Stacked Triangular Lattice Research Articles

Replica symmetry breaking in the RKKY skyrmion-crystal system

We study the RKKY Heisenberg model on a three-dimensional stacked-triangular lattice under magnetic fields by extensive Monte Carlo simulations to get insight into the chiral-degenerate symmetric skyrmion crystal (SkX) in centrosymmetric metallic magnets. The triple-$q$ SkX state and the double-$q$ states are realized, together with the single-$q$ state. We find an unexpected phenomenon of the replica-symmetry breaking (RSB) well-known in glassy systems, although the Hamiltonian and the ordered state are entirely regular. In the RSB SkX phase, the triple-$q$ SkX state macroscopically coexists with the single-$q$ state, in spite of the fact that these ordered states cannot be transformed via any Hamiltonian-symmetry operation. In the thermodynamic limit, the free energies of these states are degenerate whereas the free-energy barrier between the states diverges, breaking the ergodicity. A similar RSB is observed also in the RSB double-$q$ phase where the double-$q$ state macroscopically coexists with the single-$q$ state. Experimental implications are discussed.

Magnetic ordering and spin dynamics in theS=52staggered triangular lattice antiferromagnetBa2MnTeO6

We report studies of the magnetic properties of a staggered stacked triangular lattice Ba$_2$MnTeO$_6$ using magnetic susceptibility, specific heat, neutron powder diffraction and inelastic neutron scattering measurements, as well as first principles density functional theory calculations. Neutron diffraction measurements reveal an antiferromagnetic order with a propagated vector $\textbf{\emph{k}}=(0.5, 0.5, 0)$ and N{\'{e}}el transition temperature of $T_\text{N}\approx20$ K. The dominant interaction derived from the Curie-Weiss fitting to the inverse DC susceptibility is antiferromagnetic. Through modelling the INS spectrum with the linear spin wave theory, the magnetic exchange interactions for the nearest intralayer, nearest interlayer, and next nearest interlayer are determined to be $J_1=0.27(3),J_2=0.27(3),$ and $J_3=-0.05(1)$ meV, respectively, and a small value of easy-axis anisotropy of $D_{zz}=-0.01$ meV is introduced. We derive a magnetic phase diagram that reveals that it is the competition between $J_1, J_2$, and $J_3$ that stabilizes the collinear stripe-type antiferromagnetic order.

Monte Carlo study of the critical properties of noncollinear Heisenberg magnets: O(3)×O(2) universality class

The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\times O(2)$ symmetry. The maximum size studied is $384^3$, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicates the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yield critical exponents estimates, $\alpha=0.44(3)$, $\beta=0.26(2)$, $\gamma=1.03(5)$, $\nu=0.52(1)$, $\eta=0.02(5)$, and the chirality exponents $\beta_\kappa=0.40(3)$ and $\gamma_\kappa=0.77(6)$, supporting the existence of the $O(3)$ chiral (or $O(3)\times O(2)$) universality class governed by a new `chiral' fixed point. We also obtain an indication that the underlying fixed point is of the focus-type, characterized by the complex-valued correction-to-scaling exponent, $\omega=0.1^{+0.4}_{-0.05} + i\ 0.7^{+0.1}_{-0.4}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.

Magnetic properties in stacked triangular lattice: Monte Carlo approach

We study the magnetic properties of mixed spins σ=5∕2 and S=2 Ising in stacked triangular lattice (STL) using Monte Carlo approach. We have also give the grounds state phase diagrams of mixed spins-5/2 and 2. The different magnetic phases are detected under effect of different physical parameters. The diagrams show some key features: coexistence between regions, points where three, four and five states can coexist. The reduced critical temperatures have been also determined for different exchange interactions in each layer with a fixed value of exchange interactions between two layers. The total magnetization with reduced exchange interactions and crystal-fields of mixed spins is obtained. The multiple hysteresis and the superparamagnetic behavior are established around the reduced critical temperatures. The magnetic coercive field and magnetization remanent increase with increasing the exchange interactions and decreases with increasing the temperature values. The obtained results are similar to those obtained by experiment and theoretically results. Above reduced transition temperature the system shows superparamagnetic behavior which makes the material desirable for biomedical applications.

