Sawtooth Lattice Research Articles

Antiferromagnetism and the emergence of frustration in the sawtooth lattice chalcogenide olivines Mn2SiS4−xSex ( x=0–4 )

The magnetism in the sawtooth lattice of Mn in the olivine chalcogenides, ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4\ensuremath{-}x}{\mathrm{Se}}_{x}$ ($x=1--4$), is studied in detail by analyzing their magnetization, specific heat, and thermal conductivity properties and complemented with density functional theory calculations. The air-stable chalcogenides are antiferromagnets and show a linear trend in the transition temperature ${T}_{N}$ as a function of Se content ($x$), which shows a decrease from ${T}_{N}\ensuremath{\approx}86\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ for ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4}$ to 66 K for ${\mathrm{Mn}}_{2}{\mathrm{SiSe}}_{4}$. Additional magnetic anomalies are revealed at low temperatures for all the compositions. Magnetization irreversibilities are observed as a function of $x$. The specific heat and the magnetic entropy indicate the presence of short-range spin fluctuations in ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4\ensuremath{-}x}{\mathrm{Se}}_{x}$. A spin-flop antiferromagnetic phase transition in the presence of applied magnetic field is present in ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4\ensuremath{-}x}{\mathrm{Se}}_{x}$, where the critical field for the spin flop increases from $x=0$ towards 4 in a nonlinear fashion. Density functional theory calculations show that an overall antiferromagnetic structure with ferromagnetic coupling of the spins in the $ab$ plane minimizes the total energy. The band structures calculated for ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4}$ and ${\mathrm{Mn}}_{2}{\mathrm{SiSe}}_{4}$ reveal features near the band edges similar to those reported for Fe-based olivines suggested as thermoelectrics; however the experimentally determined thermal transport data do not support superior thermoelectric features. The transition from long-range magnetic order in ${\mathrm{Mn}}_{2}{\mathrm{SiS}}_{4}$ to short-range order and spin fluctuations in ${\mathrm{Mn}}_{2}{\mathrm{SiSe}}_{4}$ is explained using the variation of the Mn-Mn distances in the triangle units that constitutes the sawtooth lattice upon progressive replacement of sulfur with selenium. Overall, the results presented here point towards the role played by magnetic anisotropy and geometric frustration in the antiferromagnetic state of the sawtooth olivines.

Diffusion on a one-dimensional sawtooth lattice with the nearest and the next nearest neighbor interactions

The diffusion of particles adsorbed on a homogeneous one-dimensional sawtooth lattice with the nearest neighbor and the next nearest neighbor interactions between the particles is investigated using a theoretical approach and the kinetic Monte Carlo simulations. I have derived the analytical expressions for the diffusion coefficients. The calculated concentration dependencies of the coefficients have been compared with the numerical data generated by the simulations. The comparison reveals an interesting peculiarity of the particle diffusion. The lateral interactions induce correlations in the successive jumps of the particles. The thorough investigations of the dependencies clearly demonstrate how with the strengthening of the interactions the classical uncorrelated jumps are gradually replaced by the sequences of the correlated jumps. This behavior principally differs from the usual diffusion of particles in the homogeneous one-dimensional systems. It is an interesting and unexpected novelty of this system. Another uncommon feature of this model is a rich variety of the ‘diffusion coefficients’. There are four combinations of the interaction parameter signs, and every combination has its own specific behavior of the particle diffusion, qualitatively opposite from the other combinations. Also, it should be noted a quite unusual result. In the homogeneous lattice gas system (all sites have an equal depth) with the strong, repulsive nearest neighbor and attractive next nearest neighbor interactions the diffusion is perfectly described by the diffusion coefficients derived for the heterogeneous lattices with deep and shallow sites. It is a direct evidence of the specific mode of the particle migration when the correlated pairs of jumps give the main contribution to the diffusion coefficients.

Nonlinearity-dependent asymmetric transmission in a sawtooth photonic lattice with defects

We study both theoretically and numerically the asymetric transmission of a Gaussian beam in a two-dimensional nonlinear sawtooth lattice with two defects. The results show that quasi-total reflection, asymmetric propagation and asymmetric reflection can all be achieved in such a system by adjusting the input intensity, the magnitude of defects and the number of nonlinear waveguides. This study may provide a new way to realize an optical switch and optical diode.

