Zigzag Ladder Research Articles

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

Abstract It is shown that the spin- 1 2 anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- 1 2 Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang–Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang–Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang–Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang–Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang–Lee and Fisher edge singularity exponents are universal and equal to each other, σ = − 1 2 .

Unavoidable parallel minors of 4‐connected graphs

AbstractA parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4‐connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k‐partition triple fan with a complete graph on the vertices of degree k, the k‐spoke double wheel, the k‐spoke double wheel with axle, the (2k+1)‐rung Möbius zigzag ladder, the (2k)‐rung zigzag ladder, or Kk. We also find the unavoidable parallel minors of 1‐, 2‐, and 3‐connected graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 313‐326, 2009

Multidimensional mappings in antiferromagnetic models and Lyapunov exponents with the magnetization plateau

We consider the Heisenberg model with two- and three-spin exchange interactions on a recursive ladder in a strong magnetic field. Recurrent relations for branches of the partition function of the Ising model with two- and three-spin exchange interactions are deduced. As a recursive lattice the zigzag ladder is chosen. In the antiferromagnetic case magnetization plateau are observed at low temperatures. Lyapunov exponents for the three-dimensional mapping at low temperatures are calculated. It is shown that for some values of two- and three-spin exchange parameters in the antiferromagnetic case the maximum of the Lyapunov exponent approaches zero.

Magnetic ordering of weakly coupled frustrated quantum spin chains

The ordering temperature of a quasi-one-dimensional system, consisting of weakly interacting quantum spin-1/2 chains with antiferromagnetic spin-frustrating couplings (or zigzag ladder), is calculated. The results show that a quantum critical point between two phases of the one-dimensional subsystem plays a crucial role. If the one-dimensional subsystem is in the antiferromagneticlike phase in the ground state, similar to the phase of a spin chain without frustration, weak couplings yield magnetic ordering of the N\'eel type. For intrachain spin-frustrating interactions larger than the critical one (at which the quantum phase transition takes place), the quasi-one-dimensional spin system manifests a spiral magnetic incommensurate ordering. The obtained results of our quantum theory are compared with the quasiclassical approximations. The calculated features of magnetic ordering are expected to be generic for weakly-coupled quantum spin chains with gapless excitations and spin-frustrating nearest- and next-nearest-neighbor interactions.

The partition function zeros for a Potts model of helix-coil transition with three-site interactions

The partition function (Fisher) zeros of the Q -state Potts model with three-site interactions on a zigzag ladder are studied by a transfer-matrix approach. This Q -state model is proposed as a description of helix-coil transition in homopolymers. The Fisher zeros are exactly derived for arbitrary value of the system parameter Q . It is shown that there is only a quasi-phase transition for all temperatures. The dependence of the helix-coil transition on solvent is analyzed using Fisher zeros of the model. The densities of the Fisher zeros are singular at the edge singularity points and the critical exponent σ = − 1 2 .

Synthesis and structural investigation of the compounds containing HF 2− anions: Ca(HF 2) 2, Ba 4F 4(HF 2)(PF 6) 3 and Pb 2F 2(HF 2)(PF 6)

