Long-range Order Phase Research Articles

Spin-liquid behavior in the classical antiferromagnetic [formula omitted] spin system on the square-kagome lattice: Exact analysis within recursive-lattice approach

Magnetic and thermodynamic properties of the antiferromagnetic J1−J2−J3 spin system, i.e., the spin system with three different nearest-neighbor antiferromagnetic interactions, on the square-kagome lattice is investigated in the framework of the corresponding exactly solvable spin-1/2 Ising model on the recursive square-kagome lattice. The phase diagram of the model is found and it is shown that, depending on the model parameters, the model exhibits the existence of three long-range order (antiferromagnetic) phases, which are separated from the paramagnetic phase by the second order phase transitions. It is also shown that, in the zero temperature limit, the paramagnetic phase splits into three different ground states realized on the lines that separate antiferromagnetic ground states of the model. Due to the frustration, all these “paramagnetic” ground states are highly macroscopically degenerated with strict hierarchy of their residual entropies. Moreover, the high macroscopic degeneracy of the paramagnetic phase in the vicinity of these ground states together with the presence of strong correlations (given by the proximity of the second order phase transitions) allow one to consider the paramagnetic ground states as well as their immediate paramagnetic surroundings as regions with the classical spin-liquid behavior. It is also shown that the presence of the second order phase transitions in the vicinity of the paramagnetic ground states suppresses expected anomalous (Schottky-type) behavior of the specific heat related to the rapid entropy changes caused by the frustration.

Gold Nanoparticle Superlattices: Conditions for Long-Range Order, Moiré Patterns, and Binary Phase from a Single Population

Gold Nanoparticle Superlattices: Conditions for Long-Range Order, Moiré Patterns, and Binary Phase from a Single Population

Resolution of zigzag magnetic correlations in Na-deficient NaxIrO3 without long-range ordering

The materials search for Kitaev quantum spin liquids led to the discovery of many honeycomb lattice materials. Much attention has been paid to materials without magnetic order down to the lowest temperatures. The newly synthesized Na-deficient ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ has been found to bear no sign of long-range magnetic order above 1 K from physical property measurements. In this paper, we report momentum- and energy-resolved excitation spectra in Na-deficient ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ measured using a resonant inelastic x-ray scattering spectrometer. Orbital excitation spectra show that the octahedral and trigonal crystal field splittings are larger in ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ than in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$. On the other hand, the low-energy spectrum at low temperature shows a wave-vector dispersion and a spectral weight distribution that are similar to those of ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$, revealing that the two-dimensional zigzag magnetic correlations in ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ are similar to those in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ in terms of the ordered magnetic moment direction and three dynamically fluctuating zigzag orders. The azimuth angle dependence of the low-energy spectrum corroborates these results. The two-dimensional zigzag magnetic correlations rapidly weaken until 50 K. At high temperatures, the spectral weight distribution of the low-energy excitation resembles that of the pure Kitaev model, indicating that the Kitaev interaction dominates the dynamic magnetic response at high temperature. We suggest that the larger crystal field and distortion and the weakened longer-range Heisenberg exchange interactions due to the Na deficiency in ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ contribute to bring ${\mathrm{Na}}_{x}{\mathrm{IrO}}_{3}$ away from the zigzag long-range magnetic order phase.

Polaritonic excitations and Bose-Einstein condensation in the Rabi lattice model

The unitary transformation method is used to study the properties of the polaritonic states and Bose-Einstein condensation in the Rabi lattice model, where the on-site two-level systems (TLSs) are coupled with the intersite hopping photons. It is shown that the counter-rotating coupling (CRC) between TLS and photon, which breaks down the conservation of the excitation numbers, may induce a long-range Ising-like interaction among intersite TLSs. We have shown that the coupled TLSs and photons are hybridized to form the polaritonic states, and the corresponding ground state and polaritonic excitation spectra are calculated by diagonalizing our transformed Hamiltonian. When the photon hopping $J$ is weak, the excitation spectrum is gapped. But the gap decreases with increasing $J$, and at critical value $J={J}_{c}$ the gap becomes zero at the $\mathrm{\ensuremath{\Gamma}}$ point and the ground state becomes instable. Thus, ${J}_{c}$ is the phase transition point where the polaritons are condensed at the $\mathrm{\ensuremath{\Gamma}}$ point to form the delocalized superradiant phase. The results show that the larger detuning between TLS and photon favors the disorder insulator phase, which requires the stronger ${J}_{c}$ to get the long-range order phase. The phase diagram of the delocalized superradiant phase transition has been obtained where there are no Mott lobes since the CRC breaks down the number conservation. The ground state and the excitation spectra of the delocalized superradiant phase are also calculated. It is shown that in the delocalized superradiant state the order parameter is the nonzero ground-state average of the photon annihilation operator. The polaritonic excitation has a finite excitation gap because the CRC breaks down the number conservation but leaves only a discrete symmetry, the parity conservation.

