Anisotropic Triangular Lattice Research Articles

Incommensurate Order with Translationally Invariant Projected Entangled-Pair States: Spiral States and Quantum Spin Liquid on the Anisotropic Triangular Lattice.

Simulating strongly correlated systems with incommensurate order poses significant challenges for traditional finite-size-based approaches. Confining such a phase to a finite-size geometry can induce spurious frustration, with spin spirals in frustrated magnets being a typical example. Here, we introduce an Ansatz based on infinite projected entangled-pair states which overcomes these limitations and enables the direct search for the optimal spiral in the thermodynamic limit, with a computational cost that is independent of the spiral's wavelength. Leveraging this method, we simulate the Heisenberg model on the anisotropic triangular lattice, which interpolates between the square and isotropic triangular lattice limits. Besides accurately reproducing the magnetically ordered phases with arbitrary wavelength, the simulations reveal a quantum spin liquid phase emerging between the Néel and spin spiral phases.

Superconductivity and Mott Physics in Organic Charge Transfer Materials.

The phase diagrams of quasi two-dimensional organic superconductors display a plethora of fundamental phenomena associated with strong electron correlations, such as unconventional superconductivity, metal-insulator transitions, frustrated magnetism and spin liquid behavior. We analyze a minimal model for these compounds, the Hubbard model on an anisotropic triangular lattice, using cutting-edge quantum embedding methods respecting the lattice symmetry. We demonstrate the existence of unconventional superconductivity by directly entering the symmetry-broken phase. We show that the crossover from the Fermi liquid metal to the Mott insulator is associated with the formation of a pseudogap. The predicted momentum-selective destruction of the Fermi surface into hot and cold regions provides motivation for further spectroscopic studies. Our theoretical results agree with experimental phase diagrams of κ-BEDT organics.

Twisted graphene superlattices: resonating valence bond states and magnetic properties

Resonating valence bond (RVB) states are fundamental for understanding quantum spin liquids in two-dimensional (2D) systems. The RVB state is a collective phenomenon in which spins are uncoupled. 2D lattices such as triangular, honeycomb, and dice lattices were investigated using the Hubbard model and exact diagonalization method. We analyzed the total spin, spin–spin correlation functions, local magnetic moments, and spin and charge gaps as a function of on-site Coulomb repulsion, electron concentration, and electronic hopping parameters. Phase diagrams showed that RVB states can live in half-filled and hole-doped anisotropic triangular lattices. We found two types of RVB states: one in the honeycomb sublattice and the other in the centered hexagons in the triangular lattices. Owing to the novel discovery of exotic magnetic ordering in triangular moiré patterns in twisted bilayer graphene and transition metal dichalcogenide systems, our results provide physical insights into the onset of magnetism and possible spin liquid states in these layered materials.

Frustration- and doping-induced magnetism in a Fermi-Hubbard simulator.

Geometrical frustration in strongly correlated systems can give rise toa plethora of novel ordered states and intriguing magnetic phases, such as quantum spin liquids1-3. Promising candidate materials for such phases4-6 can be described by the Hubbard model on an anisotropic triangular lattice, a paradigmaticmodel capturing the interplay between strong correlations and magnetic frustration7-11. However, the fate of frustrated magnetism in the presence of itinerant dopants remains unclear, as well as its connection to the doped phases of the square Hubbard model12. Here we investigate the local spin order of a Hubbard model with controllable frustration and doping, using ultracold fermions in anisotropic optical lattices continuously tunable from a square to a triangular geometry. At half-filling and strong interactions U/t ≈ 9, we observe at the single-site level how frustration reduces the range of magnetic correlations and drives a transition from a collinear Néel antiferromagnet to a short-range correlated 120° spiral phase. Away from half-filling, the triangular limit shows enhanced antiferromagnetic correlations on the hole-doped side and a reversal to ferromagnetic correlations at particle dopings above 20%, hinting at the role of kinetic magnetism in frustrated systems. This work paves the way towards exploring possible chiral ordered or superconducting phases in triangular lattices8,13 and realizing t-t' square lattice Hubbard models that may be essential to describe superconductivity in cuprate materials14.

