The ordering of the classical Heisenberg antiferromagnet on the triangular lattice with the the bilinear-biquadratic interaction is studied by Monte Carlo simulations. It is shown that the model exhibits a topological phase transition at a finite-temperature driven by topologically stable vortices, while the spin correlation length remains finite even at and below the transition point. The relevant vortices could be of three different types, depending on the value of the biquadratic coupling. Implications to recent experiments on the triangular antiferromagnet NiGa$_2$S$_4$ is discussed.
Classical Heisenberg and XY models on zigzag ladder lattices with nearest-neighbor bilinear-biquadratic exchange: Exact solution for the ground-state problem
We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relations to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above Néel order occurs for a large range of values of the relative strength of the bilinear ($-J_1$) and biquadratic ($-J_2$) interaction terms. The proof uses the method of reflection positivity and infrared bounds. Links between spin correlations and loop correlations are proved.
We study the spin-1 bilinear-biquadratic model on the complete graph of N sites, i.e., when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of su(3) and su(2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.
The three-dimensional XY model with bilinear-biquadratic exchange interactions $J$ and $J'$, respectively, has been studied by Monte Carlo simulations. From the detailed analysis of the thermal variation of various physical quantities, as well as the order parameter and energy histogram analysis, the phase diagram including two different ordered phases has been determined. There is a single phase boundary from a paramagnetic to a dipole-quadrupole ordered phase, which is of second order in a high $J/J'$ ratio region, changing to a first-order one for $0.35 \leq J/J' \leq 0.5$. Below $J/J'=0.35…
We study thermodynamic properties as well as the dynamical spin and quadrupolar structure factors of the O(3)-symmetric spin-1 Heisenberg model with bilinear-biquadratic exchange interactions on the triangular lattice. Based on a sign-problem-free quantum Monte Carlo approach, we access both the ferromagnetic and the ferroquadrupolar ordered, spin nematic phase as well as the SU(3)-symmetric point which separates these phases. Signatures of Goldstone soft-modes in the dynamical spin and the quadrupolar structure factors are identified, and the properties of the low-energy excitations are compa…
Recent theoretical studies raised the possibility of a realization of spin nematic states in the S=1 triangular lattice compound NiGa_2S_4. We study the bilinear-biquadratic spin 1 chain in a zig-zag geometry by means of the density matrix renormalization group method and exact diagonalization. We present the phase diagram focusing on antiferromagnetic interactions. Adjacent to the known Haldane-double Haldane and the extended critical phase with dominant spin nematic correlations we find a trimerized phase with a nonvanishing energy gap. We discuss results for different order parameters, ener…
We study the pyrochlore Heisenberg antiferromagnet with additional positive biquadratic interaction in the semiclassical limit. The classical ground state manifold of the model contains an extensively large family of non-coplanar spin states known as "color ice states". Starting from a color ice state, a subset of spins may rotate collectively at no energy cost. Such excitation may be viewed in this three-dimensional system as a "membrane-like" analog of the well-known weathervane modes in the classical kagome Heisenberg antiferromagnet. We investigate the weathervane modes in detail and eluci…
We consider a finite size scaling function across a topological phase transition in 1D models. For models of non-interacting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition (Gulden et al 2016). Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/ξ$, where $L$ is t…
In a previous work a model was proposed for the phase transitions of crystals with localized magnetic moments which at low temperature have a "conical" arrangement that at higher T transforms into a more symmetrical structure (depending on the compound) before becoming totally disordered. The model assumes bilinear and biquadratic interactions between magnetic moments up to the fifth neighbours, and for any given T the structure with the least free energy is obtained by a mean-field approximation (MFA). The interaction constants are derived from ab initio energy calculations. In this work we i…
A theoretical study of the specific heat C(T) as a function of temperature in Fibonacci magnetic superlattices is presented. We consider quasiperiodic structures composed of ferromagnetic films, each described by the Heisenberg model, with biquadratic and bilinear coupling between them. We have taken the ratios between the biquadratic and bilinear exchange terms according to experimental data recently measured for different regions of their regime. Although some previous properties of the spin wave specific heat are also reproduced here, new features appear in this case, the most important of …
Using a generalized Jordan-Wigner transformation combined with the defining representation of the SO(N) spin, we map the SO(N) bilinear-biquadratic(BLBQ) spin chain into the N-color bosonic particle model. We find that, when the Jordan-Wigner transformation disentangles the symmetry-protected topological entanglement, this bosonic model becomes negative-sign free in the context of quantum Monte-Carlo simulation. For the SO(3) case, moreover, the Kennedy-Tasaki transformation for the S=1 BLBQ chain, which is also a topological disentangler, derives the same bosonic model through the dimer-R bas…
Histogram Monte Carlo simulation is used to investigate effects of biquadratic exchange $J_{2}$ on phase transitions of a 3D classical XY antiferromagnet with frustration induced by the antiferromagnetic exchange $J_{1}$ and the stacked triangular lattice geometry. The biquadratic exchange is considered negative (antiferroquadrupolar) within the triangular planes and positive (ferroquadrupolar) between the planes. The phase diagram obtained features a variety of interesting phenomena arising from the presence of both the biquadratic exchange and frustration. In a strong biquadratic exchange li…
We present a new algorithm for quantum Monte Carlo simulation based on global updating with loops. While various theoretical predictions are confirmed in one dimension, we find, for S=1 systems on a square lattice with an antiferromagnetic biquadratic interaction, that the intermediate phase between the antiferromagnetic and the ferromagnetic phases is disordered and that the two phase transitions are both of the first order in contrast to the one-dimensional case. It is strongly suggested that the transition points coincide those at which the algorithm changes qualitatively.
