Anisotropic Triangular Lattice Research Articles

A quantum fidelity study of the anisotropic next-nearest-neighbour triangular lattice Heisenberg model

Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J′ − J2 plane of this model, which connects the J1 − J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar–Ghosh point of the J1 − J2 chain (J′ = 0) are found to extend into the J′ − J2 plane and connect with points on the J2 = 0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, , χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J′, J2 ≪ J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1 − J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.

Exact diagonalization study of the anisotropic triangular lattice Heisenberg model using twisted boundary conditions

The spin-$1/2$ anisotropic triangular model, which is believed to describe the materials ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$ and ${\mathrm{Cs}}_{2}{\mathrm{CuBr}}_{4}$, among others, is dominated by incommensurate spiral physics and is thus extremely resistant to numerical analysis on small system sizes. In this paper, we use twisted boundary conditions and exact diagonalization techniques to study the ground state of this model as a function of exchange parameter strength. With these boundary conditions we are able to extract the inter- and intrachain ordering $q$ vectors for the $\frac{{J}^{\ensuremath{'}}}{J}<1$ region finding very close agreement with recent DMRG results on much larger systems. Our results suggest a phase transition between a long-range incommensurate spiral ordered phase, and a more subtle phase with short-range spiral correlations with the $q$ vector describing the incommensurate correlations varying smoothly through the transition. In the latter phase correlations between next-nearest chains exhibit an extremely close competition between predominantly antiferromagnetic and ferromagnetic correlations. Further analysis suggests that the antiferromagnetic next-nearest chain correlations may be slightly stronger than the ferromagnetic ones. This difference is found to be slight but in line with previous renormalization group predictions of a collinear antiferromagnetic ordering in this region.

Pseudogap and singlet formation in organic and cuprate superconductors

The pseudogap phase occurring in cuprate and organic superconductors is analyzed based on the dynamical cluster approximation approach to the Hubbard model. In this method a cluster embedded in a self-consistent bath is studied. With increasing Coulomb repulsion, $U$, the antinodal point [$\mathbf{k}=(\ensuremath{\pi},0)$] displays a gradual suppression of spectral density of states around the Fermi energy which is not observed at the nodal point [$\mathbf{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$]. The opening of the antinodal pseudogap is found to be related to the internal structure of the cluster and the much weaker bath-cluster couplings at the antinodal than nodal point. The role played by internal cluster correlations is elucidated from a simple four-level model. For small $U$, the cluster levels form Kondo singlets with their baths leading to a peak in the spectral density. As $U$ is increased a localized state is formed localizing the electrons in the cluster. If this cluster localized state is nondegenerate, the Kondo effect is destroyed and a pseudogap opens up in the spectra at the antinodal point. The pseudogap can be understood in terms of destructive interference between different paths for electrons hopping between the cluster and the bath. However, electrons at the nodal points remain in Kondo states up to larger $U$ since they are more strongly coupled to the bath. The strong correlation between the $(\ensuremath{\pi},0)$ and the $(0,\ensuremath{\pi})$ cluster levels in the localized state leads to a large correlation energy gain, which is important for localizing electrons and opening up a pseudogap at the antinodal point. Such a scenario is in contrast with two independent Mott transitions found in two-band systems with different bandwidths in which the localized cluster electron does not correlate strongly with any other cluster electron for intermediate $U$. The important intracluster sector correlations are associated with the resonating valence bond character of the cluster ground state containing $d$-wave singlet pairs. The low-energy excitations determining the pseudogap have suppressed $d$-wave pairing, indicating that the pseudogap can be related to breaking very short-range $d$-wave pairs. Geometrical frustration on the anisotropic triangular lattice relevant to $\ensuremath{\kappa}$-(BEDT-TTF)${}_{2}X$ leads to a switch in the character of the ground state of the cluster at intermediate hopping ratios ${t}^{\ensuremath{'}}/t\ensuremath{\sim}0.7$. Electron doping of the frustrated square lattice destroys the pseudogap, in agreement with photoemission experiments on cuprates, due to a larger Schrieffer-Wolff exchange coupling, ${J}_{K}$, and a stronger cluster-bath coupling for the antinodal point.

