Two-dimensional Heisenberg Model Research Articles

Phase diagram of the two-dimensional dipolar Heisenberg model with Dzyaloshinskii-Moriya interaction and Ising anisotropy

We study phase transitions in the two-dimensional Heisenberg model with Dzyaloshinskii-Moriya interaction, Ising anisotropy $\ensuremath{\eta}$, and dipolar interaction under zero and finite magnetic fields $H$. For three typical strengths (zero, weak, and strong) of the dipolar interaction, we present the $H\text{\ensuremath{-}}\ensuremath{\eta}$ phase diagrams by estimating order parameters for skyrmion-lattice and helical phases and in-plane magnetization using a Monte Carlo method with an $O(N)$ algorithm. We find in the phase diagrams three types of skyrmion-lattice phases, i.e., two square lattices and a triangular lattice, helical phases with diagonal and vertical (or horizontal) stripes, a canted ferromagnetic phase, and a polarized ferromagnetic phase. The effect of the dipolar interaction varies the types of the skyrmion and helical phases in a complex manner. The dipolar interaction also expands the regions of the ordered phases accompanying shifts of the phase boundaries to the positive $H$ and $\ensuremath{\eta}$ directions and causes an increase of the density of skyrmions and shortening of the pitch length (stripe width) of helical structures. We discuss the details of the features of the phase transitions.

Population control bias and importance sampling in full configuration interaction quantum Monte Carlo

Population control is an essential component of any projector Monte Carlo algorithm. This control mechanism usually introduces a bias in the sampled quantities that is inversely proportional to the population size. In this paper, we investigate the population control bias in the full configuration interaction quantum Monte Carlo method. We identify the precise origin of this bias and quantify it in general. We show that it has different effects on different estimators and that the shift estimator is particularly susceptible. We derive a re-weighting technique, similar to the one used in diffusion Monte Carlo, for correcting this bias and apply it to the shift estimator. We also show that by using importance sampling, the bias can be reduced substantially. We demonstrate the necessity and the effectiveness of applying these techniques for sign-problem-free systems where this bias is especially notable. Specifically, we show results for large one-dimensional Hubbard models and the two-dimensional Heisenberg model, where corrected FCIQMC results are comparable to the other high-accuracy results.

Cluster mean-field theory studies of the frustrated two-dimensional quantum-mechanical [formula omitted] Heisenberg model

The phase diagrams of the two-dimensional quantum-mechanical Heisenberg model with antiferromagnetic nearest and next-nearest neighbor exchange interaction is investigated with the cluster mean-field theory. The four different scenarios considered are the J1−J2 model with only S=1∕2 spins, the J1−J2 model with only S=1 spins, and the mixed S=1∕2 and S=1J1−J2 model. Within the ferrimagnetic J1−J2 model, two different configurations of the spin order are assumed and investigated: checkerboard-like order of the S=1∕2 and S=1 spins, and a line order, where the S=1∕2, respectively, S=1 sublattice spins are ordered along lines which itself are ordered in a periodic sequence. The phase diagrams are characterized by calculating the energy per atom E and the magnetization m=|〈Sz〉|. In the antiferromagnetic S=1∕2J1−J2 model, the quantum states and the entanglement (von Neumann entropy) are investigated as a function of J2∕J1.

The Two-Dimensional Infinite Heisenberg Classical Square Lattice: Zero-Field Partition Function and Correlation Length

The nonlinear σ-model has known a new interest for it allows to describe the properties of two-dimensional quantum antiferromagnets which, when properly doped, become superconductors up to a critical temperature notably high compared to other types of superconducting materials. This model has been conjectured to be equivalent at low temperatures to the two-dimensional Heisenberg model. In this article we rigorously examine 2d-square lattices composed of classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings). A general expression of the characteristic polynomial associated with the zero-field partition function is established for any lattice size. In the infinite-lattice limit a numerical study allows to select the dominant term: it is written as a l-series of eigenvalues, each one being characterized by a unique index l whose origin is explained. Surprisingly the zero-field partition function shows a very simple exact closed-form expression valid for any temperature. The thermal study of the basic l-term allows to point out crossovers between l- and (l+1)-terms. Coming from high temperatures where the l=0-term is dominant and going to zero Kelvin, l-eigen¬values showing increasing l-values are more and more selected. At absolute zero l becomes infinite and all the successive dominant l-eigenvalues become equivalent. As the z-spin correlation is null for positive temperatures but equal to unity (in absolute value) at absolute zero the critical temperature is absolute zero. Using an analytical method similar to the one employed for the zero-field partition function we also give an exact expression valid for any temperature for the spin-spin correlations as well as for the correlation length. In the zero-temperature limit we obtain a diagram of magnetic phases which is similar to the one derived through a renormalization approach. By taking the low-temperature limit of the correlation length we obtain the same expressions as the corresponding ones derived through a renor¬malization process, for each zone of the magnetic phase diagram, thus bringing for the first time a strong validation to the full exact solution of the model valid for any temperature.

