Triangular Spin Lattice Research Articles

Synthesis and characterization of a two-dimensional antiferromagnet KNiB4O6F3 with triangular spin lattice

We prepared a layered antiferromagnet KNiB4O6F31 by hydrothermal synthesis, and characterized its structure and physical properties by X-ray diffraction, magnetization and specific heat measurements as well as thermal stability, FTIR and UV-vis-NIR measurements. 1 consists of layers of composition [NiB4O6F3]- separated by layers of K+ cations. And Ni2+ ions are arranged in slightly distorted triangular pattern in each [NiB4O6F3]- layer. The magnetic susceptibility measurements indicate no divergence between the zero-field-cooled and field-cooled curves down to 7.9 K, below which a long-range order takes place as verified by specific heat measurements. The M(H) curve linearly increases when the field is weak (below ∼2 T), confirming that the ground state of 1 is antiferromagnetic. At 7 T, the magnetization of 1 is 1.67 μB which is lower than the saturation magnetization.

Observation of magnetic dimers in the diluted triangular-lattice spin liquid candidate NaYbO2

The triangular-lattice spin liquid (QSL) candidate NaYbO2 is magnetically diluted by Lu doping, forming homogeneous solid solutions NaYb1-xLuxO2 (0≤ x<1.0). A combination of structural, magnetic, and pulsed high-field electron spin resonance analysis shows that the lattice parameters and the Curie-Weiss temperature θCW vary almost linearly with x, and the crystal electric field environment of the Yb/Lu ions persists during the Lu doping. The statistical distribution analysis of clusters and the deduced linear θCW ∼ x relation indicate that the nearest neighbor exchange J does not change with magnetic dilution. Interestingly, with the magnetization measurement at 0.39 K, we have observed Seff = 1/2 antiferromagnetic Yb-Yb dimers in NaYb0.02Lu0.98O2, which is manifested as a magnetization step at μ0Hc = 2.68 T due to the transition between singlet and triplet states. Thus, the intradimer interaction is directly determined as J/kB = −5.15 K. Surprisingly, this value is close to the interaction of undoped NaYbO2, supporting the theoretical analysis of clusters. Our study provides a method of estimating the magnetic interaction of QSL materials through magnetic dilution. The statistical distribution analysis of magnetic clusters suggests that through a small amount of Lu doping (x < 0.5), the QSL state in NaYbO2 can continue to be maintained.

Finite Temperature Magnetism in the Triangular Lattice Antiferromagnet KErTe2

Abstract After the discovery of the ARECh2 (A = alkali or monovalent ions, RE = rare-earth, Ch = chalcogen) triangular lattice quantum spin liquid (QSL) family, a series of its oxide, sulfide, and selenide counterparts has been consistently reported and extensively investigated. While KErTe2 represents the initial synthesized telluride member, preserving its triangular spin lattice, it was anticipated that the substantial tellurium ions could impart more pronounced magnetic attributes and electronic structures to this material class. This study delves into the magnetism of KErTe2 at finite temperatures through magnetization and electron spin resonance (ESR) measurements. Based on the angular momentum J ^ after spin-orbit coupling (SOC) and symmetry analysis, we obtain the magnetic effective Hamiltonian to describe the magnetism of Er3+ in R 3 ¯ m space group. Applying the mean-field approximation to the Hamiltonian, we can simulate the magnetization and magnetic heat capacity of KErTe2 in paramagnetic state and determine the crystalline electric field (CEF) parameters and partial exchange interactions. The relatively narrow energy gaps between the CEF ground state and excited states exert a significant influence on the magnetism. For example, small CEF excitations can result in a significant broadening of the ESR linewidth at 2 K. For the fitted exchange interactions, although the values are small, given a large angular momentum J = 15/2 after SOC, they still have a noticeable effect at finite temperatures. Notably, the heat capacity data under different magnetic fields along the c axis direction also roughly match our calculated results, further validating the reliability of our analytical approach. These derived parameters serve as crucial tools for future investigations into the ground state magnetism of KErTe2. The findings presented herein lay a foundation for exploration of the intricate magnetism within the triangular-lattice delafossite family.