Spin-liquid phase in a spatially anisotropic frustrated antiferromagnet: A Schwinger boson mean-field approach

We explore the effect of the third-nearest neighbors on the magnetic properties of the Heisenberg model on an anisotropic triangular lattice. We obtain the phase diagram of the model using Schwinger boson mean-field theory. Competition between N\'eel, spiral, and collinear magnetically ordered phases is found as we vary the ratios of the nearest ${J}_{1}$, next-nearest ${J}_{2}$, and third-nearest ${J}_{3}$ neighbor exchange couplings. A spin-liquid phase is stabilized between the spiral and collinear ordered states when ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}1.8$, for rather small ${J}_{3}/{J}_{1}\ensuremath{\lesssim}0.1$. The lowest-energy two-spinon dispersions relevant to neutron scattering experiments are analyzed and compared to semiclassical magnon dispersions finding significant differences in the spiral and collinear phases between the two approaches. The results are discussed in the context of the anisotropic triangular materials: Cs${}_{2}$CuCl${}_{4}$ and Cs${}_{2}$CuBr${}_{4}$ and layered organic materials, $\ensuremath{\kappa}$-(BEDT-TTF)${}_{2}X$, and $Y$[Pd(dmit)${}_{2}$]${}_{2}$.

One-dimensional spin liquid, collinear, and spiral phases from uncoupled chains to the triangular lattice

We investigate the Hubbard model on the anisotropic triangular lattice with two hopping parameters $t$ and $t^\prime$ in different spatial directions, interpolating between decoupled chains ($t=0$) and the isotropic triangular lattice ($t=t^\prime$). Variational wave functions that include both Jastrow and backflow terms are used to compare spin-liquid and magnetic phases with different pitch vectors describing both collinear and coplanar (spiral) order. For relatively large values of the on-site interaction $U/t^\prime \gtrsim 10$ and substantial frustration, i.e., $0.3\lesssim t/t^\prime \lesssim 0.8$, the spin-liquid state is clearly favored over magnetic states. Spiral magnetic order is only stable in the vicinity of the isotropic point, while collinear order is obtained in a wide range of inter-chain hoppings from small to intermediate frustration.

Spin-liquid phase due to competing classical orders in the semiclassical theory of the Heisenberg model with ring exchange on an anisotropic triangular lattice

Linear spin wave theory shows that ring exchange induces a quantum disordered region in the phase diagram of the title model. Spin wave spectra show that this is a direct manifestation of competing classical orders. A spin liquid is found in the `Goldilocks zone' of frustration, where the quantum fluctuations are large enough to cause strong competition between different classical orderings but not strong enough to stabilize spiral order. We note that the spin liquid phases of $\kappa$-(BEDT-TTF)${_2}X$ and $Y$[Pd(dmit)$_2$]$_2$ are found in this Goldilocks zone.

Magnetic properties and Mott transition in the Hubbard model on the anisotropic triangular lattice

Magnetic phase diagram and Mott transition are studied in the Hubbard model on the anisotropic triangular lattice at zero temperature and half-filling by the variational cluster approximation, taking into account N\'eel, 120$^\circ$ N\'eel, and collinear orderings. Paramagnetic insulator (spin liquid) is realized above the metallic phase around the isotropic point. In general, this spin liquid state, continuously connected with the metallic state, changes to a magnetic state as the on-site Coulomb repulsion $U$ increases, but it persists up to large $U$ limit in a small window between 120$^\circ$ N\'eel and collinear phases. For very large $U$ another spin liquid state, separated from the metallic state by magnetic states, emerges around a narrow region where both N\'eel and 120$^\circ$ N\'eel orderings are highly suppressed due to the frustration and anisotropy. Implications for the $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ are discussed. As for the Mott transition, the structure of the self-energy in the spectral representation is studied in detail. As $U$ increases around the Mott transition point, single dispersion evolves in the spectral weights of the self-energy which splits the non-interacting band into the upper and lower Hubbard bands.

Quantum fluctuations in anisotropic triangular lattices with ferromagnetic and antiferromagnetic exchange

The Heisenberg model on a triangular lattice is a prime example for a geometrically frustrated spin system. However most experimentally accessible compounds have spatially anisotropic exchange interactions. As a function of this anisotropy, ground states with different magnetic properties can be realized. Motivated by recent experimental findings on Cs$_{2}$CuCl$_{4-x}$Br$_{x}$, we discuss the full phase diagram of the anisotropic model with two exchange constants $J_{1}$ and $J_{2}$, including possible ferromagnetic exchange. Furthermore a comparison with the related square lattice model is carried out. We discuss the zero-temperature phase diagram, ordering vector, ground-state energy, and ordered moment on a classical level and investigate the effect of quantum fluctuations within the framework of spin-wave theory. The field dependence of the ordered moment is shown to be nonmonotonic with field and control parameter.

Long-Period Charge Correlations in Charge-Frustrated Molecular θ-(BEDT-TTF)2X

Charge ordering (CO) with long-periodicities in a geometrically frustrated system, theta-(BEDT-TTF)2X, is numerically studied. We consider interacting fermion models on the anisotropic triangular lattice: the extended Hubbard model and the spinless fermion model, taking account of the inter-site Coulomb interactions on not only the nearest neighbor (NN) but also the next nearest neighbor (NNN) bonds. By applying Lanczos exact diagonalization method, we find that long-period charge correlations become dominant when NNN terms are included, in the charge-frustrated region. The calculated optical excitation spectra show that the charge transfer excitations in the long-period CO extend to a lower energy range compared to those in the short-period CO states including the 3-fold CO stabilized when NNN terms are absent. Our results suggest that the charge correlations experimentally observed in the title compounds near the CO insulating phase can be viewed as a tendency toward `tilted stripe'-type CO with large charge fluctuations.