Phase transitions induced by concentration and thermodynamic magnetic fluctuations in chiral ferromagnets Fe1-xCoxSi

Taking into account the results of LDA+U+SO-modeling of the electronic structure of Fe1-xCoxSi, temperature-prolonged magnetic phase transitions of the first order in strongly correlated alloys with a broken crystal structure of the B20 type are investigated. It is shown that such phase transitions are induced by concentration and thermodynamic fluctuations for compositions with 0.2<x<0.65 and are accompanied by a change in the sign of the mode-mode coupling parameter. In this case, an intermediate region appears between the long-range order phases and the paramagnetic phase, with the short-range spin order characterized by ferromagnetic and helicoidal spatial fluctuations. We have defined the intervals of external magnetic fields, at which skyrmion microstructures spatially limited by the crossover of helicoidal and ferromagnetic fluctuations appear in the considered region of the extended phase transition. The peculiarities of the temperature dependences of the magnetic susceptibility in an external magnetic field, characteristic of the appearance of skyrmions, are obtained.

Assembly, alignment and tunable optical properties of montmorillonite with superparamagnetic coating

Here we demonstrate a novel magnetic functional material composed of montmorillonite (Mt) with superparamagnetic coating Fe3O4 dispersed in the aqueous solution. Colloidal stability can be obtained through keeping 2 < pH < 3. After applying an external magnetic field, Mt@Fe3O4 exhibits long-range order and liquid crystalline phase. The transmittance of the liquid crystal is dependent on the solid content of Mt@Fe3O4, magnetic field strength and magnetic field direction. We also observe reflective phenomena when we modulate the magnetic field. The photonic bandgap can be tuned by changing the angle between incident light and magnetic field. The varying observation points also make a difference to the reflection. This facile approach for fabricating Mt@Fe3O4 with ordered periodic structures and optically tunable property is of interest for a variety of advanced optics applications with low cost and environment-friendly.

Phase stability of γ′-Ni2Cr and α-Cr in the Ni-Cr binary

The stability of the γ′-Ni2Cr long-range order (LRO) phase present on the Ni-Cr binary phase diagram was examined in a Ni-55Cr (wt. %) alloy. It is shown that the γ′-Ni2Cr phase is metastable, and that the two-phase γ-Ni + α-Cr phase field extends to room temperature. The as-cast alloy consisted of a typical dendritic microstructure, with Cr-rich primary α-Cr dendrites and a Cr-lean, γ-Ni interdendritic. Annealing the as-cast alloy at 500 °C for 1000 h (γ′-Ni2Cr + α-Cr phase field) resulted in pockets of α-Cr and γ-Ni + γ′-Ni2Cr at the dendritic/interdendritic interface, with little change to the interdendritic γ-Ni microstructure. Annealing at 900 °C for 4 h (γ-Ni + α-Cr phase field) resulted in the interdendritic precipitation of α-Cr, both as a continuous and discontinuous precipitate. Annealing at 900 °C for 4 h followed by 500 °C for 1000 h resulted in the interdendritic precipitation of α-Cr at 900 °C, but no precipitation of γ′-Ni2Cr during subsequent annealing at 500 °C. This behavior suggests that the γ′-Ni2Cr phase is not the equilibrium phase, but rather a metastable transition phase. Furthermore, it is speculated that this phase in more complex alloys (i.e. Alloy 690) is also metastable, and its metastability should be considered in applications involving long-term, high temperature exposures.

