Long-range order produced by the interaction between spin waves in classical fcc Heisenberg models (original) (raw)
Related papers
Physical Review B
We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-1 2 J1-J2 Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings J1 > 0 and J2 ≡ κJ1 > 0, respectively, in the window 0 ≤ κ < 1. The classical version of the model has a single GS phase transition at κ cl = 1 8 in this window from a phase with 3-sublattice antiferromagnetic (AFM) 120 • Néel order for κ < κ cl to an infinitely degenerate family of 4-sublattice AFM Néel phases for κ > κ cl. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case κ = 0 of the spin-1 2 model for the GS energy per spin, E/N = −0.5521(2)J1, and the GS magnetic order parameter, M = 0.198(5) (in units where the classical value is M cl = 1 2), which are among the best available. For the spin-1 2 J1-J2 model we find that the classical transition at κ = κ cl is split into two quantum phase transition at κ c 1 = 0.060(10) and κ c 2 = 0.165(5). The two quasiclassical AFM states (viz., the 120 • Néel state and the striped state) are found to be the stable GS phases in the regime κ < κ c 1 and κ > κ c 2 , respectively, while in the intermediate regimes κ c 1 < κ < κ c 2 the stable GS phase has no evident long-range magnetic order.
Quasi-long-range order in random-anisotropy Heisenberg models
1998
Monte Carlo simulations have been used to study a discretized Heisenberg ferromagnet (FM) with random uniaxial single-site anisotropy on L × L × L simple cubic lattices, for L up to 64. The spin variable on each site is chosen from the twelve [110] directions. The random anisotropy has infinite strength and a random direction on a fraction x of the sites of the lattice, and is zero on the remaining sites. In many respects the behavior of this model is qualitatively similar to that of the corresponding random-field model. Due to the discretization, for small x at low temperature there is a [110] FM phase. For x > 0 there is an intermediate quasi-long-range ordered (QLRO) phase between the paramagnet and the ferromagnet, which is characterized by a |k| −3 divergence of the magnetic structure factor S(k) for small k, but no true FM order. At the transition between the paramagnetic and QLRO phases S(k) diverges like |k| −2 . The limit of stability of the QLRO phase is somewhat greater than x = 0.5. For x close to 1 the low temperature form of S(k) can be fit by a Lorentzian, with a correlation length estimated to be 11 ± 1 at x = 1.0 and 25 ± 5 at x = 0.75.
A model for a Heisenberg antiferromagnet on a pyrochlore lattice with exchange and dipole-dipole interactions is studied via mean-field theory. In treating the dipoles by use of the Ewald method, a soft (critical) mode with a unique ordering wave vector is selected for all strengths of the dipoledipole coupling. For weak dipoles a partially ordered, three sublattice spin structure (P state), with q ord = 1 2 1 2 1 2 , is selected. A fully ordered, four sublattice spin structure (F state), with q = 000, competes with the P state and becomes the stable structure as the temperature is reduced. Our results are compared against other theoretical calculations and connection to recent experimental results for Gd2Ti2O7 are discussed.
Order by disorder and phase transitions in a highly frustrated spin model on the triangular lattice
Physical Review B, 2011
Frustration has proved to give rise to an extremely rich phenomenology in both quantum and classical systems. The leading behavior of the system can often be described by an effective model, where only the lowest-energy degrees of freedom are considered. In this paper we study a system corresponding to the strong trimerization limit of the spin 1/2 kagome antiferromagnet in a magnetic field. It has been suggested that this system can be realized experimentally by a gas of spinless fermions in an optical kagome lattice at 2/3 filling. We investigate the low-energy behavior of both the spin 1/2 quantum version and the classical limit of this system by applying various techniques. We study in parallel both signs of the coupling constant J since the two cases display qualitative differences. One of the main peculiarities of the J > 0 case is that, at the classical level, there is an exponentially large manifold of lowest-energy configurations. This renders the thermodynamics of the system quite exotic and interesting in this case. For both cases, J > 0 and J < 0, a finitetemperature phase transition with a breaking of the discrete dihedral symmetry group D6 of the model is present. For J < 0, we find a transition temperature T < c /|J| = 1.566 ± 0.005, i.e., of order unity, as expected. We then analyze the nature of the transition in this case. While we find no evidence for a discontinuous transition, the interpretation as a continuous phase transition yields very unusual critical exponents violating the hyperscaling relation. By contrast, in the case J > 0 the transition occurs at an extremely low temperature, T > c ≈ 0.0125 J. Presumably this low transition temperature is connected with the fact that the low-temperature ordered state of the system is established by an order-by-disorder mechanism in this case.
