to manipulate the fundamental properties of solids, especially the van der Waals materials. We explore the stacking-induced magnetic frustration and the spiral spin liquid on a multilayer triangular lattice antiferromagnet where the system is built from ABC stacking with competing intralayer and interlayers couplings. By combining the nematic bond theory and the self-consistent Gaussian approximation, we establish the phase diagram for this ABC-stacked multilayer magnet. It is shown that the system supports a wide regime of spiral spin liquid with multiple degenerate spiral lines in the reciprocal space, separating the low-temperature spiral order and the high-temperature featureless paramagnet. The transition to the spiral order from the spiral spin liquid regime is first order. We further show that the spiral-spin-liquid behavior persists even with small perturbations such as further neighbor intralayer exchanges. The connection to the ABC-stacked magnets, the effects of Ising or planar spin anisotropy, and the outlook on the stacking-engineered quantum magnets are discussed.

Holes and magnetic polarons in a triangular lattice antiferromagnet

The intricate interplay between charge motion and magnetic order in geometrically frustrated lattices is central for the properties of many two-dimensional quantum materials. The triangular lattice antiferromagnet is a canonical example of a frustrated system, and here we analyze the dynamics of a hole in such a lattice focusing on observables that have become accessible in a new generation of experiments. Using the $t\ensuremath{-}J$ model, we solve the problem exactly within linear spin wave theory in the limit of strong magnetic interactions, showing that the ground state is described by a coherent state of spin waves. The derivation highlights the crucial role played by the interaction between a static hole and the neighboring spins, which originates in the geometric frustration and has often been omitted in earlier works. Furthermore, we show that the nonequilibrium dynamics after a hole has abruptly been inserted at a lattice site is given by a coherent state with time-dependent oscillatory coefficients. Physically, this describes a burst of magnetic frustration propagating through only two-thirds of the lattice sites, since a destructive interference of spin waves leaves spins parallel to that removed by the hole unperturbed. After the wave has propagated through the lattice, the magnetization relaxes to that of the ground state. We then use our analytical solution to benchmark the widely used self-consistent Born approximation (SCBA) in the limit of strong magnetic interactions, showing that it is very accurate also for a triangular lattice. The magnetic polaron spectrum is analyzed for general magnetic interactions using the SCBA, and we compare our results with those for a square lattice.

Two-dimensional quantum universality in the spin-1/2 triangular-lattice quantum antiferromagnet Na2BaCo(PO4)2.

An interplay of geometrical frustration and strong quantum fluctuations in a spin-1/2 triangular-lattice antiferromagnet (TAF) can lead to exotic quantum states. Here, we report the neutron-scattering, magnetization, specific heat, and magnetocaloric studies of the recently discovered spin-1/2 TAF Na2BaCo(PO4)2, which can be described by a spin-1/2 easy axis XXZ model. The zero-field neutron diffraction experiment reveals an incommensurate antiferromagnetic ground state with a significantly reduced ordered moment of about 0.54(2) μB/Co. Different magnetic phase diagrams with magnetic fields in the ab plane and along the easy c-axis were extracted based on the magnetic susceptibility, specific heat, and elastic neutron-scattering results. In addition, two-dimensional (2D) spin dispersion in the triangular plane was observed in the high-field polarized state, and microscopic exchange parameters of the spin Hamiltonian have been determined through the linear spin wave theory. Consistently, quantum critical behaviors with the universality class of d = 2 and νz = 1 were established in the vicinity of the saturation field, where a Bose-Einstein condensation (BEC) of diluted magnons occurs. The newly discovered quantum criticality and fractional magnetization phase in this ideal spin-1/2 TAF present exciting opportunities for exploring exotic quantum phenomena.

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.

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.

Entanglement Negativity and Concurrence in Some Low-Dimensional Spin Systems.

