Sawtooth Chain Research Articles

Ferromagnetic Mn3+ Sawtooth Chains in A2(Mn2O)(SeO3)3 (A = K, Rb) Quaternary Selenites.

Two quaternary manganese selenites, A2(Mn2O)(SeO3)3 (A = K, Rb), have been synthesized by hydrothermal reactions. They both crystallize in a complex triclinic (P-1) structure built of Jahn-Teller (JT) distorted Mn3+O4+2 octahedra, connected into nearly isosceles [Mn3O14] triangles, themselves arranged into so-called "sawtooth (ST) chains". The K and Rb compounds show subtle variations in the orientations of the MnO4 planes inside the elementary triangles. The ST chains are structurally and magnetically isolated by SeO3 groups and alkali cations. In the ST chains, predominant ferromagnetic interactions were calculated and verified experimentally, which finally order antiferromagnetically between the chains around TN ≈ 22 K. The spin exchanges calculated by DFT + U and fitted by Monte Carlo simulations allow for the quantification of an effective "overall" model. The specific role of the μ3-O bridge on the ferromagnetic (FM) exchanges is discussed, together with spin reorientations observed in the ordered state. Magnetocrystalline anisotropy along the [110] direction stabilized by ∼50 meV per Mn by spin-orbit coupling (SOC) was found by DFT + U + SOC.

K2Mn3O(OH)(VO4)(V2O7) with Sawtooth Chains of Multivalent Manganese Triangular Trimer Units: Magnetic Susceptibility Shrouding a Long-Range Antiferromagnetic Order of Ferromagnetic Triangles.

ortho-Pyrovanadate (or ortho-diorthovanadate) K2Mn23+Mn2+O(OH)(VO4)(V2O7) synthesized hydrothermally crystallizes in the orthorhombic space group Pnma with a = 17.9155(5), b = 5.8940(2), c = 10.9971(3) Å, V = 1161.23(6) Å3, and Z = 4. Its crystal structure features linear chains of edge-sharing Mn3+O6 octahedra with every second pair of Mn3+O6 octahedra condensed with a Mn2+O6 octahedron on one side of a chain in a sawtooth pattern so that each sawtooth chain consists of a triangular trimer. These sawtooth chains, running parallel to the b axis and linked by the VO4 and V2O7 groups, form a framework with channels populated by K atoms. The new compound is a structural analogue of the mineral zoisite Ca2Al3O(OH)(SiO4)(Si2O7), showing a striking example of very different chemical compositions. K2Mn3O(OH)(VO4)(V2O7) undergoes a phase transition into an ordered antiferromagnetic (AFM) state at TN = 14.4 K, which was detected by high-frequency electron spin resonance as well as by both specific heat Cp and Fisher's specific heat d(χT)/dT measurements. However, this phase transition was not detected by magnetic susceptibility measurements. The origin of this puzzling observation was resolved by evaluating the spin exchanges of K2Mn3O(OH)(VO4)(V2O7), which revealed that each triangular trimer is a ferromagnetically coupled cluster, and the observed ordering involves an AFM ordering between the ferromagnetic (FM) clusters. This ordering is shrouded in magnetic susceptibility measurements due to the susceptibility contributions from the individual FM triangular trimers even below TN. We showed that the magnetic susceptibility of K2Mn3O(OH)(VO4)(V2O7) between ∼30 K and room temperature is satisfactorily described by an AFM chain made up of ferromagnetically coupled triangular clusters, as described by a few spin-exchange parameters.

Quantum spin spiral ground state of the ferrimagnetic sawtooth chain

The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions JJ and antiferromagnetic next-nearest-neighbour interactions J'J′ (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations ‹S_i›‹Si#8250;, the spiral can be detected via the connected spin-spin correlations ‹{S_i\cdot S_j}#8250;‹Si⋅Sj#8250;-‹{S_i}#8250;\cdot‹{S_j}#8250;‹Si#8250;⋅‹Sj#8250; between the apical spins. It has a long wavelength that grows with J'J′ and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems. In this work, we are able to accurately treat up to L=400L=400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to directly compute the lowest-energy state for a given total spin. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.

(Na,Li)3(Cl,OH)[Cu3OAl(PO4)3]: a first salt-inclusion aluminophosphate oxocuprate with a new type of crystal structure.