Magnetic order on a frustrated spin-1/2 Heisenberg antiferromagnet on the Union Jack lattice
Physical Review B, 2010
We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength J 1 > 0, but only half the next-nearest-neighbor pairs are connected by bonds with a strength J 2 ≡ κJ 1 > 0. The bonds are arranged such that on the 2 × 2 unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus 4-connected and 8-connected. We find strong evidence for a first phase transition between a Néel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling κ c 1 = 0.66 ± 0.02. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value M c 1 = 0.195 ± 0.005 on both sides of the transition, which is more typical of a first-order transition. The slope, dM/dκ, of the order parameter curve as a function of the coupling strength κ, also appears to be continuous, or very nearly so, at the critical point κ c 1 , thereby providing further evidence of the subtle nature of the transition between the Néel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of κ, and hence we have also used the CCM with a model collinear semi-stripeordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by J 2-bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semi-stripe-ordered ferrimagnetic states at a large value of the coupling parameter around κ c 2 ≈ 125 ± 5. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for κ c 2 is hence less reliable than that for κ c 1. Nevertheless, if this second transition at κ c 2 does exist, our results clearly indicate it to be of first-order type.
IEEE Transactions on Magnetics, 2009
Magnetization is measured in experiments for both ferromagnetic and anti-ferromagnetic materials to investigate the magnetic properties of materials, and the susceptibility of total magnetization as a function of external field is used to determine Neel or Cueri temperatures. In the Monte Carlo simulation, it is important to define the proper order parameters to describe the spin model, where the magnetization is used as an order parameter for ferromagnetic spin model and the staggered magnetization is used as an order parameter for the anti-ferromagnetic spin model without geometrical flustration. However, it is difficult to define an order parameter for flustrated spin models. We perform the Monte Carlo simulation for the anti-ferro Heigenberg spin model using the damage spreading as an order paramter, and also perform simulation using both the magnetization and the staggered magnetization as order paramters. Then, we measure the critical temperatures and the critical exponents on Random networks estimated by different order parameters, and then study the dependency of critical behaviors on different order parameters for the anti-ferro Heigenberg spin models.
Magnetic ordering in the three-dimensional site-disordered Heisenberg model
Physical Review B, 1996
simulations have been carried out on a simple cubic ferromagnet with nearest-neighbor interactions. In order to model the effects of site frustration, a fraction f of the sites are occupied at random by moments that couple antiferromagnetically ͑AF͒ to their neighbors. When the concentration of AF sites is less than ϳ 1 6 , the system has one magnetic transition from paramagnet to ferromagnet at a critical temperature T c. For f Ͼ 1 6 the system exhibits a second distinct ordering event at a lower temperature T xy , where the transverse spin components freeze out leading to an increase in total spin length. Below T xy the system is in a mixed state, in that the z components of the spins are ferromagnetically ordered while the transverse components exhibit AF correlations. The approximate magnetic phase diagram for our model is consistent with experimental results on site-disordered systems such as Eu 1Ϫx Gd x S and Fe 3Ϫx Mn x Si.