Stability of Staggered Flux State for Anisotropic Triangular Lattice

In view of pseudogap-like behavior found in organic layered superconductors κ -(BEDT-TTF) 2 X, we study the stability of correlated staggered flux state, which may bring about pseudogap behavior, with respect to the ordinary correlated Fermi sea as a low-lying normal state that underlies the unconventional superconductivity. To treat strong correlations, we apply a variational Monte Carlo method to a Hubbard model on a anisotropic triangular lattice, and construct a phase diagram of the normal state for large values of U / t . The results are qualitatively consistent with the features of non-doped κ -(BEDT-TTF) 2 salts.

Spin-current State in Anisotropic Triangular Heisenberg Model

Abstract Stability of a spin-current state with respect to the ordinary magnetically disordered (paramagnetic) state is studied for an S=1/2 Heisenberg (J1-J2) model on an anisotropic triangular lattice in a magnetic field as a low-lying excited state as well as the ground state. We use a variational Monte Carlo method in computing expectation values. As a result, the spin-current state has an appreciably lower energy than that of the paramagnetic state for J1∼J2 (nearly isotropic) and 0.5 m 0.8 (m: magnetization), but in other areas, the energy gain is small if any, in contrast to a previous argument of Chubukov and Starykh for m ∼1/3.

Majorana spin liquid and dimensional reduction in Cs2CuCl4

The low-temperature behavior of the magnetic insulator Cs${}_{2}$CuCl${}_{4}$ can be modeled by an anisotropic triangular lattice spin-$1/2$ Heisenberg antiferromagnet with two different exchange couplings $J$ and ${J}^{\ensuremath{'}}\ensuremath{\approx}J/3$. We show that in a wide range of magnetic fields the experimentally observed field dependence of the crossover temperature ${T}_{c}$ for spin-liquid behavior can be explained within a mean-field theory based on the representation of spin operators in terms of Majorana fermions. We also show that for small magnetic fields the specific heat and the spin susceptibility both exhibit a maximum as a function of temperature at ${T}_{c}=J/2$. In the spin-liquid regime, the Majorana fermions can only propagate along the direction of the strongest bond, in agreement with the dimensional reduction scenario advanced by Balents [Nature (London) 464, 199 (2010)].

Quantum phase transitions of fermionic atoms in an anisotropic triangular optical lattice

The effect of anisotropy caused by a confining potential on the properties of fermionic cold atoms in a triangular optical lattice is systematically investigated by using the dynamical cluster approximation combined with the continuous time quantum Monte—Carlo algorithm. The quantum phase diagrams which reflect the temperature—interaction relation and the competition between the anisotropic parameter and the interaction are presented with full consideration of the anisotropy of the system. Our results show that the system undergoes a transition from Fermi liquid to Mott insulator when the repulsive interaction reaches a critical value. The Kondo effect also can be observed in this system and the pseudogap is suppressed at low temperatures due to the Kondo effect. A feasible experiment protocol to observe these phenomena in an anisotropic triangular optical lattice with cold atoms is proposed, in which the hopping terms are closely related to the lattice confining potential and the atomic interaction can be adjusted via the Feshbach resonance.

Importance of anisotropy in the spin-liquid candidate Me3EtSb[Pd(dmit)2]2

Organic charge-transfer salts based on the molecule Pd(dmit)${}_{2}$ display strong electronic correlations and geometrical frustration, leading to spin-liquid, valence bond solid, and superconducting states, among other interesting phases. The low-energy electronic degrees of freedom of these materials are often described by a single band model: a triangular lattice with a molecular orbital representing a Pd(dmit)${}_{2}$ dimer on each site. We use ab initio electronic structure calculations to construct and parametrize low-energy effective model Hamiltonians for a class of Me${}_{4\ensuremath{-}n}$ Et${}_{n}X$[Pd(dmit)${}_{2}$]${}_{2}$ ($X=$ As, P, N, Sb) salts and investigate how best to model these systems by using variational Monte Carlo simulations. Our findings suggest that the prevailing model of these systems as a $t\ensuremath{-}{t}^{\ensuremath{'}}$ triangular lattice is incomplete and that a fully anisotropic triangular lattice description produces importantly different results, including a significant lowering of the critical $U$ of the spin-liquid phase.

