Ising model with solitons, phasons, and "the devil's staircase"

Bak, Per; Boehm, J. von

doi:10.1103/physrevb.21.5297

Cited by 606 publications

(283 citation statements)

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“…A periodicity, N , of the spin structure is taken up to 20 along the a, b and a±b directions, and each phase is characterized by the wave number q=M/N . The phase diagram shows a similar topological structure to that in the ANNNI model [19,20]; numerous long-range orders between the FM (q=0) and "up-up-down-down"-type AF (q=1/4) phases, that is the so-called Devil's flower. The calculated results qualitatively reproduce the phase diagram of RMnO 3 in Fig.…”

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Distorted perovskite witheg1configuration as a frustrated spin system

et al. 2003

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The evolution of spin-and orbital-ordered states has been investigated for a series of insulating perovskites RMnO3 (R=La,Pr,Nd,...). RMnO3 with a large GdFeO3-type distortion is regarded as a frustrated spin system having ferromagnetic nearest-neighbor and antiferromagnetic (AF) nextnearest-neighbor (NNN) interactions within a MnO2 plane. The staggered orbital order associated with the GdFeO3-type distortion induces the anisotropic NNN interaction, and yields unique sinusoidal and up-up-down-down AF ordered states in the distorted perovskites with e 1 g configuration. and e 1 g t 6 2g configurations, respectively, the e g orbital is doubly degenerate and the t 2g orbital degree of freedom is quenched. It is widely recognized that the layered-type (A-type) antiferromagnetic (AF) structure in LaMnO 3 is understood from the view point of the anisotropic superexchange (SE) interaction under the directional order of orbital [1,2]. On the other hand, the spin structure in nickelates (R =La) is distinct from the A-type AF; the socalled "up-up-down-down"-type one where two Ni sites of "up" spins are followed by two sites of "down" spins along the principal axes in the cubic unit cell. Origin of this unusual magnetic order has been a long-standing question, as well as its relations to metal-insulator transition, orbital order (OO), and charge disproportionation [3][4][5]. Recently, a similar spin structure, i.e. the up-up-down-down order in a MnO 2 plane (E-type AF order in the Wollan-Koehler notation [6]), is found in a manganite, HoMnO 3 [7] with a significantly distorted perovskite structure. This has to be a bridge between the well-understood A-type AF in manganites and the unique magnetic ground state in nickelates.In this Letter, we examine systematically the magnetic and orbital structures in a series of RMnO 3 as a function of the ionic radius (r R ) of R. The most significant effect on the crystal structure by decreasing r R is an enhancement of the cooperative rotation of the MnO 6 octahedra (the GdFeO 3 -type distortion) characterized by the decrease of Mn-O-Mn bond angle φ. Let us first summarize in Fig. 1 the orbital (a) and spin (b) ordering temperatures (T OO and T N , respectively) on Mn sites of RMnO 3 as a function of φ, which is based on both the present and former studies [7][8][9][10]. Here, we adopt the φ at room temperature [11]. The T OO monotonically increases with decreasing r R , while the magnetic transition occurs from the A-type AF to the E-type one through the incommensurate structure. We argue that the combination of OO and next-nearest-neighbor (NNN) SE interaction brings about a nontrivial effect on the magnetic ground state in the systems with the orbital degeneracy and the large GdFeO 3 -type distortion. Microscopic calculation shows that the magnetism in this system is mapped onto the frustrated spin model which well reproduces the phase diagram of RMnO 3 .A series of RMnO 3 (R=La−Dy) crystals were grown by the floating zone method. We made powder x-ray diffraction measurements on th...

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Distorted perovskite witheg1configuration as a frustrated spin system

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“…They crystallize into a lattice and the mean separation λ between walls determines q ∝ λ −1 . The value of λ is controlled by a balance between the chemical potential of soli- tons and an effective repulsive interaction that decays exponentially in the distance between solitons [46]. The outcome of this balance is that λ diverges logarithmically in T − T CI [our result ( Magnetization curve.-Finally, we present M(H) obtained in a simulated relaxation process in the realistic 3D model.…”

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Formation of Magnetic Microphases inCa3Co2O6

Kamiya

Batista

2012

Phys. Rev. Lett.