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 $S=1/2$.

Studying spin diffusion and quantum entanglement with LF-µSR

LF-µSR studies have previously been used to study the diffusive 1D motion of solitons and polarons in conducting polymers. This type of study was also applied to investigating the diffusive motion of spinons in spin-1/2 antiferromagnetic chains. Recently the method has been extended to examples of 2D layered triangular spin lattices which can support quantum spin liquid states, such as 1T-TaS2 and YbZnGaO4. These systems are found to show spin dynamics that matches well to 2D spin diffusion, such a model being found to provide a much better fit to the data than previously proposed models for spin correlations in such systems. In YbZnGaO4 the diffusion rate shows a clear crossover between classical and quantum regimes as T falls below the exchange coupling J. That the spin diffusion approach works well in the high T classical region might be expected, but it is found that it also works equally well in the low T quantum region where quantum entanglement controls the spin dynamics. Measurement of the diffusion rate allows a T dependent length scale to be derived from the data that can be assigned to a quantum entanglement length ξ E. Another entanglement measure, the Quantum Fisher Information F Q can also be obtained from the data and its T dependence is compared to that of ξ E.

Tracking the evolution from isolated dimers to many-body entanglement in NaLuxYb1−xSe2

We synthesize homogeneous compositions of NaLu$_x$Yb$_{1-x}$Se$_2$, connecting non-magnetic NaLuSe$_2$ to the triangular lattice spin liquid candidate NaYbSe$_2$. Thermal and magnetic properties are studied as the system evolves from one with dilute magnetic defects to one of a dense magnetic lattice. The field and temperature dependent heat capacity show the carriers of entropy crossover from isolated magnetic ions to a correlated lattice borne from spin dimers. For the dilute system we estimate the single ion anisotropy $(g_\perp/g_\parallel =3.13)$ and also the dimer exchange couplings $J_\parallel(=5.4$~K) and $J_\perp(=9.6$~K), in order to draw comparison to the half-doped and full magnetic compounds.

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.

Diluting a triangular-lattice spin liquid: Synthesis and characterization of NaYb1−xLuxS2 single crystals

Yb-based magnets, with a perfect triangular lattice of pseudospin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{\mathrm{Yb}}^{3+}$ ions, have emerged as candidates for realizing a quantum spin-liquid state, with ${\mathrm{NaYbS}}_{2}$ being a prominent example. Here, we present the solid-solution series ${\mathrm{NaYb}}_{1\ensuremath{-}x}{\mathrm{Lu}}_{x}{\mathrm{S}}_{2}$ with well-defined single crystals over the entire substitution range $0\ensuremath{\le}x\ensuremath{\le}1$. Chemical and structural analysis indicate a statistically homogeneous replacement of ${\mathrm{Yb}}^{3+}$ by ${\mathrm{Lu}}^{3+}$ ions. We magnetically characterize the relatively small single crystals using electron spin resonance (ESR). Below $30\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, the ESR intensity can be well described by a Curie-Weiss function for all $x$, with a decreasing Weiss temperature with increasing Lu content. This reduction of the average magnetic interaction upon Lu substitution is also supported by magnetization measurements. Importantly, no signs of magnetic or spin-glass order are detected down to $2\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ for any $x$. For $x&gt;0.5$, the ESR linewidth strongly increases, indicating the breakup of the magnetic system into disconnected clusters, as expected from percolation physics. The experimental magnetization data are found to be in good agreement for all $x$ with results of classical Monte Carlo simulations for a triangular-lattice Heisenberg model, amended with a small second-neighbor interaction. Taken together, our results establish ${\mathrm{NaYb}}_{1\ensuremath{-}x}{\mathrm{Lu}}_{x}{\mathrm{S}}_{2}$ as a family of diluted triangular-lattice spin liquids.