Spin s representations via sigma models* *The research of AMG and AMER was supported by the Natural Sciences and Engineering Research Council of Canada operating grant of one of the authors (AMG).

We establish and analyze a new relationship between the matrix functions describing spin fields of a spin s, where , and two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the anti-Hermitian immersion functions of their soliton surfaces in the algebra. We provide a geometric interpretation of this construction. For the spin fields which can be represented as linear combinations of the generalized Pauli matrices, we find the dynamics equation satisfied by their coefficients. This equation is identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for matrices congruent to the generalized Pauli matrices through any coordinate-independent unitary linear transformation. These properties allow for new interpretations of the spins as compositions of more elementary objects. They also open up the possibility of future applications of the sigma models to the situations which depend on spin behaviour, including spintronics, spin glasses and quantum computing.

Dynamical scaling analysis of symmetry breaking for the antiferromagnetic triangular Heisenberg model in a uniform magnetic field

Phase transitions for the two-dimensional antiferromagnetic triangular Heisenberg model in a uniform field are primarily analyzed using dynamical scaling analysis. The system has ${Z}_{3}$ and $O(2)$ symmetries. The transition temperatures and types of transition are investigated using the relaxation of order parameters and the asymptotic behaviors of relaxation time around transition temperatures. In the low-field range, the order parameters for the ${Z}_{3}$ and $O(2)$ symmetries exhibit the behavior typical of a second-order transition and Kosterlitz-Thouless (KT) transition, respectively, with distinct transition temperatures. In the high-field range, an order parameter for the ${Z}_{3}$ symmetry exhibits the behavior of a second-order transition, while another order parameter related to the components perpendicular to the field demonstrates that the relaxation time diverges with an asymptotic form, indicating a second-order transition rather than a KT transition. These transition temperatures are estimated to be close. Critical exponents for these two order parameters are estimated separately by calculating the relaxation of fluctuations in the high-field range. It was confirmed that the critical exponents for both of these order parameters change continuously as a function of field, and they are distinct from each other.

Thermal Entanglement in the Quantum XXZ Model in Triangular and Bilayer Honeycomb Lattices

Thermal entanglement is studied in the frustrated two-dimensional Heisenberg model in triangular and honeycomb lattices employing linear spin waves. Our results display a strong effect of the coupling parameters of next-nearest neighbors interaction $$\alpha =J'/J$$ on entanglement at $$T\rightarrow 0$$ , where the spin Hall conductivity is nonzero for a value of external field H. We employ linear spin waves to investigate the entanglement with T and next-nearest neighbor interaction $$J'$$. For the bilayer honeycomb lattice, we analyze the entanglement as a function of inter-chain coupling $$J^{\perp }$$, and in this case we find that the entanglement tends to zero at $$T=0$$, where it decreases with $$J^{\perp }$$ for higher temperatures.

Entanglement and quantum phase transition in the anisotropic two-dimensional XXZ model

The thermal entanglement in the spin-1 two-dimensional Heisenberg antiferromagnetic chain, with ion-single anisotropy, square lattice and in the neighborhood of critical anisotropy Dc is studied employing Schwinger bosons. In particular, we discuss the effect of the variation of the critical anisotropy Dc that separates the topological order of vortices for D<Dc from the disordered phase for D>Dc, on thermal entanglement. Furthermore, we analyze the effect of quantum phase transition described in the diagram Dc vs. J2 (second neighbors interactions ) on entanglement for a particular class of two-dimensional Heisenberg models which presents different exchange interactions Jx and Jy on x and y directions on square lattice.

Quantum-to-classical correspondence in two-dimensional Heisenberg models

The quantum-to-classical correspondence (QCC) in spin models is a puzzling phenomenon where the static susceptibility of a quantum system agrees with its classical-system counterpart, at a different corresponding temperature, within the systematic error at a sub-percent level. We employ the bold diagrammatic Monte Carlo method to explore the universality of QCC by considering three different two-dimensional spin-1/2 Heisenberg models. In particular, we reveal the existence of QCC in two-parametric models.