Monte Carlo simulation using the standard single-spin flip algorithm often fails to sample over the entire configuration space at low temperatures for frustrated spin systems. A typical example is a class of spin-ice type Ising models. In this case, the difficulty can be avoided by introducing a global-flip algorithm, the loop algorithm. Similar difficulty is encountered in O(3) Heisenberg models in the presence of biquadratic interaction. The loop algorithm, however, is not straightforwardly applied to this case, since the system does not have a priori spin-anisotropy axis for constructing th…
The ground state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected trivial phases belong to the Gaussian universality class with central charge c = 1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universa…
The classical bilinear-biquadratic nearest-neighbor Heisenberg antiferromagnet on the pyrochlore lattice does not exhibit conventional Neel-type magnetic order at any temperature or magnetic field. Instead spin correlations decay algebraically over length scales r ~ \sqrt{T}, behavior characteristic of a Coulomb phase arising from a strong local constraint. Despite this, its thermodynamic properties remain largely unchanged if Neel order is restored by the addition of a degeneracy-lifting perturbation, e.g., further neighbor interactions. Here we show how these apparent contradictions can be r…
Effect of biquadratic exchange on phase transitions of a planar classical Heisenberg (or XY) ferromagnet on a stacked triangular lattice is investigated by Standard Monte Carlo and Histogram Monte Carlo simulations in the region of a bilinear to biquadratic exchange interaction ratio $J_{1}/J_{2} \leq 1$. The biquadratic exchange is found to cause separate second-order phase transitions in a strong biquadratic exchange limit, followed by simultaneous dipole and quadrupole ordering, which is of first order for an intermediate range of the exchange ratio and changes to a second-order one again a…
We investigate the zero-temperature behavior of the classical Heisenberg model on the triangular lattice in which the competition between exchange interactions of different orders favors a relative angle between neighboring spins in the interval (0,2pi/3). In this situation, the ground states are noncoplanar and have an infinite discrete degeneracy. In the generic case, the set of the ground states is in one to one correspondence (up to a global rotation) with the non-crossing loop coverings of the three equivalent honeycomb sublattices into which the bonds of the triangular lattice can be par…
Motivated by puzzling spin-glass behaviors observed in many pyrochlore-based magnets, effects of magnetoelastic coupling to local lattice distortions were recently studied by the authors for a bond-disordered antiferromagnet on a pyrochlore lattice [Phys. Rev. Lett. 107, 047204 (2011)]. Here, we extend the analyses with focusing on the critical property of the spin-glass transition which occurs concomitantly with a nematic transition. Finite-size scaling analyses are performed up to a larger system size with 8192 spins to estimate the transition temperature and critical exponents. The exponent…
We investigate the isotropic two-leg S=1/2 ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians which have a matrix product wavefunction as exact ground state. We demonstrate that a smooth change of parameters leads one from the S=1/2 dimer and Majumdar-Ghosh chains to the S=1 chain with biquadratic exchange. This proves that these model systems are in the same phase. We also present a new set of models of frustrated S=1/2 spin chains (including only bilinear NN and NNN interactions) whose ground states can be fou…
The quasi-one-dimensional S=1 Heisenberg antiferromagnet with a biquadratic term is investigated at zero temperature by quantum Monte Carlo simulation. As the magnitude of the inter-chain coupling is increased, the system undergoes a phase transition from a spontaneously dimerized phase to a Néel ordered or spin nematic phase. The numerical results suggest the possibility of an unconventional second-order transition in which the symmetry group characterizing one phase is not a subgroup of the other.