Modulating thermal robustness and entanglement distribution in a two-dimensional dissipative spin system using impurities

We consider a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian environment at finite temperature in the presence of an external uniform magnetic field. We show how inserting a magnetic impurity in the spin system can be used to effectively control the dynamics and asymptotic state of the system. A strong impurity, at a border or central site, was found to enhance its entanglement with the other spins and their thermal robustness to the dissipative environment reaching an asymptotic state that is independent of the system initial state. However, it reduces the entanglement among the other spins in the lattice and their thermal robustness and may diminish them depending on its strength and the environment temperature. Moreover, the effect of the impurity increases significantly as the degree of anisotropy of the spin system increases. Therefore, the entanglement distribution over the different sites of the lattice can be modulated by tuning the impurity strength, system anisotropy and environment temperature. The impurity can be used as a switch that turns the entanglement on among specific spins and off among others simultaneously.

Heterospin frustration in a metal-fullerene-bonded semiconductive antiferromagnet

Lithium-ion-encapsulated fullerenes (Li+@C60) are 3D superatoms with rich oxidative states. Here we show a conductive and magnetically frustrated metal–fullerene-bonded framework {[Cu4(Li@C60)(L)(py)4](NTf2)(hexane)}n (1) (L = 1,2,4,5-tetrakis(methanesulfonamido)benzene, py = pyridine, NTf2− = bis(trifluoromethane)sulfonamide anion) prepared from redox-active dinuclear metal complex Cu2(L)(py)4 and lithium-ion-encapsulated fullerene salt (Li+@C60)(NTf2−). Electron donor Cu2(L)(py)2 bonds to acceptor Li+@C60 via eight Cu‒C bonds. Cu–C bond formation stems from spontaneous charge transfer (CT) between Cu2(L)(py)4 and (Li+@C60)(NTf2−) by removing the two-terminal py molecules, yielding triplet ground state [Cu2(L)(py)2]+(Li+@C60•−), evidenced by absorption and electron paramagnetic resonance (EPR) spectra, magnetic properties and quantum chemical calculations. Moreover, Li+@C60•− radicals (S = ½) and Cu2+ ions (S = ½) interact antiferromagnetically in triangular spin lattices in the absence of long-range magnetic ordering to 1.8 K. The low-temperature heat capacity indicated that compound 1 is a potential candidate for an S = ½ quantum spin liquid (QSL).

Thermal Robustness of Entanglement in a Dissipative Two-Dimensional Spin System in an Inhomogeneous Magnetic Field.

Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

Hidden magnetic order in the triangular-lattice magnet Li2MnTeO6

The manganese tellurate Li2MnTeO6 consists of trigonal spin lattices made up of Mn4+ (d3, S = 3/2) ions. The magnetic properties of this compound were characterized by several experimental techniques, which include magnetic susceptibility, specific heat, dielectric permittivity, electron spin resonance (ESR), nuclear magnetic resonance (NMR) and neutron powder diffraction (NPD) measurements, and by density functional calculations (DFT). The magnetic susceptibility chi(T) demonstrates very unusual behavior. It isdescribed by the Curie-Weiss law at high temperature with Curie-Weiss temperature of Theta = -74 K, exhibits no obvious anomaly indicative of a long-range magnetic ordering at low magnetic fields. At high magnetic fields, however, the character of chi(T) changes showing a maximum at about 9 K. That this maximum of chi(T) reflects the onset of an antiferromagnetic order was confirmed by specific heat measurements, which exhibit a clear lambda-type anomaly at TN around 8.5 K even at zero magnetic field, and by 7Li NMR and dielectric permittivity measurements. The magnetic structure of Li2MnTeO6, determined by neutron powder diffraction measurements at 1.6 K, is described by the 120-degree non-collinear spin structure with the propagation vector k = (1/3, 1/3, 0). Consistent with this finding, the spin exchange interactions evaluated for Li2MnTeO6 by density functional calculations are dominated by the nearest-neighbor antiferromagnetic exchange within each triangular spin lattice. This spin lattice is strongly spin frustrated with f = |Theta|/TN around 8 and exhibits a two-dimensional magnetic character in a broad temperature range above TN.