The influence of magnon bands on entanglement in the antiferromagnetic XXZ model on a triangular lattice, which models the bilayer structure consisting of an antiferromagnetic insulator and normal metal, is investigated. This effect was studied in ferromagnetic as well as antiferromagnetic triangular lattices. Quantum entanglement measures given by the entanglement negativity have been studied, where a magnon current is induced in the antiferromagnet due to interfacial exchange coupling between localized spins in the antiferromagnet and itinerant electrons in a normal metal. Moreover, quantum correlations in other frustrated models, namely the metal-insulation antiferromagnetic bilayer model and the Heisenberg model with biquadratic and bicubic interactions, are analyzed.

Magnetic phase diagram of a 2-dimensional triangular lattice antiferromagnet Na2BaMn(PO4)2

We report the magnetic phase transitions of a spin-5/2, 2-dimensional triangular lattice antiferromagnet (AFM) Na2BaMn(PO4)2. From specific heat measurements, we observe two magnetic transitions at temperatures 1.15 and 1.30 K at zero magnetic field. Detailed AC magnetic susceptibility measurements reveal multiple phases including the (up-up-down)-phase between 1.9 and 2.9 T at 47 mK when magnetic field is applied along the c axis, implying that Na2BaMn(PO4)2 is a classical 2d TL Heisenberg AFM with easy-axis anisotropy. However, it deviates from an ideal model as evidenced by a hump region with hysteresis between the and V-phases and weak phase transitions. Our work provides another experimental example to study frustrated magnetism in 2d TL AFM which also serves as a reference to understand the possible quantum spin liquid behavior and anomalous phase diagrams observed in sibling systems Na2BaM(PO4)2 (M = Co, Ni).

Field-orientation dependence of quantum phase transitions in the S=12 triangular-lattice antiferromagnet Ba3CoSb2O9

Ba$_3$CoSb$_2$O$_9$ approximates the two-dimensional spin-1/2 triangular-lattice Heisenberg antiferromagnet. This compound displays magnetic-field-induced quantum phase transitions, including the 1/3-magnetization-plateau, but its magnetization processes for the magnetic field $H$ parallel and perpendicular to the $c$ axis are different due to the weak easy-plane anisotropy and the weak interlayer antiferromagnetic exchange interaction. To elucidate how the quantum phase transitions change between these two field directions, we measured the field-angle dependence of the magnetization process in Ba$_3$CoSb$_2$O$_9$ using pulsed high magnetic fields. We compared obtained magnetic field-field angle phase diagram with those obtained by the large-size cluster mean-field method combined with a scaling scheme and the semiclassical theory. We also found a narrow 1/3-magnetization plateau and a high-field transition with a small magnetization jump for $H\,{\parallel}\,c$, not observed in the previous studies.

Spin supersolidity in nearly ideal easy-axis triangular quantum antiferromagnet Na2BaCo(PO4)2

Prototypical models and their material incarnations are cornerstones to the understanding of quantum magnetism. Here we show theoretically that the recently synthesized magnetic compound Na2BaCo(PO4)2 (NBCP) is a rare, nearly ideal material realization of the S = 1/2 triangular-lattice antiferromagnet with significant easy-axis spin exchange anisotropy. By combining the automatic parameter searching and tensor-network simulations, we establish a microscopic model description of this material with realistic model parameters, which can not only fit well the experimental thermodynamic data but also reproduce the measured magnetization curves without further adjustment of parameters. According to the established model, the NBCP hosts a spin supersolid state that breaks both the lattice translation symmetry and the spin rotational symmetry. Such a state is a spin analog of the long-sought supersolid state, thought to exist in solid Helium and optical lattice systems, and share similar traits. The NBCP therefore represents an ideal material-based platform to explore the physics of supersolidity as well as its quantum and thermal melting.