2010
We study the ground state phases of the S=1/2S=1/2S=1/2 Heisenberg quantum antiferromagnet on the spatially anisotropic triangular lattice and on the square lattice with up to next-next-nearest neighbor coupling (the J1J2J_3J_1J_2J_3J_1J_2J_3 model), making use of Takahashi's modified spin-wave (MSW) theory supplemented by ordering vector optimization. We compare the MSW results with exact diagonalization and projected-entangled-pair-states calculations, demonstrating their qualitative and quantitative reliability. We find that MSW theory correctly accounts for strong quantum effects on the ordering vector of the magnetic phases of the models under investigation: in particular collinear magnetic order is promoted at the expenses of non-collinear (spiral) order, and several spiral states which are stable at the classical level, disappear from the quantum phase diagram. Moreover, collinear states and non-collinear ones are never connected continuously, but they are separated by parameter regions in which MSW breaks down, signaling the possible appearance of a non-magnetic ground state. In the case of the spatially anisotropic triangular lattice, a large breakdown region appears also for weak couplings between the chains composing the lattice, suggesting the possible occurrence of a large non-magnetic region continuously connected with the spin-liquid state of the uncoupled chains.
Magnetic order in a spin-12 interpolating square-triangle Heisenberg antiferromagnet
Physical Review B, 2009
Using the coupled cluster method ͑CCM͒ we study the zero-temperature phase diagram of a spin-half Heisenberg antiferromagnet ͑HAF͒, the so-called J 1-J 2 Ј model, defined on an anisotropic two-dimensional lattice. With respect to an underlying square-lattice geometry the model contains antiferromagnetic ͑J 1 Ͼ 0͒ bonds between nearest neighbors and competing ͑J 2 ЈϾ0͒ bonds between next-nearest neighbors across only one of the diagonals of each square plaquette, the same diagonal in every square. Considered on an equivalent triangular-lattice geometry the model may be regarded as having two sorts of nearest-neighbor bonds, with J 2 ЈϵJ 1 bonds along parallel chains and J 1 bonds providing an interchain coupling. Each triangular plaquette thus contains two J 1 bonds and one J 2 Ј bond. Hence, the model interpolates between a spin-half HAF on the square lattice at one extreme ͑ =0͒ and a set of decoupled spin-half chains at the other ͑ → ϱ͒, with the spin-half HAF on the triangular lattice in between at = 1. We use a Néel state, a helical state, and a collinear stripe-ordered state as separate starting model states for the CCM calculations that we carry out to high orders of approximation ͑up to eighth order, n = 8, in the localized subsystem set of approximations, LSUBn͒. The interplay between quantum fluctuations, magnetic frustration, and varying dimensionality leads to an interesting quantum phase diagram. We find strong evidence that quantum fluctuations favor a weakly first-order or possibly second-order transition from Néel order to a helical state at a first critical point at c 1 = 0.80Ϯ 0.01 by contrast with the corresponding second-order transition between the equivalent classical states at cl = 0.5. We also find strong evidence for a second critical point at c 2 = 1.8Ϯ 0.4 where a first-order transition occurs, this time from the helical phase to a collinear stripe-ordered phase. This latter result provides quantitative verification of a recent qualitative prediction of and Starykh and Balents ͓Phys. Rev. Lett. 98, 077205 ͑2007͔͒ based on a renormalization group analysis of the J 1-J 2 Ј model that did not, however, evaluate the corresponding critical point.
Physical review. B, Condensed matter, 1989
We have calculated the probability distribution for staggered magnetization at T =0 for the twodimensional (2D) antiferromagnetic Heisenberg model (2D AFH) on a series of finite lattices up to 26 sites. We find that the singlet ground state of the 2D AFH possesses long-range magnetic order without broken symmetry. We also study the lowest triplet state and find that it becomes degenerate with the ground state in the thermodynamic limit. This state does exhibit broken symmetry on a finite lattice. The value of the staggered magnetization in the thermodynamic limit is also obtained by extrapolation. We compare our results with the results obtained by other methods and discuss the relevance to the high-T, superconductive oxide compounds.