Signatures of a Spin-1/2 Cooperative Paramagnet in the Diluted Triangular Lattice of Y_{2}CuTiO_{6}.

We present a combination of thermodynamic and dynamic experimental signatures of a disorder driven dynamic cooperative paramagnet in a 50% site diluted triangular lattice spin-1/2 system: Y_{2}CuTiO_{6}. Magnetic ordering and spin freezing are absent down to 50mK, far below the Curie-Weiss scale (-θ_{CW}) of ∼134 K. We observe scaling collapses of the magnetic field and temperature dependent magnetic heat capacity and magnetization data, respectively, in conformity with expectations from the random singlet physics. Our experiments establish the suppression of any freezing scale, if at all present, by more than 3 orders of magnitude, opening a plethora of interesting possibilities such as disorder stabilized long range quantum entangled ground states.

Triangular-Lattice Organic Mott Insulator with a Disorder-Free Polyanion.

We report the synthesis, crystal and band structures, and transport properties of organic conductor κ-(ET)2Cu[Au(CN)2]Cl [ET = bis(ethylenedithio)tetrathiafulvalene], which has a triangular spin-lattice (S = 1/2) composed of (ET)2•+ dimers and polyanions with no disorder. The anisotropy of triangular lattice t'/t = 1.19 and physical properties indicate that this material is the first ET-based quantum-spin-liquid candidate having a nearly regular triangular lattice with a disorder-free anion.

Exploring Magnetic XY Behavior in a Quasi-2D Anisotropic Triangular Lattice of Cu(II) by Functionalized Graphene.

Study on magnetism in two-dimensional (2D) spin-lattices is advancing rapidly. In this work, phase-pure botallackite (Bo) (Cu2(OH)3Br), a quasi-2D S = 1/2 anisotropic triangular spin-lattice is stabilized over 2D reduced graphene oxide (rGO) nanosheets via simple oxidation-reduction reaction chemistry. In comparison to polycrystalline Bo, such an anchoring resulted in the oriented growth of Bo crystallites in the Bo-rGO system. The Bo-rGO nanocomposite was found to be magnetically active with a Néel transition at ∼8.9 K, crossing over to possible XY anisotropy at ∼5 K-as revealed by complementary dc and ac susceptibility measurements-an unprecedented observation in the field assigned to an interfacial effect. This work demonstrates the potential usage of nonmagnetic 2D functionalized graphene to significantly modulate the magnetic properties of 2D spin-lattices.

Simulating Spin Chains Using a Superconducting Circuit: Gauge Invariance, Superadiabatic Transport, and Broken Time‐Reversal Symmetry

AbstractSimulation of materials by using quantum processors is envisioned to be a major direction of development in quantum information science. Here, the mathematical analogies between a triangular spin lattice with Dzyaloshinskii–Moriya coupling on one edge and a three‐level system driven by three fields in a loop configuration are exploited to emulate spin‐transport effects. It is shown that the spin transport efficiency, seen in the three‐level system as population transfer, is enhanced when the conditions for superadiabaticity are satisfied. It is demonstrated experimentally that phenomena characteristic to spin lattices due to gauge invariance, non‐reciprocity, and broken time‐reversal symmetry can be reproduced in the three‐level system.

Specific Heat Study of 1D and 2D Excitations in the Layered Frustrated Quantum Antiferromagnets Cs_{2}CuCl_{4-x}Br_{x}.