Frustrated antiferromagnetic triangular lattice with Dzyaloshinskii–Moriya interaction: Ground states, spin waves, skyrmion crystal, phase transition

We study in this article a triangular lattice with Heisenberg spins interacting with each other via an antiferromagnetic exchange interaction J and a Dzyaloshinskii–Moriya (DM) interaction D, between nearest-neighbors (NN). We consider two cases: the first case in which the DM vector D is perpendicular to the lattice plane and the second case where it lies in the plane. A magnetic field H is applied perpendicular to the spin plane in both cases. The ground state (GS) of this system is calculated by minimizing the energy using the very fast steepest-descent method in the two cases. In the case of perpendicular D with H=0, the GS is periodic. We analytically determine the GS configuration which is characterized by two well-defined angles. We calculate the spin-wave spectrum in this case which shows that for small wave vectors the spin waves are forbidden in the system. When H≠0, the GS in the perpendicular D case shows no skyrmions. However, in the case of in-plane D with H≠0, we find a crystal of skyrmions at T=0 composed of three interpenetrating skyrmion sublattice crystals, in agreement with low-T spin textures found in earlier works. We show by Monte Carlo simulations that this skyrmion crystal is stable at finite temperatures below a critical temperature.

Varying critical behaviour of stacked triangular lattice Ising antiferromagnets in the presence of spin–lattice coupling

We perform Monte Carlo simulations on the multilayered triangular-lattice Ising antiferromagnet with six sublattices in the frustrated abab stacking and quantify the impact of spin–lattice coupling on its critical properties in the vicinity of the antiferromagnetic phase transition. By invoking the Einstein site spin–phonon model and by varying the strengths of both antiferromagnetic interlayer exchange interactions and spin–lattice coupling, we compute various thermodynamic observable that characterize how the geometrical frustration and the spin–phonon coupling affect the critical behaviour of the system. Using finite-size scaling, we determine the nature of phase transition and the critical exponents, the latter being found to vary with varying strength of couplings. We find that the transition temperature is more sensitive to the interlayer coupling strength rather than that of magnetoelastic coupling. The variation of critical exponents with the strength of spin–lattice coupling hints to the importance of magnetoelastic interaction in the vicinity of magnetic phase transition in strongly correlated magnetic systems, as already observed in several experiments.

Successive Phase Transitions and Multiferroicity in Deformed Triangular-Lattice Antiferromagnets Ca3MNb2O9 (M=Co, Ni) with Spatial Anisotropy

We constructed the magnetic field-temperature phase diagrams of new quasi-two-dimensional isosceles triangular lattice antiferromagnets (TLAF) Ca3MNb2O9 (M=Co, Ni) from dc and ac magnetic susceptibilities, specific heat, dielectric constant, and electric polarization measurements on single crystalline samples. Ca3CoNb2O9 with effective spin-1/2 Co2+ ions undergoes a two-step antiferromagnetic phase transition at T N1 = 1.3 K and T N2 = 1.5 K and enters a stripe ordered state at zero magnetic field. With increasing field, successive magnetic phase transitions, reminiscent of the up-up-down (uud) and the oblique phases, are observed. The dielectric constant of Ca3CoNb2O9 shows anomalies related to the magnetic phase transitions, but clear evidence of ferroelectricity is absent. Meanwhile, Ca3NiNb2O9 with spin-1 Ni2+ ions also shows a two-step antiferromagnetic transition at T N1 = 3.8 K and T N2 = 4.2 K at zero field. For Ca3NiNb2O9, the electric polarization in the magnetic ordered phases was clearly observed from the pyroelectric current measurements, which indicates its coexistence of magnetic ordering and ferroelectricity.

Imaginary-time hierarchical equations of motion for thermodynamic variables.

The partition function (PF) plays a key role in the calculation of quantum thermodynamic properties of a system that interacts with a heat bath. The imaginary-time hierarchical equations of motion (imHEOM) approach was developed to evaluate in a rigorous manner the PF of a system strongly coupled to a non-Markovian bath. In this paper, we present a numerically efficient scheme to evaluate the imHEOM utilizing the β-differentiated imHEOM (BD-imHEOM) that are obtained by differentiating the elements of the imHEOM with respect to the inverse temperature. This approach allows us to evaluate the system, system-bath interaction, and heat-bath parts of the PF efficiently. Moreover, we employ a polyharmonic decomposition method to construct a concise hierarchical structure with better convergence, thus reducing the cost of numerical integrations. We demonstrate the proposed approach by compute thermodynamic quantities of a spin-boson system and a 2 × 2 antiferromagnetic triangular spin lattice system with an Ohmic spectral distribution.