Three new compounds Ca(HF 2) 2, Ba 4F 4(HF 2)(PF 6) 3 and Pb 2F 2(HF 2)(PF 6) were obtained in the system metal(II) fluoride and anhydrous HF (aHF) acidified with excessive PF 5. The obtained polymeric solids are slightly soluble in aHF and they crystallize out of their aHF solutions. Ca(HF 2) 2 was prepared by simply dissolving CaF 2 in a neutral aHF. It represents the second known compound with homoleptic HF environment of the central atom besides Ba(H 3F 4) 2. The compounds Ba 4F 4(HF 2)(PF 6) 3 and Pb 2F 2(HF 2)(PF 6) represent two additional examples of the formation of a polymeric zigzag ladder or ribbon composed of metal cation and fluoride anion (MF +) n besides PbF(AsF 6), the first isolated compound with such zigzag ladder. The obtained new compounds were characterized by X-ray single crystal diffraction method and partly by Raman spectroscopy. Ba 4F 4(HF 2)(PF 6) 3 crystallizes in a triclinic space group P1¯ with a=4.5870(2) Å, b=8.8327(3) Å, c=11.2489(3) Å, α=67.758(9)°, β=84.722(12), γ=78.283(12)°, V=413.00(3) Å 3 at 200 K, Z=1 and R=0.0588. Pb 2F 2(HF 2)(PF 6) at 200 K: space group P1¯, a=4.5722(19) Å, b=4.763(2) Å, c=8.818(4) Å, α=86.967(10)°, β=76.774(10)°, γ=83.230(12)°, V=185.55(14) Å 3, Z=1 and R=0.0937. Pb 2F 2(HF 2)(PF 6) at 293 K: space group P1¯, a=4.586(2) Å, b=4.781(3) Å, c=8.831(5) Å, α=87.106(13)°, β=76.830(13)°, γ=83.531(11)°, V=187.27(18) Å 3, Z=1 and R=0.072. Ca(HF 2) 2 crystallizes in an orthorhombic Fddd space group with a=5.5709(6) Å, b=10.1111(9) Å, c=10.5945(10) Å, V=596.77(10) Å 3 at 200 K, Z=8 and R=0.028.

Quantum phases of trapped ions in an optical lattice

We propose loading trapped ions into microtraps formed by an optical lattice. For harmonic microtraps, the Coulomb coupling of the spatial motions of neighboring ions can be used to construct a broad class of effective short-range Hamiltonians acting on an internal degree of freedom of the ions. For large anharmonicities, on the other hand, the spatial motion of the ions itself represents a spin-1/2 model with frustrated dipolar XY interactions. We illustrate the latter setup with three systems: the linear chain, the zigzag ladder and the triangular lattice. In the frustrated zigzag ladder with dipolar interactions we find chiral ordering beyond what was predicted previously for a next-nearest-neighbor model. In the frustrated anisotropic triangular lattice with nearest-neighbor interactions we find that the transition from the one-dimensional (1D) gapless spin-liquid phase to the 2D spiraling ordered phase passes through a gapped spin-liquid phase, similar to what has been predicted for the same model with Heisenberg interactions. Further, a second gapped spin-liquid phase marks the transition to the 2D Néel-ordered phase.

Antiferromagnetic model and magnetization plateaus on the zigzag ladder with two- and three-site exchanges

The Ising approximation is proposed for the Heisenberg model with two- and three-spin exchange interactions on a zigzag ladder placed in a strong magnetic field. Using the transfer matrix technique the magnetization is calculated and compared with the results obtained in the dynamical recursive approach. Magnetization plateaus are found at m = 0 and m = 1 3 with the two- and three-spin exchanges. The magnetic susceptibility in small and large zigzag ladders for various exchange parameters and temperatures is also analyzed. As temperature decreases the magnetic susceptibility versus magnetic field shows the formation of spin pseudogaps and transformation from double to four peak structure. Small clusters also exhibit characteristic features of large thermodynamic systems.

Amine-Templated Open-Framework Zinc Arsenates of Varying Dimensionalities: Synthesis, Structure, Polymorphism, and Transformation Reactions