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We study a frustrated quantum Ising model relevant for Ca 3 Co 2 O 6 that consists of a triangular lattice of weakly-coupled ferromagnetic (FM) chains. According to our quantum Monte Carlo (QMC) simulations, the chains become FM and form a three-sublattice "up-up-down" structure for T ≤ T CI . In contrast, long-period spin-density-wave (SDW) microphases are stabilized along the chains for T CI < T < T c . Our mean field solutions reveal a quasi-continuous temperature dependence of the SDW wavelength, implying the existence of metastable states that explain the very slow dynamics observed in Ca 3 Co 2 O 6 . We also discuss implications of microphases for the related multiferroic compounds Ca 3 CoMnO 6 and Lu 2 MnCoO 6 . [3][4][5], the spin-driven "nematic" transition in pnictides [6][7][8][9], and dimensional reduction in BaCuSi 2 O 6 [10,11]. Ca 3 Co 2 O 6 is another example comprising a triangular lattice of FM Ising chains coupled by weak antiferromagnetic (AFM) exchanges. This compound exhibits field-induced magnetization steps whose heights depend on the field sweep history and rate [12][13][14][15][16]. We will show that this out-of-equilibrium behavior has its roots in exotic equilibrium properties that can be extended to the related multiferroic compound Ca 3 Co 2−x Mn x O 6 [17][18][19][20][21][22].The Co 3+ ions (Co II) on the trigonal prism sites of Ca 3 Co 2 O 6 contain 3d 6 localized electrons that generate an S = 2 spin with large Ising-like anisotropy [23][24][25][26]. These ions form a triangular lattice of FM Ising chains along the caxis (Fig. 1) and the structure comprises three sublattices of layers stacked along the c-axis in an ABCABC . . . configuration. Although the AFM inter-chain couplings J 2 and J 3 [27] [ Fig. 1(a)] are an order of magnitude smaller than the intrachain FM exchange, |J 1 | = 2 × 10 K [23, 27], we will show that they strongly affect the intra-chain spin correlations over a window of temperatures below T c .The initial interest in Ca 3 Co 2 O 6 was triggered by the observation of out-of-equilibrium magnetization steps measured below ∼ 8 K and ∼ 3.6 T that appear at regular field intervals. Previous works invoked a "rigid-chain model": every chain is replaced by a single Ising spin by assuming [28][29][30][31][32]. Each spin of the resulting triangular lattice Ising model (TLIM), represents the magnetization of the whole chain and it is flipped if gµ B H overcomes its molecular field. Within this simplified framework, the regular field intervals result from the equally-spaced discrete molecular field spectrum [28]. However, this 2D scenario was challenged by the recent discovery of long-wavelength intra-chain spindensity-wave (SDW) ordering below T c 25 K [16,33,34]. Motivated by this discovery, Chapon initiated the study of a more realistic 3D lattice model by using a random-phase approximation (RPA) which is only valid close to T = T c [35].By combining QMC simulations and mean field (MF) solu-

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“…Thus, we restrict our model to six non-zero coupling constants Kn, Ln, n = 1, 2, 3. This is a generalization of the so-called anisotropic next-nearest neighbor Ising (ANNNI) model [11], [12]. Our model differs from the standard ANNNI one by two features: the third-neighbor interaction and the 4 positions of the hand instead of two in the Ising model.…”

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Microscopic model for the magnetic subsystem inHoNi2B2C

Kalatsky

Pokrovsky

1998

Phys. Rev. B

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We demonstrate that the system of localized magnetic moments in HoNi2B2C can be described by the 4-positional clock model. This model, at a proper choice of the coupling constants, yields several metamagnetic phases in magnetic field at zero temperature in full agreement with the experimental phase diagram. The model incorporates couplings between not nearest neighbors in the direction perpendicular to the ferromagnetic planes. The same model leads to a c-modulated magnetic phase near the Curie temperature. The theoretical value of the modulation wave-vector agrees surprisingly well with that observed by the neutron diffraction experiment without new adjustable parameters.In the works by Rathnayaka et al. In this compound easy magnetization axes are identified with crystallographic directions 110 and 110 . The low temperature magnetization data show the existence of 4 metamagnetic phases. The low field phase has been identified by neutron diffraction experiments [3,5] and magnetic measurements [2] with the antiferromagnetic phase, which we denote symbolically ↑↓. The phase boundaries and magnetization in other phases versus magnetic field found in the experiment [2] can be readily explained by assuming that the remaining three phases are as follows: phase 2 -↑↑↓, phase 3 -↑↑→, and the high-field phase 4 -↑. It means that 2 3 of the spins in the phases 2 and 3 are parallel to one of the easy axes whereas the remaining 1 3 is antiparallel and perpendicular, respectively, to the same axis. Note that all metamagnetic phases are stoichiometric in the meaning that the concentrations of spins parallel, antiparallel, or perpendicular to the reference axis are rational numbers. The phase diagram of HoNi2B2C at zero temperature is especially simple if the components of magnetic field Hx, Hy are chosen as variables. The experimental phase diagram of HoNi2B2C is shown in Fig. 1 (in the original work [2] it has been presented in polar coordinates h, θ h ).Siegrist et al.[9] and Huang et al.[6] determined the structure of Lu and Ho 1:2:2:1 compounds as the body-centered tetragonal lattice with the space group I4/mmm. The xray structure analysis and the neutron-scattering experiments performed by Goldman et al. [3,4] and Grigereit et al. [5][6][7]. showed that an incommensurate modulated magnetic structures with the wave-vectors Kc = 0.915c * and Ka = 0.585a * occur in the temperature range 4.7-6 K. At temperatures below 4.7 K they vanish and an antiferromagnetic reflections corresponding to alternating ferromagnetic ab-planes of Ho 3+ localizad moments appear. Though the spacial arrangement of the phases ↑↑↓ and ↑↑→ can not bee directly derived from the magnetization measurements, it is unplausible that the ferromagnetic in-plane interaction changes suddenly by switching on of the magnetic field. Therefore, we believe that our symbols ↑↑↓ and ↑↑→ correspond to the real spatial sequences of in-plane magnetic moments. In this article we present a simple microscopic model for magnetic subsystem in the 1:2:2:1 compound which expl...

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Ising model with solitons, phasons, and "the devil's staircase"

Cited by 606 publications

References 22 publications

Distorted perovskite witheg1configuration as a frustrated spin system

Distorted perovskite witheg1configuration as a frustrated spin system

Formation of Magnetic Microphases inCa3Co2O6

Microscopic model for the magnetic subsystem inHoNi2B2C

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