Field-tunable toroidal moment and anomalous Hall effect in noncollinear antiferromagnetic Weyl semimetal Co1/3TaS2

Combining magnetism with band topology provides various novel phases that are otherwise impossible. Among several cases, noncollinear metallic antiferromagnets can reveal particularly rich topological physics due to their diverse magnetic ground states. However, there are only a few experimental studies due to the lack of suitable materials, especially with triangular lattice antiferromagnets. Here, we report that metallic triangular antiferromagnet Co1/3TaS2 exhibits a substantial anomalous Hall effect (AHE) related to its noncollinear magnetic order. Our first-principles calculations found that hourglass Weyl fermions from the non-symmorphic symmetry trigger AHE. We further show that AHE in Co1/3TaS2 can be characterized by the toroidal moment, a vortex-like multipole component that arises from a combination of chiral lattice and geometrical frustration. Finally, the unusual field-tunability of the toroidal moment puts Co1/3TaS2 as a rare example of a noncollinear metallic antiferromagnet filled with interesting magnetic and topological properties.

Collinear order in the spin- 52 triangular-lattice antiferromagnet Na3Fe(PO4)2

We set forth the structural and magnetic properties of the frustrated spin-$5/2$ triangle lattice antiferromagnet ${\mathrm{Na}}_{3}\mathrm{Fe}{({\mathrm{PO}}_{4})}_{2}$ examined via x-ray diffraction, magnetization, heat capacity, and neutron diffraction measurements on the polycrystalline sample. No structural distortion was detected from the temperature-dependent x-ray diffraction down to 12.5 K, except a systematic lattice contraction. The magnetic susceptibility at high temperatures agrees well with the high-temperature series expansion for a spin-$5/2$ isotropic triangular lattice antiferromagnet with an average exchange coupling of $J/{k}_{\mathrm{B}}\ensuremath{\simeq}1.8$ K rather than a one-dimensional spin-$5/2$ chain model. This value of the exchange coupling is consistently reproduced by the saturation field of the pulse field magnetization data. It undergoes a magnetic long-range order at ${T}_{\mathrm{N}}\ensuremath{\simeq}10.4$ K. Neutron diffraction experiments elucidate a collinear antiferromagnetic ordering below ${T}_{\mathrm{N}}$ with the propagation vector $k=(1,0,0)$. An intermediate value of frustration ratio ($f\ensuremath{\simeq}3.6$) reflects moderate frustration in the compound which is corroborated by a reduced ordered magnetic moment of $\ensuremath{\sim}1.52{\ensuremath{\mu}}_{\mathrm{B}}$ at 1.6 K, compared to its classical value ($5{\ensuremath{\mu}}_{\mathrm{B}}$). Magnetic isotherms exhibit a change of slope envisaging a field induced spin-flop transition at ${H}_{\mathrm{SF}}\ensuremath{\simeq}3.2$ T. The magnetic field vs temperature phase diagram clearly unfold three distinct phase regimes, reminiscent of a frustrated magnet with in-plane ($XY$-type) anisotropy.

Spin Dynamics Simulation of the Z2-vortex Fluctuations

Motivated by the recent quasi-elastic neutron scattering experiment, we extend the spin-dynamics simulation on the triangular-lattice Heisenberg antiferromagnet, to observe a sharp central peak of its energy width $\sim 0.001J$ ($J$ the exchange coupling) of the $Z_2$-vortex origin, consistently with the experiment.