GPU-Accelerated Population Annealing Algorithm: Frustrated Ising Antiferromagnet on the Stacked Triangular Lattice

The population annealing algorithm is a novel approach to study systems with rough free-energy landscapes, such as spin glasses. It combines the power of simulated annealing, Boltzmann weighted differential reproduction and sequential Monte Carlo process to bring the population of replicas to the equilibrium even in the low-temperature region. Moreover, it provides a very good estimate of the free energy. The fact that population annealing algorithm is performed over a large number of replicas with many spin updates, makes it a good candidate for massive parallelism. We chose the GPU programming using a CUDA implementation to create a highly optimized simulation. It has been previously shown for the frustrated Ising antiferromagnet on the stacked triangular lattice with a ferromagnetic interlayer coupling, that standard Markov Chain Monte Carlo simulations fail to equilibrate at low temperatures due to the effect of kinetic freezing of the ferromagnetically ordered chains. We applied the population annealing to study the case with the isotropic intra- and interlayer antiferromagnetic coupling (J 2 /|J 1 | = −1). The reached ground states correspond to non-magnetic degenerate states, where chains are antiferromagnetically ordered, but there is no long-range ordering between them, which is analogical with Wannier phase of the 2D triangular Ising antiferromagnet.

Low-Temperature Properties of Ising Antiferromagnet on a Stacked Triangular Lattice

An Ising antiferromagnet on a stacked triangular lattice in zero field is studied by Monte Carlo simulations, focusing on the character of the low-temperature phase and the effect of the relative strength of the exchange interaction in the stacking direction {\alpha}. Our results support the presence of the 3D Wannier phase, with the sublattice magnetization structure (m, -m, 0) and power-law decaying m with the lattice size. The extent of this low-temperature phase shrinks with decreasing {\alpha}, however, it appears even at very low values if it is accessed from higher temperatures by sufficiently slow cooling.

Bosonict–JModel in a Stacked Triangular Lattice and Its Phase Diagram

In this paper, we study phase diagram of a system of two-component hard-core bosons with nearest-neighbor (NN) pseudo-spin antiferromagnetic (AF) interactions in a stacked triangular lattice. Hamiltonian of the system contains three parameters one of which is the hopping amplitude $t$ between NN sites, and the other two are the NN pseudo-spin exchange interaction $J$ and the one that measures anisotropy of pseudo-spin interactions. We investigate the system by means of the Monte-Carlo simulations and clarify the low-temperature phase diagram. In particular, we are interested in how the competing orders, i.e., AF order and superfluidity, are realized, and also whether supersolid forms as a result of hole doping into the state of the $\sqrt{3}\times \sqrt{3}$ pseudo-spin pattern with the $120^o$ structure.

Transitions in three-dimensional XY magnets with two chiral order parameters

Two models of classic XY antiferromagnets in three dimensions are studied by Monte Carlo simulation: the model on a simple cubic lattice with two extra intralayer exchanges and the model on a stackedtriangular lattice with an extra interlayer exchange. In suggested models, the order parameters are magnetization and two chiral parameters. A transition corresponds to breaking ℤ2 ⊗ ℤ2 ⊗ SO(2) symmetry. A distinct first order transition is found in both models.

First order transition in three-dimensional systems with fully broken O(3) symmetry

Systems with fully broken O(3) symmetry are studied by Monte Carlo simulations. Two models of classical Heisenberg magnets in three dimensions are considered: the stacked three-exchange model on a simple cubic lattice and the model on a stacked-triangular lattice with two competing interlayer exchanges. The first order transition is found in both models. Pseudouniversal behavior of the models is found for small size lattices.

Microcanonical Simulation on the First Order Phase Transition of the XY Antiferromagnet on the Stacked Triangular Lattice

The nature of the phase transition of antiferromagnetic vector spin models on the stacked triangular lattice (STA) has been an issue of debate. In order to account for the coplanar non-collinear order in the ground state one needs two vector fields in a Ginzburg–Landau–Wilson Hamiltonian. The field theoretical renormalization group analysis shows that there is a threshold value Nc for the number of spin component N, and for N > Nc the transition is of second order belonging to a new ‘‘chiral’’ universality class, while for N < Nc the chiral fixed points ceases to exist in the real parameter space and the transition is of first order. According to a recent estimate based on a Monte Carlo renormalization group study, 3 < Nc < 8. 2) Therefore a first order transition is expected for the physically relevant XY and Heisenberg spin systems. The phase transition of the XY or Heisenberg STA is of very weak first order and is said to be of ‘‘almost second order’’ with large (but finite) correlation lengths due to a slow velocity region in the renormalization group flow. Monte Carlo simulations on systems with moderate size appear to show second order transitions with pseudo-critical behavior. Strong first order transitions were found for variants of the model on which a constraint of local rigidity was imposed which still belong to the same universality class as the original STA. Recently the first order nature of the phase transition in XY antiferromagnet on the stacked triangular lattice (XY-STA model) was demonstrated directly by canonical Monte Carlo simulations on large systems, in which the energy probability distribution shows a double peak structure near the transition temperature. The purpose of the present paper is to show the first order transition of the XY-STA model by applying a novel technique: a microcanonical method. To distinguish a weak first order transition from a second order one can be difficult, especially if the energy jump, the characteristic of a first order transition, is tiny as is the case for frustrated spin systems. Limited to accessible system sizes one may analyze the different power law behavior of the maximum of specific heat or susceptibility, with an exponent equal to three for a first order transition and less than three for a second order transition. The obvious solution to increase the size of the system runs into difficulties for the canonical simulations (at constant temperature) because of the growing autocorrelation times. Also the histogram method is then restricted to smaller intervals of temperature shrinking as the inverse power of the system size. In this short note we will show that the microcanonical simulations (at constant energy) can help to solve the problems encountered in canonical simulations. The method has been tested for strong first order transitions and also for second order ones. Before presenting the method and the results we have to describe the model we study. It is given by the usual Hamiltonian