Mechanical properties of an aged Ni-Cr-Mo alloy and effect of long-range order phase on deformation behavior

Mechanical properties of a Ni-Cr-Mo alloy subjected to an aging treatment at 600 °C and effect of long-range order (LRO) phase on deformation behavior of the alloy are systematically investigated. Results show that peak-aging conditions are reached after 600 h of aging at 600 °C. Meanwhile, the aging treatment at 600 °C in the aging time range of 100–700 h can lead to the precipitation of the coherent Pt2Mo-type LRO phase in the C2000 alloy. Fine scale Pt2Mo-type LRO phase exists in the sample aged for 100 h and provides an enhanced yield strength. Because the solution treated sample and the sample aged for 100 h show the almost identical strain hardening behaviors, it can be suggested that the ordered structure of the fine scale LRO phase is destroyed during the quite small plastic deformation. Besides, the ordered structure of the relatively large LRO phase is destroyed after the large deformation. Mechanical twinning occurs in the tensile-failed sample which is aged for 500 h. And, mechanical twinning is responsible for the high ductility of the alloy aged for more than 100 h and the change in the shape of the strain hardening rate vs true strain curve.

Topological orders and quantum phase transitions in a one-dimensional extended quantum compass model

By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.

Disorder trapping by rapidly moving phase interface in an undercooled liquid

Non-equilibrium phenomena such as the disappearance of solute drag, the origin of solute trapping and evolution of disorder trapping occur during fast transformations with originating metastable phases [D.M. Herlach, P.K. Galenko, D. Holland-Moritz, Metastable solids from undrercooled melts (Elsevier, Amsterdam, 2007)]. In the present work, a theoretical investigation of disorder trapping by a rapidly moving phase interface is presented. Using a model of fast phase transformations, a system of governing equations for the diffusion of atoms, and the evolution of both long-range order parameter and phase field variable is formulated. First numerical solutions are carried out for a congruently melting binary alloy system.

Antiferromagnetic triangular Blume-Capel model with hard-core exclusions.

Using Monte Carlo simulation, we analyze phase transitions of two antiferromagnetic (AFM) triangular Blume-Capel (BC) models with AFM interactions between third-nearest neighbors. One model has hard-core exclusions between the nearest-neighbor (1NN) particles (3NN1 model) and the other has them between the nearest-neighbor and next-nearest-neighbor particles (3NN12 model). Finite-size scaling analysis reveals that in these models, the transition from the paramagnetic to long-range order (LRO) AFM phase is either of the first order or goes through an intermediate phase which might be attributed to the Berezinskii-Kosterlitz-Thouless (BKT) type. The properties of the low-temperature phase transition to the AFM phase of the 1NN, 3NN1, and 3NN12 models are found to be very similar for almost all values of a normalized single-ion anisotropy parameter, 0 < δ < 1.5. Higher temperature behavior of the 3NN12 and 3NN1 models is rather different from that of the 1NN model. Three phase transitions are observed for the 3NN12 model: from the paramagnetic phase to the phase with domains of the LRO AFM phase at T(c), from this structure to the diluted frustrated BKT-type phase at T(2), and from the frustrated phase to the AFM LRO phase at T(1). For the 3NN12 model, T(c) > T(2) > T(1) at 0 < δ < 1.15 (range I), T(c) ≈ T(2) > T(1) at 1.15 < δ < 1.3 (range II), and T(c) = T(2) = T(1) at 1.3 < δ < 1.5 (range III). For the 3NN1 model, T(c) ≈ T(2) > T(1) at 0 < δ < 1.2 (range II) and T(c) = T(2) = T(1) at 1.2 < δ < 1.5 (range III). There is only one first-order phase transition in range III. The transition at T(c) is of the first order in range II and either of a weak first order or a second order in range I.

Computational studies of history dependence in nematic liquid crystals in random environments.