Wave-packet dynamics of Bogoliubov quasiparticles: Quantum metric effects

We study the dynamics of the Bogoliubov wave packet in superconductors and calculate the supercurrent carried by the wave packet. We discover an anomalous contribution to the supercurrent, related to the quantum metric of the Bloch wave function. This anomalous contribution is most important for flat or quasiflat bands, as exemplified by the attractive Hubbard models on the Creutz ladder and sawtooth lattice. Our theoretical framework is general and can be used to study a wide variety of phenomena, such as spin transport and exciton transport.

Haldane phase on the sawtooth lattice: Edge states, entanglement spectrum, and the flat band

Using density matrix renormalization group numerical calculations, we study the phase diagram of the half filled Bose-Hubbard system in the sawtooth lattice with strong frustration in the kinetic energy term. We focus in particular on values of the hopping terms which produce a flat band and show that, in the presence of contact and near neighbor repulsion, three phases exist: Mott insulator (MI), charge density wave (CDW), and the topological Haldane insulating (HI) phase which displays edge states and particle imbalance between the two ends of the system. We find that, even though the entanglement spectrum in the Haldane phase is not doubly degenerate, it is in excellent agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state built in the Wannier basis associated with the flat band. This emphasizes that the absence of degeneracy in the entanglement spectrum is not necessarily a signature of a non-topological phase, but rather that the (hidden) protecting symmetry involves non-local states. Finally, we also show that the HI phase is stable against small departure from flatness of the band but is destroyed for larger ones.

Quantum collision theory in flat bands

We consider quantum scattering of particles in media exhibiting strong dispersion degeneracy. In particular, we study flat-banded lattices and linearly dispersed energy bands. The former constitute a prime example of single-particle frustration while the latter show degeneracy at the few- and many-particle level. We investigate both impurity and two-body scattering and show that, quite generally, scattering does not occur, which we relate to the fact that transition matrices vanish on the energy shell. We prove that scattering is instead replaced by projections onto band-projected eigenstates of the interaction potential. We then use the general results to obtain localized flat band states that are eigenstates of impurity potentials with vanishing eigenvalues in one-dimensional flat bands and study the particular case of a sawtooth lattice. We also uncover the relation between certain solutions of one-dimensional systems that have been categorized as ‘strange’, and the scattering states in linearly dispersed continuum systems.

Transport in Sawtooth photonic lattices.

We investigate, theoretically and experimentally, a photonic realization of a Sawtooth lattice. This special lattice exhibits two spectral bands, with one of them experiencing a complete collapse to a highly degenerate flat band for a special set of inter-site coupling constants. We report the observation of different transport regimes, including strong transport inhibition due to the appearance of the non-diffractive flat band. Moreover, we excite localized Shockley surface states residing in the gap between the two linear bands.

Frustration-induced supersolids in the absence of intersite interactions

We discuss a mechanism for the realization of supersolids in lattices in the absence of intersite interactions that surprisingly works as well at unit filling. This mechanism, that we study for the case of the sawtooth lattice, is based on the existence of frustrated and unfrustrated plaquettes. For sufficiently large interactions and frustration the particles gather preferentially at unfrustrated plaquettes breaking spontaneously translational invariance, resulting in a supersolid. We show that for the sawtooth lattice the supersolid exists for a large region of parameters for densities above half filling. Our results open a feasible path for realizing supersolids in existing ultracold atomic gases in optical lattices without the need for long-range interactions.

One-dimensional sawtooth and zigzag lattices for ultracold atoms.

We describe tunable optical sawtooth and zigzag lattices for ultracold atoms. Making use of the superlattice generated by commensurate wavelengths of light beams, tunable geometries including zigzag and sawtooth configurations can be realised. We provide an experimentally feasible method to fully control inter- (t) and intra- (t′) unit-cell tunnelling in zigzag and sawtooth lattices. We analyse the conversion of the lattice geometry from zigzag to sawtooth, and show that a nearly flat band is attainable in the sawtooth configuration by means of tuning the lattice parameters. The bandwidth of the first excited band can be reduced up to 2% of the ground bandwidth for a wide range of lattice setting. A nearly flat band available in a tunable sawtooth lattice would offer a versatile platform for the study of interaction-driven quantum many-body states with ultracold atoms.

Compactification tuning for nonlinear localized modes in sawtooth lattices.