An exactly solvable variant of mixed spin-(1/2,1) Ising-Heisenberg diamond chain is considered. Vertical spin-1 dimers are taken as quantum ones with Heisenberg bilinear and biquadratic interactions and with single-ion anisotropy, while all interactions between spin-1 and spin-1/2 residing on the intermediate sites are taken in the Ising form. The detailed analysis of the $T=0$ ground state phase diagram is presented. The phase diagrams have shown to be rather rich, demonstrating large variety of ground states: saturated one, three ferrimagnetic with magnetization equal to 3/5 and another four…
In frustrated antiferromagnets with isotropic exchange interactions, there is typically a manifold of degenerate classical ground states. This degeneracy is broken by the (free) energy of quantum or thermal fluctuations, or the uniform effects of bond disorder. We derive effective Hamiltonians to express each kind of selection effect, in both exact forms and convenient approximate forms. It is argued that biquadratic terms, representing the collinear-selecting effects of quantum fluctuations, should be included in classical simulations of large-$S$ frustrated magnets at low temperatures
The dynamic phase transitions have been studied, within a mean-field approach, in the kinetic spin-1 Ising model Hamiltonian with arbitrary bilinear and biquadratic pair interactions in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The nature (first- or second-order) of the transition is characterized by investigating the behavior of the thermal variation of the dynamic order parameters. The dynamic phase transitions (DPTs) are obtained and the phase diagrams are constructed in the temperature and magnetic field amplitude plane and fo…
For Ising spin models which bear the spin-ice type macroscopic (quasi-)degeneracy, conventional classical Monte Carlo (MC) simulation using single spin flips suffers from dynamical freezing at low temperatures ($T$). A similar difficulty is seen also in a family of Heisenberg spin models with easy-axis anisotropy or biquadratic interactions. In the Ising case, the difficulty is avoided by introducing a non-local update based on the loop algorithm. We present an extension of the loop algorithm to the Heisenberg case. As an example of its application, we review our recent study on spin-glass (SG…
We analyze the onset of incommensurabilities around the VBS point of the S=1 bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point $θ_{\rm vbs}$. Moreover, the theory predicts another special point $θ_{\rm disp}$, distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense.
We theoretically investigate the Heisenberg antiferromagnet on a triangular lattice doped with nonmagnetic impurities. Two nontrivial effects resulting from collective impurity behavior are predicted. The first one is related to presence of uncompensated magnetic moments localized near vacancies as revealed by the low-temperature Curie tail in the magnetic susceptibility. These moments exhibit an anomalous growth with the impurity concentration, which we attribute to the clustering mechanism. In an external magnetic field, impurities lead to an even more peculiar phenomenon lifting the classic…
We consider a classical XY-like Hamiltonian on a body-centered tetragonal lattice, focusing on the role of interlayer frustration. A three-dimensional (3D) ordered phase is realized via thermal fluctuations, breaking the mirror-image reflection symmetry in addition to the XY symmetry. A heuristic field-theoretical model of the transition has a decoupled fixed point in the 3D XY universality, and our Monte Carlo simulation suggests that there is such a temperature region where long-wavelength fluctuations can be described by this fixed point. However, it is shown using scaling arguments that th…
The spin-1 Ising (BEG) model with the nearest-neighbour bilinear and biquadratic interactions and single-ion anisotropy is simulated on a cellular automaton which improved from the Creutz cellular automaton(CCA) for a simple cubic lattice. The simulations have been made for several sets of parameters $K/J$ and $D/J$ in the $-3<D/J\leq 0$ and $-1\leq K/J\leq 0$ parameter regions. The re-entrant and double re-entrant phase transitions of the BEG model are determined from the temperature variations of the thermodynamic quantities ($M$, $Q$ and $χ$). The phase diagrams characterizing phase transit…
The incipient quantum phase transitions of relevance to nonzero fluctuations and entanglement in Heisenberg clusters are studied in this paper by exploiting negativity as a measure in bipartite and frustrated spin-1 anisotropic Heisenberg clusters with bilinear-biquadratic exchange, single-ion anisotropy and magnetic field. Using the exact diagonalization technique, it is shown that quantum critical points signaled by qualitative changes in behavior of magnetization and particle number are ultimately related to microscopic entanglement and collective excitations. The plateaus and peaks in spin…