The synthesis and characterization of a first salt-inclusion aluminophosphate oxocuprate, (Na,Li)3(Cl,OH)[Cu3OAl(PO4)3], obtained as single crystals, is reported. A novel phase, with a strongly pseudo-orthorhombic structure, is described as a monoclinic crystal structure established by the study of a pseudomerohedric microtwin. It was investigated using scanning electron microscopy, microprobe analysis and low-temperature X-ray diffraction. The composite crystal structure represents an original framework assembled from Cu-centered polyhedra, AlO6 octahedra and PO4 tetrahedra with channels, which incorporate the Na/Li salt component [(Na,Li)3(Cl,OH)]2+ that ensures electroneutrality of the compound. Layers of strongly corrugated chains of Cu-centered octahedra with shared edges and linked by PO4 tetrahedra are shown to be topologically identical with the layers also built from Cu-centered polyhedra and AsO4/VO4 tetrahedra forming the crystal structure of a fumarolic mineral aleutite, (M0.5Cl)[Cu5O2(AsO4)(VO4)] [Siidra et al. (2019). MinMag, 83, 847-853]. `Sawtooth chains' and pairs of Cu-centered octahedra inherent in the title structure may be of interest in solid-state physics, engaging studies in the field of low-dimensional and frustrated magnetism.

Field tunable magnetic transitions of CsCo2(MoO4)2(OH): a triangular chain structure with a frustrated geometry

We present a comprehensive series of magnetic and neutron scattering measurements of the sawtooth chain compound, CsCo2(MoO4)2(OH). The magnetic properties of CsCo2(MoO4)2(OH) can be easily manipulated by applied magnetic fields.

Electric field driven flat bands: Enhanced magnetoelectric and electrocaloric effects in frustrated quantum magnets

The $J_1$-$J_2$ quantum spin sawtooth chain is a paradigmatic one-dimensional frustrated quantum spin system exhibiting unconventional ground-state and finite-temperature properties. In particular, it exhibits a flat energy band of one-magnon excitations accompanied by an enhanced magnetocaloric effect for two singular ratios of the basal interactions $J_1$ and the zigzag interactions $J_2$. In our paper, we demonstrate that one can drive the spin system into a flat-band scenario by applying an appropriate electric field, thus overcoming the restriction of fine-tuned exchange couplings $J_1$ and $J_2$ and allowing one to tune more materials towards flat-band physics, that is to show a macroscopic magnetization jump when crossing the magnetic saturation field, a residual entropy at zero temperature as well as an enhanced magnetocaloric effect. While the magnetic field acts on the spin system via the ordinary Zeeman term, the coupling of an applied electric field with the spins is given by the sophisticated Katsura-Nagaosa-Balatsky (KNB) mechanism, where the electric field effectively acts as a Dzyaloshinskii-Moriya spin-spin interaction. The resulting novel features are corresponding reciprocal effects: We find a magnetization jump driven by the electric field as well as a jump of the electric polarization driven by the magnetic field, i.e.\ the system exhibits an extraordinarily strong magnetoelectric effect. Moreover, in analogy to the enhanced magnetocaloric effect the system shows an enhanced electrocaloric effect.

NaCo2(SeO3)2(OH): competing magnetic ground states of a new sawtooth structure with 3d7 Co2+ ions

We report the high-pressure synthesis and a comprehensive study of NaCo2(SeO3)2(OH) sawtooth chain structure using bulk magnetic properties and neutron scattering.

On the failure of effective-field theory in predicting a spurious spontaneous ordering and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires

The present work clarifies a failure of the effective-field theory in predicting a false spontaneous long-range order and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires with either zero- or one-dimensional magnetic dimensionality. It is conjectured that the standard formulation of the effective-field theory due to Honmura and Kaneyoshi generally predicts for the Ising spin systems a spurious spontaneous long-range order with nonzero critical temperature regardless of their magnetic dimensionality whenever at least one Ising spin has coordination number greater than two. The failure of the effective-field theory is exemplified on a few paradigmatic exactly solved examples of zero- and one-dimensional Ising nanosystems: star, cube, decorated hexagon, star of David, branched chain, sawtooth chain, two-leg and hexagonal ladders. The presented exact solutions illustrate eligibility of a few rigorous analytical methods for exact treatment of the Ising nanosystems: exact enumeration, graph-theoretical approach, transfer-matrix method and decoration-iteration transformation. The paper also provides a substantial survey of the scientific literature, in which the effective-field theory led to a false prediction of the spontaneous long-range ordering and phase transition.

Symmetry-protected topological phases in spinful bosons with a flat band

We theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on the lattices which have a flat band at the bottom of the band structure. Ground states of such systems are not conventional Mott insulators in the sense that the ground states possess not only spin fluctuations but also non-negligible charge fluctuations. The SPT phases in such systems are determined by both spin and charge fluctuations at zero temperature. We find that the many-body ground states of such systems can be exactly obtained in some special cases, and these exact ground states turn out to serve as representative states of the SPT phases. As a concrete example, we demonstrate that spin-1 bosons on a sawtooth chain can be in an SPT phase protected by $\mathbb{Z}_2 \times \mathbb{Z}_2$ spin rotation symmetry or time-reversal symmetry, and this SPT phase is a result of spin fluctuations. We also show that spin-3 bosons on a kagome lattice can be in an SPT phase protected by $D_2$ point group symmetry, but this SPT phase is, however, a result of charge fluctuations.