We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schrödinger model with general on-site nonlinearity. Analytic conditions for existence of exact compact three-site solutions are obtained, and explicitly illustrated for the cases of power-law (cubic) and saturable nonlinearities. These nonlinear compact modes appear as continuations of linear compact modes belonging to a flat dispersion band. While for the linear system a compact mode exists only for one specific ratio of the two different coupling constants, nonlinearity may lead to compactification of otherwise noncompact localized modes for a range of coupling ratios, at some specific power. For saturable lattices, the compactification power can be tuned by also varying the nonlinear parameter. Introducing different on-site energies and anisotropic couplings yields further possibilities for compactness tuning. The properties of strongly localized modes are investigated numerically for cubic and saturable nonlinearities, and in particular their stability over large parameter regimes is shown. Since the linear flat band is isolated, its compact modes may be continued into compact nonlinear modes both for focusing and defocusing nonlinearities. Results are discussed in relation to recent realizations of sawtooth photonic lattices.

Low-energy behavior of strongly interacting bosons on a flat-band lattice above the critical filling factor

Bosons interacting repulsively on a lattice with a flat lowest band energy dispersion may, at sufficiently small filling factors, enter into a Wigner-crystal-like phase. This phase is a consequence of the dispersionless nature of the system, which in turn implies the occurrence of single-particle localized eigenstates. We investigate one of these systems---the sawtooth lattice---filled with strongly repulsive bosons at filling factors infinitesimally above the critical point where the crystal phase is no longer the ground state. We find, in the hard-core limit, that the crystal retains its structure in all but one of its cells, where it is broken. The broken cell corresponds to an exotic kind of repulsively bound state, which becomes delocalized. We investigate the excitation spectrum of the system analytically and find that the bound state behaves as a single particle hopping on an effective lattice with reduced periodicity, and is therefore gapless. Thus, the addition of a single particle to a flat-band system at critical filling is found to be enough to make kinetic behavior manifest.

Spin-orbit coupling and electronic charge effects in Mott insulators

We derive the effective charge- and current-density operators for the strong-coupling limit of a single-band Mott insulator in the presence of spin-orbit coupling and show that the spin-orbit contribution to the effective charge density leads to novel mechanisms for multiferroic behavior. In some sense, these mechanisms are the electronic counterpart of the ionic-based mechanisms, which have been proposed for explaining the electric polarization induced by spiral spin orderings. The new electronic mechanisms are illustrated by considering cycloidal and proper-screw magnetic orderings on sawtooth and kagome lattices. As for the isotropic case, geometric frustration is crucial for achieving this purely electronic coupling between spin and charge degrees of freedom.

Quantum dynamics in driven sawtooth lattice under uniform magnetic field

We study the Bloch-Zener oscillation, which is a superposition of Bloch oscillation and Landau-Zener tunneling between Bloch bands, for a quantum particle in a frustrated sawtooth lattice with and without uniform magnetic field. Under the single-band tight-binding approximation, the sawtooth lattice is a two-miniband system and may have a flat band structure. The presence of magnetic field can make the gap between two minibands close, and around the touch point the dispersion is an asymmetric Dirac cone. We analyze in detail the Landau-Zener tunneling and Bragg scattering in the Bloch-Zener oscillation and the effect of magnetic field. Our results also give a clear picture of the dynamical localization in real space induced by the flat band structure of the lattice.

Bose condensation in flat bands

We derive effective Hamiltonians for lattice bosons with strong geometrical frustration of the kinetic energy by projecting the interactions on the flat lowest Bloch band. Specifically, we consider the Bose Hubbard model on the one dimensional sawtooth lattice and the two dimensional kagome lattice. Starting from a strictly local interaction the projection gives rise to effective long-range terms stabilizing a supersolid phase at densities above nu_c=1/9 of the kagome lattice. In the sawtooth lattice on the other hand we show that the solid order, which exists at the magic filling nu_c=1/4, is unstable to further doping. The universal low-energy properties at filling 1/4+delta nu are described by the well known commensurate-incommensurate transition. We support the analytic results by detailed numerical calculations using the Density Matrix Renormalization Group and exact diagonalization. Finally, we discuss possible realizations of the models using ultracold atoms as well as frustrated quantum magnets in high magnetic fields. We compute the momentum distribution and the noise correlations, that can be extracted from time of flight experiments or neutron scattering, and point to signatures of the unique supersolid phase of the kagome lattice.