Random fields and the partially paramagnetic state of CsCo0.83Mg0.17Br3: critical scattering study.

We have performed measurements of the critical neutron scattering on CsCo0.83Mg0.17Br3, a dilute stacked triangular lattice (STL) Ising antiferromagnet (AF). A two component line shape associated with the critical fluctuations appears at a temperature coincident with T(N1) observed in pure CsCoBr3. Such scattering is indicative of fluctuations in prototypical random field Ising model (RFIM) systems. The random field domain state arises in this case due to geometrical frustration within the STL Ising AF, which gives rise to a three sublattice Néel state, in which one sublattice is disordered. Magnetic vacancies nucleate AF domains in which the vacancies reside on the disordered sublattice thereby generating a RFIM state in the absence of an applied magnetic field.

Monte Carlo Renormalization Group Study of the Heisenberg and the XY Antiferromagnet on the Stacked Triangular Lattice and the Chiral φ4Model

With the help of the improved Monte Carlo renormalization-group scheme, we numerically investigate the renormalization group flow of the antiferromagnetic Heisenberg and XY spin model on the stacked triangular lattice (STA-model) and its effective Hamiltonian, 2N-component chiral $\phi^4$ model which is used in the field-theoretical studies. We find that the XY-STA model with the lattice size $126\times 144 \times 126$ exhibits clear first-order behavior. We also find that the renormalization-group flow of STA model is well reproduced by the chiral $\phi^4$ model, and that there are no chiral fixed point of renormalization-group flow for N=2 and 3 cases. This result indicates that the Heisenberg-STA model also undergoes first-order transition.

Tricritical behavior in a stacked triangular lattice Ising antiferromagnetCsCoBr3

${\mathrm{CsCoBr}}_{3}$ is well described as a spin-1/2 Ising-like antiferromagnet (AF) on a stacked triangular lattice (STL). The combination of AF interactions and triangular coordination leads to geometrical frustration, which manifests itself in an unusual partially paramagnetic, long-range ordered state at low temperatures. We have studied both the order parameter as well as the fluctuations in the order parameter (the critical fluctuations), associated with this unusual magnetic phase transition at ${T}_{N}=(28.23\ifmmode\pm\else\textpm\fi{}0.04)\mathrm{K},$ in a high quality single crystal sample of ${\mathrm{CsCoBr}}_{3}$ using neutron scattering techniques. Critical scattering measurements were performed through two equivalent ordering wave vectors $(\ensuremath{-}\frac{1}{3},\ensuremath{-}\frac{1}{3},1)$ and $(\ensuremath{-}\frac{2}{3},\ensuremath{-}\frac{2}{3},1).$ These measurements are well described by a single, anisotropic Lorentzian line shape over a wide range of about two decades in reduced temperature. The magnetic phase transition is characterized by a set of three critical exponents $\ensuremath{\nu}=(0.54\ifmmode\pm\else\textpm\fi{}0.05),$ $\ensuremath{\gamma}=(1.05\ifmmode\pm\else\textpm\fi{}0.08),$ and $\ensuremath{\beta}=(0.28\ifmmode\pm\else\textpm\fi{}0.02),$ which are consistent with the scaling laws. These results are very close to expectations of tricritical behavior. They are inconsistent with expectations from both analytical theory and Monte Carlo simulation results relevant to the classical Ising model on a hexagonal lattice with isotropic AF interactions, which predict that this phase transition belongs to the 3D $\mathrm{XY}$ universality class. In light of experimental results, we have reexamined heat capacity measurements on the same sample and show that these results are also consistent with tricritical behavior.