Magnetization Process of Atacamite: A Case of Weakly Coupled S=1/2 Sawtooth Chains.

We present a combined experimental and theoretical study of the mineral atacamite Cu_{2}Cl(OH)_{3}. Density-functional theory yields a Hamiltonian describing anisotropic sawtooth chains with weak 3D connections. Experimentally, we fully characterize the antiferromagnetically ordered state. Magnetic order shows a complex evolution with the magnetic field, while, starting at 31.5T, we observe a plateaulike magnetization at about M_{sat}/2. Based on complementary theoretical approaches, we show that the latter is unrelated to the known magnetization plateau of a sawtooth chain. Instead, we provide evidence that the magnetization process in atacamite is a field-driven canting of a 3D network of weakly coupled sawtooth chains that form giant moments.

Current jumps in flat-band ladders with Dzyaloshinskii-Moriya interactions

Localized magnons states, due to flat bands in the spectrum, is an intensely studied phenomenon and can be found in many frustrated magnets of different spatial dimensionality. The presence of Dzyaloshinskii-Moriya (DM) interactions may change radically the behavior in such systems. In this context, we study a paradigmatic example of a one-dimensional frustrated antiferromagnet, the sawtooth chain in the presence of DM interactions. Using both path integrals methods and numerical Density Matrix Renormalization Group, we revisit the physics of localized magnons and determine the consequences of the DM interaction on the ground state. We have studied the spin current behavior, finding three different regimes. First, a Luttinger-liquid regime where the spin current shows a step behavior as a function of parameter $D$, at a low magnetic field. Increasing the magnetic field, the system is in the Meissner phase at the $m = 1/2$ plateau, where the spin current is proportional to the DM parameter. Finally, further increasing the magnetic field and for finite $D$ there is a small stiffness regime where the spin current shows, at fixed magnetization, a jump to large values at $D = 0$, a phenomenon also due to the flat band.

Lifshitz phase transitions in a one-dimensional Gamma model.

In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize the semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at k=π change from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy, and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.

Flat-band physics in the spin-1/2 sawtooth chain

We consider the strongly anisotropic spin-1/2 XXZ model on the sawtooth-chain lattice with ferromagnetic longitudinal interaction Jzz = ΔJ and aniferromagnetic transversal interaction Jxx = Jyy = J > 0. At Δ = −1∕2 the lowest one-magnon excitation band is dispersionless (flat) leading to a massively degenerate set of ground states. Interestingly, this model admits a three-coloring representation of the ground-state manifold [H.J. Changlani et al., Phys. Rev. Lett. 120, 117202 (2018)]. We characterize this ground-state manifold and elaborate the low-temperature thermodynamics of the system. We illustrate the manifestation of the flat-band physics of the anisotropic model by comparison with two isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is complemented by exact diagonalization and finite-temperature Lanczos method calculations.Graphical abstract

Enhancement of magnetization plateaus in low-dimensional spin systems

We study the low-energy properties and, in particular, the magnetization process of a spin-1/2 Heisenberg $J_1-J_2$ sawtooth and frustrated chain (also known as zig-zag ladder) with a spatially anisotropic $g$-factor. We treat the problem both analytically and numerically while keeping the $J_2/J_1$ ratio generic. Numerically, we use complete and Lanczos diagonalization as well as the infinite time-evolving block decimation (iTEBD) method. Analytically we employ (non-)Abelian bosonization. Additionally for the sawtooth chain, we provide an analytical description in terms of flat bands and localized magnons. By considering a specific pattern for the $g$-factor anisotropy for both models, we show that a small anisotropy significantly enhances a magnetization plateau at half saturation. For the magnetization of the frustrated chain, we show the destruction of the $1/3$ of the full saturation plateau in favor of the creation of a plateau at half-saturation. For large anisotropies, the existence of an additional plateau at zero magnetization is possible. Here and at higher magnetic fields, the system is locked in the half-saturation plateau, never reaching full saturation.

Reentrance of the topological phase in a spin-1 frustrated Heisenberg chain

For the Haldane phase, the magnetic field usually tends to break the symmetry and drives the system into a topologically trivial phase. Here, we report a novel reentrance of the Haldane phase at zero temperature in the spin-1 antiferromagnetic Heisenberg model on sawtooth chain. A partial Haldane phase is induced by the magnetic field, which is the combination of the Haldane state in one sublattice and a ferromagnetically ordered state in the other sublattice. Such a partial topological order is a result of the zero-temperature entropy due to quantum fluctuations caused by geometrical frustration.