Eight new open-framework zinc arsenates, encompassing the entire hierarchy of open-framework structures, have been prepared hydrothermally. The structures include zero-dimensional, one-dimensional chains, two-dimensional layers, and three-dimensional structures formed through the transformation of the molecular zinc arsenates. The structure of [C6N4H21][Zn(HAsO4)2(H2AsO4)], I, is composed of ZnO4 and H2AsO4 units connected through the vertices forming four-membered rings with HAsO4 units hanging from the Zn center. The four-membered rings are connected through the corners forming the one-dimensional chain structures in [C4N2H12][Zn(HAsO4)2] x H2O, II, and [C5N2H14][Zn(HAsO4)2] x H2O, III. ZnO4 and AsO4 units form a fully four-connected two-dimensional structure in [C4N2H12][Zn(AsO4)]2, IV. One-dimensional zigzag ladders are connected through HAsO4 units forming two-dimensional layers in [C4N2H12]1.5[Zn2(AsO4)(HAsO4)2] x H2O, V, while the similar building units form a layer with hanging HAsO4 units in the layered arsenate [C6N4H21]6[Zn12(HAsO4)21], VI. Hanging HAsO4 units are also observed in the polymorphic structures of [C6N3H20][Zn2(AsO4)(HAsO4)2] x 2H2O, VII and VIII. Formation of zero-dimensional monomer, I, a fully four-connected layer, IV, and the polymorphic structures, VII and VIII, are important and noteworthy. The transformation reactions of I indicate that the monomer is reactive and gives rise to structures of higher dimensionalities, indicating a possible Aufbau-type building-up process in these structures.

Exact dimer ground states for a continuous family of quantum spin chains and zigzag ladders

Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.

MAGNETIZATION PLATEAUS ON THE ZIGZAG LADDER WITH TWO-, AND THREE-SITE EXCHANGES

We consider the Heisenberg model with two-, and three-spin exchange interactions on a zigzag ladder in a strong magnetic field. Using the recursion method for the partition function in Ising approximation we have found exact results for the magnetization. We have shown the existence of magnetization plateaus in the case of mutual two-, and three-spin exchanges at low temperatures and different exchange parameters. The system exhibits different magnetic behaviors, depending on the values of the exchange parameters.

Entangled 3D metal-organic architectures from the self-assembly of mixed ligands and transition-metal ions

Two new entangled 3D metal-organic frameworks, [Ni 2(TP) 2(BPP) 2(H 2O) 2] n ( 1) and {[Co(TP)(BPP)] 4·6H 2O} n ( 2), have been hydrothermally synthesized with mixed ligands of terephthalate (TP) and 1,3-bis(4-pyridyl)propane (BPP) and transition-metal nitrates. Both compounds were characterized by elemental analysis, TGA, IR and single crystal X-ray diffraction. Compound 1 crystallizes in the orthorhombic space group Pnna with a = 18.350(4), b = 13.447(3), c = 16.394(3) Å, V = 4045.2(14) Å 3, Z = 4, R1(w R2) = 0.0481(0.1072). Compound 2 crystallizes in the triclinic space group P 1 ¯ with a = 11.860(2), b = 18.029(4), c = 20.069(4) Å, α = 106.94(3), β = 92.27(3), γ = 97.02(3), V = 4061.2(14) Å 3, Z = 2, R1(w R2) = 0.0895(0.2279). Compound 1 contains a new type of 1D zigzag ladders which are further interlocked with the other two identical ones, forming a 3D polycatenated framework. Compound 2 displays a new 3D “inclined” interpenetration architecture based on 2D undulated layers.

Quantum phase transitions in trimerized zigzag spin ladders

We analyze the effects of a trimerized modulation in a quantum spin $S=\frac{1}{2}$ zigzag ladder at the magnetization plateau $M=1∕3$. Such periodicity is argued to be stemmed from lattice deformations by phonons. The interplay between frustration and exchange modulation is well described by an effective triple sine-Gordon field theory close to the homogeneous ladder and by block-spin perturbation theory in the weakly coupled trimer regime. The characteristic triple degeneracy of the ground state for homogeneous ladders gives place to modulation driven quantum phase transitions, leading to a rich phase diagram including up-up-down, quantum plateau, and gapless plateau states.

Mean-Field Theory for Heisenberg Zigzag Ladder: Ground State Energy and Spontaneous Symmetry Breaking

The spin-1/2 zig-zag Heisenberg ladder (J1 − J2 model) is considered. A new representation for the model is found and a saddle point approximation over the spin-liquid order parameter \({\left\langle {\vec{\sigma }_{{{{n - 1}} }} {\left( {\vec{\sigma }_{n} \times \vec{\sigma }_{{n + 1}} } \right)}} \right\rangle }\) is performed. Corresponding effective action is derived and analytically analyzed. We observe the presence of phase transitions at values J2/J1 = 0.230971 and J2/J1 = 1/2.