Magnetic phases of the anisotropic triangular lattice Hubbard model

The Hubbard model on an anisotropic triangular lattice in two dimensions, a fundamental model for frustrated electron physics, displays a wide variety of phases and phase transitions. This work investigates the model using the ladder dual fermion approximation which captures local correlations non-perturbatively but approximates non-local correlations. We find metallic, one-dimensional antiferromagnetic, non-collinear antiferromagnetic, square-lattice antiferromagnetic, and spiral phases but no evidence of collinear antiferromagnetic order in different parts of the phase diagram. Analyzing the spin susceptibility in detail, we see both regions of agreement and of discrepancy with previous work. The case of Cs$_2$CuCl$_4$ is discussed in detail.

Anomalously field-susceptible spin clusters emerging in the electric-dipole liquid candidate κ-(ET)2Hg(SCN)2Br.

Mutual interactions in many-body systems bring about various exotic phases, among which liquid-like states failing to order due to frustration are of keen interest. The organic system with an anisotropic triangular lattice of molecular dimers, κ-(ET)2Hg(SCN)2Br, has been suggested to host a dipole liquid arising from intradimer charge-imbalance instability, possibly offering an unprecedented stage for the spin degrees of freedom. Here, we show that an extraordinary unordered/unfrozen spin state having soft matter-like spatiotemporal characteristics emerges in this system. 1H nuclear magnetic resonance (NMR) spectra and magnetization measurements indicate that gigantic, staggered moments are nonlinearly and inhomogeneously induced by a magnetic field, whereas the moments vanish in the zero-field limit. The analysis of the NMR relaxation rate signifies that the moments fluctuate at a characteristic frequency slowing down to below megahertz at low temperatures. The inhomogeneity, local correlation, and slow dynamics indicative of middle-scale dynamical correlation length of several nanometers suggest novel frustration-driven spin clusterization.

Ingredients for Generalized Models of κ-Phase Organic Charge-Transfer Salts: A Review

The families of organic charge-transfer salts κ-(BEDT-TTF)2X and κ-(BETS)2X, where BEDT-TTF and BETS stand for the organic donor molecules C10H8S8 and C10H8S4Se4, respectively, and X for an inorganic electron acceptor, have been proven to serve as a powerful playground for the investigation of the physics of frustrated Mott insulators. These materials have been ascribed a model character, since the dimerization of the organic molecules allows to map these materials onto a single band Hubbard model, in which the dimers reside on an anisotropic triangular lattice. By changing the inorganic unit X or applying physical pressure, the correlation strength and anisotropy of the triangular lattice can be varied. This has led to the discovery of a variety of exotic phenomena, including quantum-spin liquid states, a plethora of long-range magnetic orders in proximity to a Mott metal-insulator transition, and unconventional superconductivity. While many of these phenomena can be described within this effective one-band Hubbard model on a triangular lattice, it has become evident in recent years that this simplified description is insufficient to capture all observed magnetic and electronic properties. The ingredients for generalized models that are relevant include, but are not limited to, spin-orbit coupling, intra-dimer charge and spin degrees of freedom, electron-lattice coupling, as well as disorder effects. Here, we review selected theoretical and experimental discoveries that clearly demonstrate the relevance thereof. At the same time, we outline that these aspects are not only relevant to this class of organic charge-transfer salts, but are also receiving increasing attention in other classes of inorganic strongly correlated electron systems. This reinforces the model character that the κ-phase organic charge-transfer salts have for understanding and discovering novel phenomena in strongly correlated electron systems from a theoretical and experimental point of view.

Possible coexistence of short-range resonating valence bond and long-range stripe correlations in the spatially anisotropic triangular-lattice quantum magnet Cu2 (OH) 3NO3