We report an experimental and theoretical study of the low-temperature specific heat C and magnetic susceptibility χ of the layered anisotropic triangular-lattice spin-1/2 Heisenberg antiferromagnets Cs_{2}CuCl_{4-x}Br_{x} with x=0, 1, 2, and 4. We find that the ratio J^{'}/J of the exchange couplings ranges from 0.32 to ≈0.78, implying a change (crossover or quantum phase transition) in the materials' magnetic properties from one-dimensional (1D) behavior for J^{'}/J<0.6 to two-dimensional (2D) behavior for J^{'}/J≈0.78. For J^{'}/J<0.6, realized for x=0, 1, and 4, we find a magnetic contribution to the low-temperature specific heat, C_{m}∝T, consistent with spinon excitations in 1D spin-1/2 Heisenberg antiferromagnets. Remarkably, for x=2, where J^{'}/J≈0.78 implies a 2D magnetic character, we also observe C_{m}∝T. This finding, which contrasts the prediction of C_{m}∝T^{2} made by standard spin-wave theories, shows that Fermi-like statistics also plays a significant role for the magnetic excitations in spin-1/2 frustrated 2D antiferromagnets.

Density matrix renormalization group study of superconductivity in the triangular lattice Hubbard model

With the discovery of strong coupling physics and superconductivity in Moir\'e superlattices, it's essential to have an understanding of strong coupling driven superconductivity in systems with trigonal symmetry. The simplest lattice model with trigonal symmetry is the triangular lattice Hubbard model. Although the triangular lattice spin model is a heavily studied model in the context of frustration, studies of the hole-doped triangular lattice Hubbard model are rare. Here we use density matrix renormalization group (DMRG) to investigate the domininant superconducting channels in the hole-doped triangular lattice Hubbard model over a range of repulsive interaction strengths. We find a clear transition from $p$-wave superconductivity at moderate on-site repulsion strength ($U/t = 2$) at filling above 1/4 ($n \sim 0.65$) to $d$-wave superconductivity at strong on-site repulsion strength ($U/t = 10$) at filling below 1/4 ($n \sim 0.4$). The unusual tunability that Moir\'e superlattices offer in controlling $U/t$ would open up the opportunity to realize this transition between $d$-wave and $p$-wave superconductivity.

The Effects of Three Magnons Interactions in the Magnon-Density Waves of Triangular Spin Lattices

We investigate the magnon-density waves proposed as the longitudinal excitations in triangular lattice antiferromagnets by including the cubic and quartic corrections in the large-s expansion. The longitudinal excitation spectra for the two-dimensional (2D) triangular antiferromagnetic model and quasi-one dimensional (quasi-1D) antiferromagnetic materials have been obtained for a general quantum spin number $s$. For the 2D triangular lattice model, we find a significant reduction (about 40 %) in the energy spectra at the zone boundaries due to both the cubic and quartic corrections. For the quasi-1D antiferromagnets, since the cubic term comes from the very weak couplings on the hexagonal planes, they make very little correction to the energy spectra, whereas the major correction contribution comes from the quartic terms in the couplings along the chains with the numerical values for the energy gaps in good agreement with the experimental results as reported earlier (Ref.~41).

Magnetic-field-induced incommensurate to collinear spin order transition in NiBr2

The triangular spin lattice of NiBr2 is a canonical example of the frustrated helimagnet that shows a collinear commensurate antiferromagnetic to an incommensurate spin helix phase transition on cooling. Herein, we have studied a self-flux grown NiBr2 single crystal by neutron diffraction and low temperature magnetization measurements at fields up to 14 T. Experimental findings enable the deduction of the driving force responsible for the spin spiral ordering. The neutron diffraction data reveal satellite peaks representing characteristic features of an incommensurate magnetic state. The satellites develop symmetrically below TN=44.0(1)K, replacing the main magnetic reflections. Interestingly, a field-induced incommensurate to commensurate spin phase transition has been successfully demonstrated, which enforces the spin helix to restore the high temperature compensated antiferromagnetic structure. This reorientation can be described by a spin-flop in the (a–b) basal plane of a triangular spin lattice system. The findings offer a novel way for spin helix control of incommensurate phases, having immense scientific and technological implications in the next-generation data storage devices.