Magnetic Structures of Co2TeO3Cl2

Measurements of the magnetization and specific heat have been carried out for Co 2 TeO 3 Cl 2 with a sawtooth lattice having two crystallographically distinct Co 2+ sites. A powder neutron diffraction study has revealed that two magnetic transitions take place at T N1 ∼73 K and T N2 ∼15 K. Below T N1 , Co spins at the apical sites of the sawtooth lattice order antiferromagnetically with their directions parallel to the b -axis. Co spins at the basal sites order antiferromagnetically at T N2 with their directions canted in the a b -plane from the a -axis by Dzyaloshinski–Moriya interaction. At this transition, no change in the spin structure at the apical sites has been observed. Discussion on possible origins of these complicated spin structure is presented.

Enhanced low-temperature entropy and flat-band ferromagnetism in the [formula omitted] model on the sawtooth lattice

Using the example of the sawtooth chain, we argue that the t – J model shares important features with the Hubbard model on highly frustrated lattices. The lowest single-fermion band is completely flat (for a specific choice of the hopping parameters t i , j in the case of the sawtooth chain), giving rise to single-particle excitations which can be localized in real space. These localized excitations do not interact for sufficient spatial separations such that exact many-electron states can also be constructed. Furthermore, all these excitations acquire zero energy for a suitable choice of the chemical potential μ . This leads to: (i) a jump in the particle density at zero temperature, (ii) a finite zero-temperature entropy, (iii) a ferromagnetic ground state with a charge gap when the flat band is fully occupied and (iv) unusually large temperature variations when μ is varied adiabatically at finite temperature.

First-principles determination of exchange interactions in delafossiteYCuO2.5

We perform a first-principles calculation within the local density approximation $(\mathrm{LDA})+\mathrm{U}$ approximation for the delafossite $\mathrm{Y}\mathrm{Cu}{\mathrm{O}}_{2.5}$, which up to now was considered as a sawtooth lattice. The magnetic interactions are mapped onto a Heisenberg model whose exchange interactions are fitted to first principles total energy calculations for different spin configurations. Four interactions appear to have a significant value. While interplane interactions can be safely neglected, interchain interactions are much smaller than intrachain ones, confirming the low-dimensional character of this system. Besides the nearest-neighbor intrachain antiferromagnetic interactions, we find two unexpected ferromagnetic interactions that can play an important role. The results are discussed in relation with recent experimental data.

The Δ chain with different base–base and base–vertex interactions

On the basis of the recently determined crystallographic structure of the delafossiteY CuO2.5, we argue that the Cu–O network has nearly independentΔ chains, with however different interactions between thes = 1/2 spins, owing to the different angles, distances and coordinations of the Cu ions. Althoughband-structure calculations are still lacking, motivated by this observation we study herethe sawtooth lattice for different ratios of the base–base and base–vertex interactions,Jbb/Jbv. By exact diagonalization and extrapolation to the infinite-size limit, weshow that the elementary excitation spectrum is the same for total spinsStot = 0 and 1,but not for Stot = 2, and has a gap only in the interval . The gap, dispersionless for Jbb = Jbv, acquires increasing k-dependence as the ratio Jbb/Jbv moves away from unity, with the minimum energy excitations fork = 0 (k = π)when Jbb/Jbv<1 (Jbb/Jbv>1). Finally, we show that the gap closure is related to the instability of the dimers in theground state as the difference between the interactions increases.

Alternating spin-1/spin-12sites in a diamond lattice: Ground state and excitations

We consider a diamond lattice with spin-1 and spin-½ sites in alternating positions. The system is shown to be mapped on to a spin-1 sawtooth lattice for a range of parameter values. With varying strengths of interactions between spin sites, we find a number of phases, including, gapped spin-1, dimerized, and gapless spin-1 phases. In the gapped phase, there exist two special lines, namely, disorder and the Lifshitz lines, which differentiate spin ordering from the Neel type to spiral type, in real and in the momentum space, respectively. The dimerized phase, which ends at disorder line, is gapless in this case and corresponds to nearest-neighbor spin-½ sites forming alternate singlet and triplet.

Quantum solitons in the sawtooth lattice

We study the sawtooth lattice of a coupled spin $1/2$ Heisenberg system, a variant of the railroad trestle lattice. The ground state of this system is two-fold degenerate with periodic boundary conditions and supports kink antikink excitations, which are distinct in this case, unlike the railroad trestle lattice. The resulting low temperature thermodynamics is compared with the recently discovered Delafossites $Y Cu O_{2.5} ~$.