We show that short-range resonating valence bond correlations and long-range order can coexist in the ground state (GS) of a frustrated spin system. Our study comprises a comprehensive investigation of the quantum magnetism on the structurally disorder-free single crystal of ${\mathrm{Cu}}_{2}{(\mathrm{OH})}_{3}{\mathrm{NO}}_{3}$, which realizes the $s$ = 1/2 Heisenberg model on a spatially anisotropic triangular lattice. Competing exchange interactions determined by fitting the magnetization measured up to 55 T give rise to an exotic GS wave function with the coexistence of the dominant short-range resonating valence bond correlations and weak long-range stripe order (ordered moment ${M}_{0}=|\ensuremath{\langle}{s}_{i}^{z}\ensuremath{\rangle}|\ensuremath{\sim}0.02$). At low temperatures, a first-order spin-flop transition is visible at $\ensuremath{\sim}$1--3 T. As the applied field further increases, another two magnetic field induced quantum phase transitions are observed at $\ensuremath{\sim}$14--19 and $\ensuremath{\sim}$46--52 T, respectively. Simulations of the Heisenberg exchange model show semiquantitative agreement with the magnetic-field modulation of these unconventional phases, as well as the absence of visible magnetic reflections in neutron diffraction, thus supporting the GS of the spin system of ${\mathrm{Cu}}_{2}{(\mathrm{OH})}_{3}{\mathrm{NO}}_{3}$ may be approximate to a quantum spin liquid. Our study establishes structurally disorder-free magnetic materials with spatially anisotropic exchange interactions as a possible arena for spin liquids.

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.

Isothermal and adiabatic magnetization processes of the spin- 12 Heisenberg model on an anisotropic triangular lattice

In this study, we investigate the magnetic susceptibility, entropy, and isothermal magnetization curve of the spin-1/2 Heisenberg model on an anisotropic triangular lattice using the orthogonalized finite-temperature Lanczos method. In addition, we investigate the adiabatic magnetization curve and magnetocaloric effect. We estimate these physical quantities with sufficient accuracy in the thermodynamic limit, except at low temperatures. We observe a 1/3 magnetization plateau in the isothermal magnetization process, whereas the plateau is observed to have a slope in the adiabatic process. We show that the magnetocaloric effect can be used to detect the signature of phase transitions. We believe that these results will be useful for understanding the magnetism of anisotropic triangular lattice compounds through a comparison with experimental results in the future.

Спиральное магнитное упорядочение и переход металл--диэлектрик в модели Хаббарда на треугольной решeтке

The magnetic phase diagrams of the two-dimensional Hubbard model for isotropic and anisotropic triangular lattices are constructed within the Hartree-Fock and slave boson approximations. The triangular lattice specific non-collinear and spiral magnetic states, as well as phase separation between them, are shown to be realized in a wide range of model parameters along with collinear magnetic states (stripe antiferromagnetic and ferromagnetic). Phase transitions of the first and second order are found, and the boundaries of the phase separation regions are determined. A comparison of the two approximations, Hartree-Fock and slave boson, shows that electronic correlations suppress magnetic states, the region of paramagnetism being expand, for values U/t>5. At the same time, when the Fermi level is near the van Hove singularity, electron correlations do not change the diagrams qualitatively, which is consistent with the previously obtained result for square and cubic lattices. The results are compared with the data available in the literature for other methods and approaches.

Spiral magnetic order and metal-insulator transition in the Hubbard model on a triangular lattice

The magnetic phase diagrams of the two-dimensional Hubbard model for isotropic and anisotropic triangular lattices are constructed within the Hartree--Fock and slave boson approximations. The triangular lattice specific non-collinear and spiral magnetic states, as well as phase separation between them, are shown to be realized in a wide range of model parameters along with collinear magnetic states (stripe antiferromagnetic and ferromagnetic). Phase transitions of the first and second order are found, and the boundaries of the phase separation regions are determined. A comparison of the two approximations, Hartree--Fock and slave boson, shows that electronic correlations suppress magnetic states, the region of paramagnetism being expand, for values U/t>~=5. At the same time, when the Fermi level is near the van Hove singularity, electron correlations do not change the diagrams qualitatively, which is consistent with the previously obtained result for square and cubic lattices. The results are compared with the data available in the literature for other methods and approaches. Keywords: Hubbard model, phase separation, spiral magnetic order, triangular lattice, metal-insulator transition

Simultaneous Control of Bandfilling and Bandwidth in Electric Double-Layer Transistor Based on Organic Mott Insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl

The physics of quantum many-body systems have been studied using bulk correlated materials, and recently, moiré superlattices formed by atomic bilayers have appeared as a novel platform in which the carrier concentration and the band structures are highly tunable. In this brief review, we introduce an intermediate platform between those systems, namely, a band-filling- and bandwidth-tunable electric double-layer transistor based on a real organic Mott insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl. In the proximity of the bandwidth-control Mott transition at half filling, both electron and hole doping induced superconductivity (with almost identical transition temperatures) in the same sample. The normal state under electric double-layer doping exhibited non-Fermi liquid behaviors as in many correlated materials. The doping levels for the superconductivity and the non-Fermi liquid behaviors were highly doping-asymmetric. Model calculations based on the anisotropic triangular lattice explained many phenomena and the doping asymmetry, implying the importance of the noninteracting band structure (particularly the flat part of the band).