Anisotropy-temperature phase diagram for the two-dimensional dipolar Heisenberg model with and without magnetic field

We investigate phase transitions in the two-dimensional dipolar Heisenberg model with uniaxial anisotropy with a specific ratio between the exchange and dipolar constants, $\delta=1$. We obtain the $\eta$--$T$ (anisotropy vs. temperature) phase diagrams for typical values of magnetic field by a Monte Carlo method with an $O(N)$ algorithm. We find that at lower fields, the $\eta$--$T$ phase diagram consists of the planar ferromagnetic (F), (perpendicular) stripe-ordered (SO), and paramagnetic (P) phases, and is characterized by the triple point. In the SO phase realized at larger $\eta$ and smaller $T$, the SO pattern changes depending on the field. On the other hand, we find that at higher fields, the SO phase does not exist, while the planer F phase robustly remains. We study the properties of the phase boundaries by employing finite-size-scaling analyses. We find that the slope of the spin-reorientation-transition line is positive with and without field, i.e., $\frac{d\eta}{dT}>0$, which implies that the planar F phase changes to the SO phase with lowering temperature. In the phase diagrams we observe a characteristic shape of the P--planer F phase-transition line, whose maximum point of $\eta$ is located at an intermediate temperature. This structure leads to the temperature-induced reentrant transition associated with P and planar F phases, which appears in successive phase transitions with lowering temperature: P $\rightarrow$ planar F $\rightarrow$ P $\rightarrow$ SO phase at lower fields and P $\rightarrow$ planar F $\rightarrow$ P phases at higher fields.

Numerical Investigation of Magnetic Bubble Types in a Two-Dimensional Ferromagnetic System with Dipole–Dipole Interactions

Landau–Lifshitz equation for a two-dimensional anisotropic Heisenberg model with dipole–dipole interactions is solved numerically to investigate magnetic bubbles. Type I magnetic bubbles (magnetic ...

Criticality and scaling corrections for two-dimensional Heisenberg models in plaquette patterns with strong and weak couplings

We use the stochastic series expansion quantum Monte Carlo method to study the Heisenberg models on the square lattice with strong and weak couplings in the form of three different plaquette arrangements known as checkerboard models C$2\times2$, C$2\times4$ and C$4\times4$. The $a\times b$ here stands for the shape of plaquette consisting with spins connected by strong couplings. Through detailed analysis of finite-size scaling study, the critical point of C$2\times2$ model is improved as $g_{c}=0.548524(3)$ compared with previous studies with $g$ to be the ratio of weak and strong couplings in the models. For C$2\times4$ and C$4\times4$ we give $g_{c}=0.456978(2)$ and $0.314451(3)$. We also study the critical exponents $\nu$, $\eta$, and the universal property of Binder ratio to give further evidence that all quantum phase transitions in these three models are in the three-dimensional O(3) universality class. Furthermore, our fitting results show the importance of effective corrections in the scaling study of these models.

Strongly fluctuating moments in the high-temperature magnetic superconductor RbEuFe4As4

The iron-based superconductor $\mathrm{Rb}\mathrm{Eu}{\mathrm{Fe}}_{4}{\mathrm{As}}_{4}$ undergoes a magnetic phase transition deep in the superconducting state. We investigate the calorimetric response of $\mathrm{Rb}\mathrm{Eu}{\mathrm{Fe}}_{4}{\mathrm{As}}_{4}$ single crystals of the magnetic and the superconducting phase and its anisotropy to in-plane and out-of-plane magnetic fields. Whereas the unusual cusplike anomaly associated with the magnetic transition is suppressed to lower temperatures for fields along the crystallographic $c$ axis, it rapidly transforms to a broad shoulder shifting to higher temperatures for in-plane fields. We identify the cusp in the specific-heat data as a Berezinskii-Kosterlitz-Thouless transition with fine features caused by the three-dimensional effects. The high-temperature shoulder in high magnetic fields marks a crossover from a paramagnetically disordered to an ordered state. This observation is further supported by Monte Carlo simulations of an easy-plane two-dimensional Heisenberg model and a fourth-order high-temperature expansion, both of which agree qualitatively and quantitatively with the experimental findings.