Finite-field ground state of an [formula omitted] zigzag ladder

Using numerical methods, we study the finite-field ground state of an S = 1 zigzag ladder. Special attention is paid to the magnetization plateau which appears at a half of the saturation magnetization in the ground-state magnetization curve.

Charge Ordering and Spin Gap Transitions in Quarter-Filled Ladders

Several new families of charge transfer solids with ladder-like structure show unusually high spin gap transition temperatures. We argue that these materials are best described by a 1/4-filled ladder with zigzag interstack bonds. We find that in the 1/4-filled zigzag ladder, a cooperative bond-charge ordering leads to the formation of local singlets and a spin gap. The spin gap for the zigzag ladder is an order of magnitude larger than that found in equivalent one dimensional systems.

Multi-dimensional mapping and folding properties for non-classical helix-stabilizing interactions of proteins

Abstract By using a dynamical systems approach we investigate a model for helix-coil phase transition for polypeptides and proteins in the thermodynamic limit on recursive zigzag ladder with three-site interaction. We explicitly take into account a non-classical helix-stabilizing term (hydrogen bonds between C α H⋯O). We obtain folding and quasi-unfolding of the order parameter (degree of helicity Θ ).

Solvothermal preparation, X-ray structural characterization and properties of two novel 3D copper(I) halide/ N-allyl imidazole coordination polymers

The solvothermal reactions of N-allyl imidazole with Cu IX (X=Cl, Br) afford two unprecedented 3D copper(I)–olefin coordination polymer Cu 2X 2( N-allyl imidazole) (X=Cl for 1; Br for 2). This is the first 3D copper(I)–olefin coordination polymer synthesized in directed method, compared to most of olefin–Cu(I) coordination polymers prepared by electrochemical method. Of practical interest is that both of them contain 1D zigzag ladder chain composed of basic Cu 2X 2 dimer unit. Photoluminescent measurements show that 1 displays blue fluorescent emission while 2 has red fluorescent emission at room temperature.

Structural, Magnetic and Electronic Transport Properties of Novel Hollandite-Type Molybdenum Oxide, Rb1.5Mo8O16

Structural, magnetic, and transport properties of a novel hollandite-type molybdenum oxide, Rb 1.5 Mo 8 O 16 have been investigated. Whereas a typical hollandite structure is characterized by uniform zigzag ladder chains of edge-shared MO 6 (M=transition metal) octahedra, the zigzag ladders in Rb 1.5 Mo 8 O 16 are strongly distorted, and consequently, “Mo 4 ” clusters composed of four edge-shared MoO 6 octahedra are formed. A semiquantitative band calculation based on an extended Huckel method shows that the band structure assumes the characteristics of the molecular orbitals of an isolated Mo 4 cluster. The magnetic susceptibility reveals the existence of spins spread over each Mo 4 cluster rather than localized on each Mo atom. The electrical resistivity and the thermoelectric power indicate that the electric conduction is governed by cluster-to-cluster electron hopping. Rb 1.5 Mo 8 O 16 undergoes a phase transtion at T 1 = 208 K, accompanied by a change in the number of spins and carriers. X-ray diffra...

High-Field NMR Insights into Quantum Spin Systems

Possibilities of NMR technique will be illustrated by a selection of high-field NMR results on antiferromagnetic quantum spin systems, obtained at the Grenoble High Magnetic Field Laboratory. We present results for two well known systems, doped Haldane spin chain Y2BaNi1-xMgxO5 and spin-Peierls compound CuGeO3, as well as two newly studied compounds, the NaCU2O2 chain (or zigzag ladder) and the 2D dimer compound BaCuSi2O6 (Han purple).