ChemInform Abstract: Theories on Frustrated Electrons in Two‐Dimensional Organic Solids

AbstractReview: 112 refs.

Magnetic ordering and charge dynamics in κ-(BEDT-TTF)2Cu[N(CN)2]Cl

The Mott insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl consists of molecular dimers arranged on an anisotropic triangular lattice. At low temperatures a pronounced dielectric anomaly is observed, and eventually a canted antiferromagnetic ground state forms. Optical spectroscopy clearly rules out charge imbalance and the existence of quantum electric dipoles with a dipolar-spin coupling. Here we suggest a novel form of spin–charge coupling where the prominent in-plane dielectric response in κ-(BEDT-TTF)2Cu[N(CN)2]Cl is explained by short-range discommensurations of the antiferromagnetic phase in the temperature range 30 K < T < 50 K, and by relaxation of charged domain walls in the ferromagnetic structure at lower temperatures.

Ground states of the Ising model on an anisotropic triangular lattice: stripes and zigzags

A complete solution of the ground-state problem for the Ising model on an anisotropic triangular lattice with the nearest-neighbor interactions in a magnetic field is presented. It is shown that this problem can be reduced to the ground-state problem for an infinite chain with the interactions up to the second neighbors. In addition to the known ground-state structures (which correspond to full-dimensional regions in the parameter space of the model), new structures are found (at the boundaries of these regions), in particular, zigzagging stripes similar to those observed experimentally in colloidal monolayers. Though the number of parameters is relatively large (four), all the ground-state structures of the model are constructed and analyzed and therefore the paper can be considered as an example of a complete solution of a ground-state problem for classical spin or lattice-gas models. The paper can also help to verify the correctness of some results obtained previously by other authors and concerning the ground states of the model under consideration.

Quasi Long Range Order of Defects in Frustrated Antiferromagnetic Ising Models on Spatially Anisotropic Triangular Lattices

It is known that there is no phase transition down to zero temperature in the antiferromagnetic Ising model on spatially anisotropic triangular lattices, in which the exchange coupling of one direction is stronger than those of other two directions. In the model, the low-temperature physics is governed by domain-wall excitations (defects) residing on bonds of the strong-coupling direction. In this letter, we show that an additional small attractive interaction between defects (a ferromagnetic next-nearest-neighbor interaction in the weak-coupling direction) leads to a Berezinskii-Kosterlitz-Thouless (BKT) transition at a finite temperature, by performing the Monte Carlo simulation. The BKT phase can be viewed as the phase with a quasi long-range order of defects. We determine the phase diagram in a wide parameter regime and argue the phase structure from statistical-mechanics and field-theory viewpoints.

Disorder line and incommensurate floating phases in the quantum Ising model on an anisotropic triangular lattice

We present a quantum Monte Carlo study of the Ising model in a transverse field on a square lattice with nearest-neighbor antiferromagnetic exchange interaction $J$ and one diagonal second-neighbor interaction ${J}^{\ensuremath{'}}$, interpolating between square-lattice (${J}^{\ensuremath{'}}=0$) and triangular-lattice (${J}^{\ensuremath{'}}=J$) limits. At a transverse field of ${B}_{x}=J$, the disorder line first introduced by Stephenson [J. Math. Phys. 5, 1009 (1964)], where the correlations go from Neel to incommensurate, meets the zero-temperature axis at ${J}^{\ensuremath{'}}\ensuremath{\approx}0.7J$. Strong evidence is provided that the incommensurate phase at larger ${J}^{\ensuremath{'}}$, at finite temperatures, is a floating phase with power-law decaying correlations. We sketch a general phase-diagram for such a system and discuss how our work connects with the previous quantum Monte Carlo work by Isakov and Moessner [Phys. Rev. B 68, 104409 (2003)] for the isotropic triangular lattice (${J}^{\ensuremath{'}}=J$). For the isotropic triangular lattice, we also obtain the entropy function and constant entropy contours using a mix of quantum Monte Carlo, high-temperature series expansions and high-field expansion methods and show that phase transitions in the model in the presence of a transverse field occur at very low entropy.

Spin-Current Order in Anisotropic Triangular Antiferromagnets

We analyze instabilities of the collinear up-up-down state of a two-dimensional quantum spin-S spatially anisotropic triangular lattice antiferromagnet in a magnetic field. We find, within the large-S approximation, that near the end point of the plateau, the collinear state becomes unstable due to the condensation of two-magnon bound pairs rather than single magnons. The two-magnon instability leads to a novel two-dimensional vector chiral phase with alternating spin currents but no magnetic order in the direction transverse to the field. This phase breaks a discrete Z(2) symmetry but preserves a continuous U(1) one of rotations about the field axis. It possesses orbital antiferromagnetism and displays a magnetoelectric effect.