Magnon Hall conductivity and thermal transport in frustrated antiferromagnets

The spin Hall conductivity, σxy(ω) is studied in the quantum frustrated two-dimensional bilayer anisotropic Heisenberg model in the triangular lattice in an external magnetic field employing linear spin waves. Our results show a spin Hall conductivity nonzero when the external field hz is different from zero. Moreover, we present the study of the thermal transport by spin excitations for the two-dimensional frustrated monolayer Heisenberg model in the triangular lattice. We also employ the linear spin wave theory to investigate the dependence on T of the thermal Drude weight in this case. We present our results for the dependence on T of the thermal Drude weight in the gapless regime. Our results show an ideal thermal conductivity for T > 0 as expected for gapless spin systems, and an insulator thermal transport at T=0.

Antiferromagnetic Long-Range Order for S = 1/2 Two-Dimensional Site-Diluted Heisenberg Model Coupled to Lattice Degree of Freedom

The appearance of the antiferromagnetic long-range order (AFLRO) for an S = 1/2 two-dimensional site-diluted antiferromagnetic Heisenberg model coupled to a lattice degree of freedom is investigate...

On the Ordered States in a Two-Dimensional Heisenberg Model with Dipole–Dipole Interactions

In the two-dimensional Heisenberg model with dipole–dipole interactions and uniaxial anisotropy, the ordered state is known to be an in-plane ferromagnetic state or a state with an out-of-plane str...

Order and excitations in site diluted quantum antiferromagnet in the triangular lattice

Abstract In this work we use the linear spin-wave approximation to study the quantum two-dimensional Heisenberg model on triangular lattice in the presence of nonmagnetic impurities. In particular we discuss how site disorder changes the ground-state properties of this important frustrated antiferromagnetic spin system. Our main focus is on the role of the nonmagnetic impurities which generates an influence of the quantum order-disorder transition and the effect of the ground-state spin structure in many spin systems.

Discrete lattice symmetry breaking in a two-dimensional frustrated spin-1 Heisenberg model

Spontaneous discrete symmetry breaking can be described in the framework of Projected Entangled Pair States (PEPS) by linearly superposing local tensors belonging to two (or more) symmetry classes of tensors. This is illustrated in the case of a frustrated spin-1 Heisenberg model on the square lattice, which hosts a nematic spin liquid spontaneously breaking lattice $\pi/2$-rotation symmetry. A superposition of SU(2)-symmetric PEPS tensors belonging to two irreducible representations of the lattice point group is shown to capture accurately the properties of the nematic phase, as shown from a comparison to Exact Diagonalisations and Density Matrix Renormalization Group results.

Monte Carlo determination of the low-energy constants for a two-dimensional spin-1 Heisenberg model with spatial anisotropy

The low-energy constants, namely the spin stiffness $\rho_s$, the staggered magnetization density ${\cal M}_s$ per area, and the spinwave velocity $c$ of the two-dimensional (2D) spin-1 Heisenberg model on the square and rectangular lattices are determined using the first principles Monte Carlo method. In particular, the studied models have antiferromagnetic couplings $J_1$ and $J_2$ in the spatial 1- and 2-directions, respectively. For each considered $J_2/J_1$, the aspect ratio of the corresponding linear box sizes $L_2/L_1$ used in the simulations is adjusted so that the squares of the two spatial winding numbers take the same values. In addition, the relevant finite-volume and -temperature predictions from magnon chiral perturbation theory are employed in extracting the numerical values of these low-energy constants. Our results of $\rho_{s1}$ are in quantitative agreement with those obtained by the series expansion method over a broad range of $J_2/J_1$. This in turn provides convincing numerical evidence for the quantitative correctness of our approach. The ${\cal M}_s$ and $c$ presented here for the spatially anisotropic models are new and can be used as benchmarks for future related studies.

Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field: Unconventional phase transitions in a two-dimensional isotropic Heisenberg model

We consider the spin-1/2 antiferromagnetic Heisenberg model on a bilayer honeycomb lattice including interlayer frustration in the presence of an external magnetic field. In the vicinity of the saturation field, we map the low-energy states of this quantum system onto the spatial configurations of hard hexagons on a honeycomb lattice. As a result, we can construct effective classical models (lattice-gas as well as Ising models) on the honeycomb lattice to calculate the properties of the frustrated quantum Heisenberg spin system in the low-temperature regime. We perform classical Monte Carlo simulations for a hard-hexagon model and adopt known results for an Ising model to discuss the finite-temperature order-disorder phase transition that is driven by a magnetic field at low temperatures. We also discuss an effective-model description around the ideal frustration case and find indications for a spin-flop-like transition in the considered isotropic spin model.