Magnetic Structures of Electron Systems on the Extended Spatially Completely Anisotropic Triangular Lattice near Quantum Critical Points

We examine magnetic structures of electron systems on an extended triangular lattice that consists of two types of bond triangles with electron transfer energies t_l and t'_l (l = 1, 2, and 3), respectively. We examine the ground state in the mean-field theory when t_1 = t'_1, focusing on collinear states with two sublattices. It is shown that when the imbalance of the spatial anisotropies of the two triangles is large, up-up-down-down (uudd) phases are stable, and the most likely ground states of the lambda-(BETS)_2FeCl_4 system are the Neel state with the modulation vector (\pi/c,\pi/a) and a uudd state, where c = a_1 = a'_1 and a = (a_2+a'_2)/2, with a_1, a_1', a_2, and a_2' being the lattice constants of the bonds with t_1, t'_1, t_2, and t'_2, respectively. These results are consistent with those from the classical spin system. In addition, this study reveals behaviors near the quantum critical point, which cannot be reproduced in the localized spin model. As the imbalance of the spatial anisotropies increases, the U_c of the Neel state increases, and that of the uudd state decreases. In the phase diagrams containing areas of the paramagnetic state, the Neel state with (\pi/c,\pi/a), and a uudd state, their boundaries terminate at a triple point, near which all the transitions are of the first order. The phase boundary between the antiferromagnetic phases does not depend on U, and the transition is of the first order everywhere on the boundary. By contrast, the transitions from the two antiferromagnetic phases to the paramagnetic phase are of the second order, unless the system is close to the triple point.

Gapped ground state in a spin- 12 frustrated square lattice

We present a model compound with a spin-1/2 frustrated square lattice, in which three ferromagnetic (F) interactions and one antiferromagnetic (AF) compet. Considering the effective spin-1 formed by the dominant F dimer, this square lattice can be mapped to a spin-1 spatially anisotropic triangular lattice. The magnetization curve exhibits gapped behavior indicative of a dominant one-dimensional (1D) AF correlation. In the field-induced gapless phase, the specific heat and magnetic susceptibility show a phase transition to an ordered state with 2D characteristics. These results indicate that the spin-1 Haldane state is extended to the 2D system. We demonstrate that the gapped ground state observed in the present spin-1/2 frustrated square lattice originates from the one-dimensionalization caused by frustration.

Phase diagram of the anisotropic triangular lattice Hubbard model

In a recent study [Phys. Rev. X 10, 021042 (2020)], we showed using large-scale density matrix renormalization group (DMRG) simulations on infinite cylinders that the triangular lattice Hubbard model has a chiral spin liquid phase. In this work, we introduce hopping anisotropy in the model, making one of the three distinct bonds on the lattice stronger or weaker compared with the other two. We implement the anisotropy in two inequivalent ways, one which respects the mirror symmetry of the cylinder and one which breaks this symmetry. In the full range of anisotropy, from the square lattice to weakly coupled one-dimensional chains, we find a variety of phases. Near the isotropic limit we find the three phases identified in our previous work: metal, chiral spin liquid, and 120$^\circ$ spiral order; we note that a recent paper suggests the apparently metallic phase may actually be a Luther-Emery liquid, which would also be in agreement with our results. When one bond is weakened by a relatively small amount, the ground state quickly becomes the square lattice N\'{e}el order. When one bond is strengthened, the story is much less clear, with the phases that we find depending on the orientation of the anisotropy and on the cylinder circumference. While our work is to our knowledge the first DMRG study of the anisotropic triangular lattice Hubbard model, the overall phase diagram we find is broadly consistent with that found previously using other methods, such as variational Monte Carlo and dynamical mean field theory.