Glassy liquid crystalline systems are expected to show significant history-dependent effects. Two model glassy systems are the RAN and SSS (sprinkled silica spin) lattice models. The RAN model is a Lebwohl-Lasher lattice model with locally coupled nematic spins, together with uncorrelated random anisotropy fields at each site, while the SSS model has a finite concentration of impurity spins frozen in random directions. Here Brownian simulation is used to study the effect of different sample histories in the low temperature regime in a three-dimensional (d = 3) model intermediate between SSS and RAN, in which a finite concentration p < p(c) (p(c) the percolation threshold) of frozen spins interacts with neighboring nematic spins with coupling W. Simulations were performed at temperature T ∼ T(NI)/2 (T(NI) the bulk nematic-isotropic transition temperature) for temperature-quenched and field-quenched histories (TQH and FQH, respectively), as well as for temperature-annealed histories (AH). The first two of these limits represent extreme histories encountered in typical experimental studies. Using long-time averages for equilibrated systems, we calculate orientational order parameters and two-point correlation functions. Finite-size scaling was used to determine the range of the orientational ordering, as a function of coupling strength W,p and sample history. Sample history plays a significant role; for given concentration p, as disorder strength W is increased, TQH systems sustain quasi-long-range order (QLRO) and short-range order (SRO). The data are also consistent with a long-range order (LRO) phase at very low disorder strength. By contrast, for FQH and p ≤ 0.1, only LRO and QLRO occur within the range of parameters investigated. The crossover between regimes depends on history, but in general, the FQH phase is more ordered than the AH phase, which is more ordered than the TQH phase. However, at temperatures close to the isotropic-nematic phase transition of pure samples we observe SRO for p = 0.1 even for FQH. We detect also in the QLRO phase a domain-type structural pattern, consistent with ideas introduced by Giamarchi and Doussal [Phys. Rev. B 52, 1242 (1995)] on superconducting flux lattices. In the weak-disorder limit the orientational correlation length obeys the Larkin-Imry-Ma scaling ξ ∼ D(-2/(4-d)).

Frustrated quantum two-dimensional J 1-J 2-J 3 antiferromagnet in a spherically symmetric self-consistent approach

In the framework of a spherically symmetric self-consistent approach to two-time retarded spin-spin Green’s functions, we develop the theory of a two-dimensional frustrated J1-J2-J3 quantum S=1/2 antiferromagnet. We show that taking the damping of spin fluctuations into account is decisive in forming both the spin-liquid state and the state with long-range order. In particular, the existence of damping allows explaining the scaling behavior of the susceptibility χ(q, ω) of the CuO2 cuprate plane, the behavior of the spin spectrum in the two-plane case, and the occurrence of an incommensurable χ(q, ω) peak. In the case of the complete J1-J2-J3 model, in a single analytic approach, we find continuous transitions between three phases with long-range order (“checkerboard,” stripe, and helical (q, q) phases) through the spin-liquid state. We obtain good agreement with cluster computations for the J1-J2-J3 model and agreement with the neutron scattering data for the J1-J2 model of cuprates.

High-pressureV51NMR study of the magnetic phase diagram and metal-insulator transition in quasi-one-dimensionalβ-Na0.33V2O5

High-pressure $^{51}\text{V}$ NMR measurements are conducted up to 9 GPa using a modified Bridgman anvil cell in a quasi-one-dimensional conductor $\ensuremath{\beta}{\text{-Na}}_{0.33}{\text{V}}_{2}{\text{O}}_{5}$ which shows pressure-induced superconductivity from a charge order state. We determine the magnetic and Na order transition temperatures and construct the phase diagram of the metal-insulator transition in $\ensuremath{\beta}{\text{-Na}}_{0.33}{\text{V}}_{2}{\text{O}}_{5}$. We demonstrate that the antiferromagnetic long-range order phase persists up to a critical pressure ${P}_{c}\ensuremath{\sim}8\text{ }\text{GPa}$ where the charge order completely melts. The local spin susceptibility shows distinct site dependence and a crossover from a Curie-Weiss one-dimensional liquid to a Fermi-liquid metal with enhanced density of states above ${P}_{c}$.

Analysis of the conditions favoring the onset of long-range order in disordered relaxors of different types

The possibility of realizing a stable relaxor behavior in PbSc1/2Nb1/2O3 (PSN) compounds and the conditions favoring the formation of a phase with long-range order in PbZn1/3Nb2/3O3 (PZN) are studied using optical methods. It is found that the replacement of only 6% of Pb ions with Ba ions (PBSN-6) freezes the relaxor state in PSN compounds, while adding even 7% of PbTiO3 to the PZN relaxor (PZN-7PT) leads to the formation of a phase with long-range order and a phase transition occurring at Tmaxe through the percolation mechanism. Common features in the formation and destruction of the long-range order phase in relaxors of different types are revealed. The differences associated with the presence of spontaneously polarized ferroelectric regions in the cubic nonpolar matrix of PSN-type compounds are determined.