Spin- and chirality-orderings of frustrated magnets – stacked-triangular anti-ferromagnets and spin glasses

"Chirality" is a multispin quantity representing the sense or the handedness of the noncollinear spin structures induced by spin frustration. Recent studies have revealed that the chirality often plays an important role in the ordering of certain frustrated magnets. Here I take up two such examples, stacked-triangular anti-ferromagnets and spin glasses, where the inherent chiral degree of freedom affects underlying physics and might lead to novel ordering phenomena. The first topic is the criticality of the magnetic-phase transition of vector (i.e., XY or Heisenberg) anti-ferromagnets on the three-dimensional stacked-triangular lattice. The second topic is the nature of the spin-glass ordering. I will review the recent theoretical and experimental works on these topics, with particular emphasis on the important role played by the chirality. PACS Nos.: 67.70+n, 67.57Lm

Quantum Phase Transitions in the Triangular-Lattice Bilayer Heisenberg Model

We study the triangular lattice bilayer Heisenberg model with antiferromagnetic interplane coupling $J_\perp$ and nearest neighbour intraplane coupling $J= \lambda J_\perp$, which can be ferro- or antiferromagnetic, by expansions in $\lambda$. For negative $\lambda$ a phase transition is found to an ordered phase at a critical $\lambda_c=-0.2636 \pm 0.0001$ which is in the 3D classical Heisenberg universality class. For $\lambda>0$, we find a transition at a rather large $\lambda_c\approx 1.2$. The universality class of the transition is consistent with that of Kawamura's 3D antiferromagnetic stacked triangular lattice. The spectral weight for the triplet excitations, at the ordering wavevector, remains finite at the transition, suggesting that a phase with free spinons does not exist in this model.

Extended Mean-Field Analysis of the Stacked-Triangular Ising Antiferromagnet

Phase transitions of the frustrated Ising model on a stacked-triangular lattice with the antiferromagnetic nearest-neighbor and the ferromagnetic next-nearest-neighbor interactions are studied. For this system, an earlier mean-field analysis yielded successive phase transitions, with a partial disordered phase stabilized in an intermediate temperature range and ferrimagnetic phases stabilized at lower temperatures. We have performed an extended mean-field analysis based on the variational principle in which correlation effects in the frustrated triangular layer are taken into account. It is found that the partial disordered phase is largely suppressed relative to the ferrimagnetic phase due to the correlation effects.

Low temperature phase of the antiferromagnetic Ising model on a stacked-triangular lattice

The spin structure in the low temperature phase of a stacked-antiferromagnetic-triangular Ising lattice is investigated by the use of Monte Carlo simulations. We conclude that the structure, ( M, − M 2 , − M 2 ), proposed by Blankschtein et al., is one of the solutions but may not be realized as a stable solution.

Geometrically-frustrated magnetism on triangular and tetrahedral lattices

The combination of simple, nearest neighbor antiferromagnetic interactions and certain lattice symmetries leads to phenomena known broadly as frustration. In such systems, the formation of a collinear magnetic structure cannot occur at low temperature. The material may undergo a phase transition to an unusual magneticallyordered state, or it may not undergo a conventional transition at all-entering a glass-like state as the temperature is lowered. This paper will review recent work, mostly from neutron scattering studies, on three such magnetic materials. CsMnBr3 and CsCoBr3 are XY-like and Ising-like stacked-triangular lattice antiferromagnets, respectively. Both undergo magnetic phase transitions with interesting properties due to the presence of geometricalfrustration within the triangular layers. TD2Mo2O7 is a pyrochlore antiferromagnet, in which the magnetic moments reside on a network of corner-sharing tetrahedra. This material enters a magnetic glass-like state at low temperature.

Enhancement of anisotropy due to fluctuations in quasi-one-dimensional antiferromagnets.

It is shown that the observed anisotropy of magnetization at high magnetic fields in ${\mathrm{RbMnBr}}_{3}$, a quasi-one-dimensional antiferromagnet on a distorted stacked triangular lattice, is due to quantum and thermal fluctuations. These fluctuations are taken into account in the framework of linear spin-wave theory in the region of strong magnetic fields. In this region the divergent one-dimensional integrals are cut off by magnetic field and the bare easy-plane anisotropy. Logarithmical dependence on the cutoff leads to the ``enhancement'' of the anisotropy in magnetization. Comparison between magnetization data and our theory with parameters obtained from neutron scattering experiments has been done.

Spin structure of the antiferromagnetic continuous-spin Ising model on a stacked-triangular lattice

The spin structure of the antiferromagnetic continuous-spin ( S=∞) Ising model on a stacked-triangular lattice is investigated by the use of Monte Carlo simulations. We find that the spin structure is the ferrimagnetic structure ( M, − m, − m) in the low temperature region and the partial-disorder structure ( M′, 0, −- M′) in the intermediate temperature region.