Hierarchy of energy scales and field-tunable order by disorder in dipolar-octupolar pyrochlores

Dipolar-octupolar pyrochlore magnets in a strong external magnet field applied in the [110] direction are known to form a `chain' state, with subextensive degeneracy. Magnetic moments are correlated along one-dimensional chains carrying effective Ising degrees of freedom which are noninteracting on the mean-field level. Here, we investigate this phenomenon in detail, including the effects of quantum fluctuations. We identify two distinct types of chain phases, both featuring distinct subextensive, classical ground state degeneracy. Focussing on one of the two kinds, we discuss lifting of the classical degeneracy by quantum fluctuations. We map out the ground-state phase diagram as a function of the exchange couplings, using linear spin wave theory and real-space perturbation theory. We find a hierarchy of energy scales in the ground state selection, with the effective dimensionality of the system varying in an intricate way as the hierarchy is descended. We derive an effective two-dimensional anisotropic triangular lattice Ising model with only three free parameters which accounts for the observed behavior. Connecting our results to experiment, they are consistent with the observation of a disordered chain state in Nd$_2$Zr$_2$O$_7$. We also show that the presence of two distinct types of chain phases has consequences for the field-induced breakdown of the apparent $U(1)$ octupolar quantum liquid phase recently observed in Ce$_2$Sn$_2$O$_7$.

Bound spinon excitations in the spin- 12 anisotropic triangular antiferromagnet Ca3ReO5Cl2

This work presents the inelastic neutron scattering spectrum on the spin-1/2 anisotropic triangular lattice antiferromagnetCa_3 ReO_5 Cl_2 , and show that the spinon-like continuum coexists with dispersive modes due to the formation of bound spinon pairs.

Anomalous Hall effect in κ -type organic antiferromagnets

We theoretically propose a mechanism for the anomalous Hall effect (AHE) in an antiferrromagnetic (AFM) state of $\kappa$-type organic conductors. We incorporate the spin-orbit coupling in the effective Hubbard model on the $\kappa$-type lattice structure taking into account the orientation of the molecules and their arrangement with dimerization. Treating this model by means of the Hartree-Fock approximation and the linear response theory, we find that an intrinsic contribution to the Hall conductivity becomes nonzero in the electron-doped AFM metallic phase with a small canted ferromagnetic moment. We show that, contrary to the conventional wisdom, the spin canting is irrelevant to the Hall response; the nonzero Hall conductivity originates from the collinear component of the AFM order in the presence of the spin-orbit coupling. These features are well explained analytically in the limit of strong dimerization on the anisotropic triangular lattice. Furthermore, we present an intuitive picture for the present AHE by considering the real-space configuration of emergent magnetic fluxes. We also find that the Hall response appears even in the undoped AFM insulating phase at nonzero frequency as the magneto-optical Kerr effect, which is enhanced around the charge transfer excitations. We discuss possible detections of the AHE in ET based compounds.

Anisotropic Triangular Lattice Realized in Rhenium Oxychlorides A3ReO5Cl2 (A = Ba, Sr).

We report the synthesis, crystal structure, and magnetic properties of the two new quantum antiferromagnets A3ReO5Cl2 (A = Sr, Ba). The crystal structure is isostructural with the mineral pinalite Pb3WO5Cl2, in which the Re6+ ion is square pyramidally coordinated by five oxide atoms and forms an anisotropic triangular lattice (ATL) made of S = 1/2 spins. The magnetic interactions J and J' in the ATL are estimated from magnetic susceptibilities to be 19.5 (44.9) and 9.2 (19.3) K, respectively, with J'/J = 0.47 (0.43) for A = Ba (Sr). For each compound, the heat capacity at low temperatures shows a large T-linear component with no signature of long-range magnetic order above 2 K, which suggests a gapless spin liquid state of one-dimensional character of the J chains in spite of the significantly large J' couplings. This is a consequence of one-dimensionalization by geometrical frustration in the ATL magnet; a similar phenomenon has been observed in two compounds with slightly smaller J'/J values: Cs2CuCl4 (J'/J = 0.3) and the related compound Ca3ReO5Cl2 (0.32). Our findings demonstrate that 5d mixed-anion compounds provide a unique opportunity to explore novel quantum magnetism.