Non-magnetic ion site disorder effects on the quantum magnetism of a spin-1/2 equilateral triangular lattice antiferromagnet

With the motivation to study how non-magnetic ion site disorder affects the quantum magnetism of Ba3CoSb2O9, a spin-1/2 equilateral triangular lattice antiferromagnet, we performed DC and AC susceptibility, specific heat, elastic and inelastic neutron scattering measurements on single crystalline samples of Ba2.87Sr0.13CoSb2O9 with Sr doping on non-magnetic Ba2+ ion sites. The results show that Ba2.87Sr0.13CoSb2O9 exhibits (i) a two-step magnetic transition at 2.7 K and 3.3 K, respectively; (ii) a possible canted 120 degree spin structure at zero field with reduced ordered moment as 1.24 μ B/Co; (iii) a series of spin state transitions for both H∥ab-plane and H∥c-axis. For H∥ab-plane, the magnetization plateau feature related to the up–up–down phase is significantly suppressed; (iv) an inelastic neutron scattering spectrum with only one gapped mode at zero field, which splits to one gapless and one gapped mode at 9 T. All these features are distinctly different from those observed for the parent compound Ba3CoSb2O9, which demonstrates that the non-magnetic ion site disorder (the Sr doping) plays a complex role on the magnetic properties beyond the conventionally expected randomization of the exchange interactions. We propose the additional effects including the enhancement of quantum spin fluctuations and introduction of a possible spatial anisotropy through the local structural distortions.

Structures of magnetic excitations in the spin- 12 kagome-lattice antiferromagnets Cs2Cu3SnF12 and Rb2Cu3SnF12

We show the structures of magnetic excitations in spin-1/2 kagome-lattice antiferromagnets Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$ investigated by inelastic neutron scattering in wide energy and momentum ranges. For Cs$_2$Cu$_3$SnF$_{12}$, four single-magnon excitation modes were observed. Low-energy three modes are assigned to be transverse modes and the high-energy fourth mode is suggested to be an amplitude mode. It was found that the broad excitation continuum without a marked structure spreads in a wide energy range from $0.15J$ to approximately $2.5J$ in contrast to the clearly structured excitation continuum observed in the spin-1/2 triangular-lattice Heisenberg antiferromagnet. These findings strongly suggest spinon excitations as elementary excitations in Cs$_2$Cu$_3$SnF$_{12}$. In Rb$_2$Cu$_3$SnF$_{12}$, singlet-triplet excitations from the pinwheel VBS state and their ghost modes caused by the enlargement of the chemical unit cell were clearly confirmed. It was found that the excitation continuum is structured in the low-energy region approximately below $J_{\mathrm{avg}}$ and the almost structureless high-energy excitation continuum extends to approximately $2.6J_{\mathrm{avg}}$. The characteristics of the high-energy excitation continuum are common to both Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$, irrespective of their ground states. The experimental results strongly suggest that the spin liquid component remains in the ground state as quantum fluctuations in Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$.

Doping-induced structural transformation in the spin-1/2 triangular-lattice antiferromagnet Na2Ba1-xSrxCo(PO4)2

The effects of Sr doping on the structural properties of Na2BaCo(PO4)2, a spin-1/2 triangular-lattice antiferromagnet as a quantum spin liquid candidate, are investigated by complementary x-ray and neutron powder diffraction measurements. It is found that in Na2Ba1−xSrxCo(PO4)2 (NBSCPO), the trigonal phase (space group P3̅m1) with a perfect triangular lattice of Co2+ ions is structurally stable when the doping level of Sr is below 30% (x≤ 0.3), while a pure monoclinic phase (space group P21/a) with slight rotations of CoO6 octahedra and displacements of Ba2+/Sr2+ ions will be established when the Sr doping level is above 60% (x≥ 0.6). Such a doping-induced structural transformation in NBSCPO is supported by first-principles calculations and Raman spectroscopy. Na2SrCo(PO4)2, a novel spin-1/2 antiferromagnet with glaserite-type structure, although monoclinically distorted, exhibits no long-range magnetic order down to 2 K and a similar negative Curie-Weiss temperature as Na2BaCo(PO4)2 with a perfect triangular lattice, suggesting the robustness of magnetic exchange interaction against the Ba/Sr substitutions.