Magnetic hyperfine interactions in a sawtooth chain iron oxoselenite Fe2O(SeO3)2: Experimental and theoretical Investigation

The iron oxoselenite Fe2O(SeO3)2 hosts unique tilted saw-tooth chains of vertex-sharing iron triangles. It experiences long-range magnetic order below TN ≈ 105 K characterized by a sharp dichotomy in zero-field-cooled and field-cooled behavior (T < TN). The spin-flop transition at low temperatures, BSF = 5.5 T at T = 2 K, suggests the predominance of antiferromagnetic interactions. The 57Fe Mössbauer spectra at T » TN confirm the presence of three non-equivalent crystal sites for high-spin Fe3+ ions. Below TN, the well-resolved Zeeman patterns demonstrate a commensurate magnetic ordering in iron sublattices. The anomalously low value of the saturation hyperfine fields Bhf,Fei at 57Fe nuclei is discussed in terms of the spin reduction (<ΔS> ≈ 0.31), which results from a spin-wave treatment. The mean-field analysis of the temperature evolution of the fields Bhf,Fei(T) allowed evaluation of exchange interactions between different iron sublattices. A spin relaxation behavior of Fe2O(SeO3)2 in the temperature range T* < T < TN (T* ≈ 90 K), which is characteristic for frustrated systems, was shown by the Mössbauer measurements. Density functional theory (DFT) calculations provide the values of exchange parameters within and between the saw-tooth chains.

Thermal density matrix renormalization group for highly frustrated quantum spin chains: A user perspective

Thermal DMRG is investigated with emphasis of employability in molecular magnetism studies. To this end magnetic observables at finite temperature are evaluated for two one-dimensional quantum spin systems: a Heisenberg chain with nearest-neighbor antiferromagnetic interaction and a frustrated sawtooth (delta) chain. It is found that thermal DMRG indeed accurately approximates magnetic observables for the chain as well as for the sawtooth chain, but in the latter case only for sufficiently high temperatures. We speculate that the reason is due to the peculiar structure of the low-energy spectrum of the sawtooth chain induced by frustration.

Thermodynamics of a delta chain with ferromagnetic and antiferromagnetic interactions

Motivated by a novel cyclic compound $Fe_{10}Gd_{10}$ with record ground state spin in which the arrangement of magnetic ions with $s=\frac{5}{2}$ and $s=\frac{7}{2}$ corresponds to a saw-tooth chain we investigate the thermodynamics of the delta-chain with competing ferro- and antiferromagnetic interactions. We study both classical and quantum versions of the model. The classical model is exactly solved and quantum effects are studied using full diagonalization and a finite temperature Lanczos technique for finite delta-chains as well as modified spin wave theory. It is shown that the main features of the magnetic susceptibility of the quantum spin delta chain are correctly described by the classical spin model, while quantum effects significantly change the low-temperature behavior of the specific heat. The relation of the obtained results to the $Fe_{10}Gd_{10}$ system is discussed.

Resonant frequencies and spatial correlations in frustrated arrays of Josephson type nonlinear oscillators

We present a theoretical study of resonant frequencies and spatial correlations of Josephson phases in frustrated arrays of Josephson junctions. Two types of one-dimensional arrays, namely, the diamond and sawtooth chains, are discussed in detail. For these arrays in the linear regime the Josephson phase dynamics is characterized by multiband dispersion relation , and the lowest band becomes completely flat at a critical value of frustration, f = f c. In a strongly nonlinear regime such critical value of frustration determines the crossover from non-frustrated (0 < f < f c) to frustrated (f c < f < 1) regimes. The crossover is characterized by the thermodynamic spatial correlation functions of phases on vertices, , i.e. displaying the transition from long- to short-range spatial correlations. We find that higher-order correlations functions, e.g. p = 2 and p = 3, restore the long-range behavior deeply in the frustrated regime, . Monte-Carlo simulations of the thermodynamics of frustrated arrays of Josephson junctions are in good agreement with analytical results. We also outline the extension of our results to the case of kagome lattice, prototypical 2D frustrated lattice, and other higher dimensional lattices.

Heisenberg–Ising delta-chain with bond alternation

The spin- delta-chain (sawtooth chain) with antiferromagnetic Heisenberg basal chain and Ising apical-basal interactions is studied. The basal-apical interactions involve the bond alternation. The limiting cases of the model include the symmetrical delta-chain and the antiferromagnetic chain in the staggered magnetic field. We study ground state properties of the model by the exact diagonalization and density matrix renormalization group methods. The ground state phase diagram as a function of the bond alternation consists of magnetic and various non-magnetic phases. All phases excluding the ferrimagnetic phase are gapped and an origin of the gaps is cleared.