Doping change and distortion effect on double-exchange ferromagnetism

Doping change and distortion effect on the double-exchange ferromagnetism are studied within a simplified double-exchange model. The presence of distortion is modelled by introducing the Falicov-Kimball interaction between itinerant electrons and classical variables. By employing the dynamical mean-field theory the charge and spin susceptibility are exactly calculated. It is found that there is a competition between the double-exchange induced ferromagnetism and disorder-order transition. At low temperature various long-range order phases such as charge ordered and segregated phases coexist with ferromagnetism depending on doping and distortion. A rich phase diagram is obtained.

Direct synthesis of vinyl-functionalized cubic mesoporous silica SBA-1

Well-ordered cubic mesoporous silicas SBA-1 functionalized with vinyl groups have been synthesized via room temperature co-condensation of tetraethoxysilane (TEOS) and trimethoxyvinylsilane (TMVS) templated by cetyltriethylammonium bromide (CTEABr), a large headgroup surfactant, under strongly acidic conditions. The materials thus obtained were characterized by a variety of techniques including powder X-ray diffraction (XRD), solid-state NMR, FTIR, thermogravimetric analysis, scanning electron microscopy (SEM), and nitrogen sorption measurements. The HCl concentration was found to be a key factor that determines the stability of vinyl-functionalized SBA-1 towards the solvent extraction treatment. Direct evidence of the presence of chemically attached vinyl moieties was provided by solid-state 29Si and 13C NMR. SEM results revealed that the particle of vinyl-functionalized SBA-1 exhibited a highly isotropic morphology with more crystal faces than that of pure silica SBA-1. In contrast to the direct synthesis of vinyl-functionalized SBA-15, the concentration of TMVS that can co-condense with TEOS can be up to 25% without observing a significant loss in the long-range structure order and uncontrollable phase transformation of the cubic SBA-1 mesostructure.

Microheterogeneity and field-cooling effects on Pb[(Zn1∕3Nb2∕3)0.955Ti0.045]O3 single crystals probed by micro-Brillouin scattering

Microheterogeneity and field-cooling effects were investigated on Pb[(Zn1∕3Nb2∕3)0.955Ti0.045]O3 (PZN-4.5%PT) single crystals by using a high-resolution micro-Brillouin scattering. The temperature dependence of Brillouin shift showed a typical relaxor behavior with marked softening on approaching the diffuse phase transition, but also revealed a clear microareal variation in a rhombohedral phase below 150°C which means a heterogeneity exists over a length scale of at least a few microns in PZN-4.5%PT. These two features seem to correlate with the coexistence of both micronsized domains and irregular nanosized domains, recently confirmed by high-resolution domain studies. This complex domain structure may make each microdomain represent different relaxor behaviors due to its own polar nanoregions and their dynamics. When the crystal was cooled under the electric field along the [001] direction from a cubic phase, two field-induced changes were observed in the Brillouin shift at around 143°C and 106°C. This observation is in good agreement with the dielectric measurements, meaning a medium-range ordered phase exists between short-range ordered and long-range order phases.

Role of ferroquadrupolar exchange in critical behavior of planar Heisenberg antiferromagnet with STL geometry

Effects of ferroquadrupolar exchange interaction J 2 on critical behavior of a frustrated classical XY model with antiferromagnetic exchange interaction J 1 are investigated on stacked triangular lattice by means of histogram Monte Carlo simulations. We identify four long-range order phases separated by the boundaries which can be either of second or first order, depending on the kind of transition and magnitude of J 1/ J 2. Characteristics of the respective phases, mechanisms of their creation, and natures of the transitions between the respective phases are elaborated on.

Planar Heisenberg antiferromagnet with antiferroquadrupolar exchange on stacked triangular lattice: finite-size scaling

We examined the effects of antiferroquadrupolar exchange interaction J 2 on phase transitions of a frustrated classical XY model with antiferromagnetic exchange interaction J 1 on stacked triangular lattice by histogram Monte Carlo simulations. Variation of the J 1/ J 2 ratio produces phase diagram featuring three long-range order (LRO) phases separated by the boundaries which can be either of second- or first-order, depending on J 1/ J 2. Besides the phases of simultaneous antiferromagnetic and antiferroquadrupolar LRO and only antiferroquadrupolar LRO, a new incommensurate phase arises in the region 0< J 1/ J 2<2 due to the inter-plane competition between the two exchange interactions. This phase is characterized by a non-universal turn angle between spins of two successive layers in the stacking direction, the absolute value of which is given by J 1/ J 2 but its sign is arbitrary.