Low temperature static behavior of the two-dimensional quantum easy-axis Heisenberg model

Comparative study between a two-dimensional anisotropic Heisenberg antiferromagnet with easy-axis single-ion anisotropy and one with easy-axis exchange anisotropy

Using a classical Monte Carlo simulation on the two-dimensional Heisenberg model with exchange anisotropy, we have obtained reliable results for several properties: spontaneous magnetization, total energy, self-correlation, susceptibility, and correlation functions. An improved algorithm has been used in order to eliminate artifacts in such quantities below the critical temperature. The present results allow us to confirm the validity of spin-wave theory up to temperatures of 0.5 T c and, simultaneously, to predict the phase-transition temperature according to an Ising-like model with a renormalized exchange constant

It is known that magnetic vortices in two-dimensional Heisenberg models with easy-plane anisotropy exhibit an instability depending on the anisotropy strength. In this paper, we study the statistic behavior of the two-dimensional easy-plane Heisenberg models with distance-dependent interactions ${J}_{xy}(r)$ and ${J}_{z}(r)$ for in-plane and out-of-plane components. We develop analytical and numerical methods for accurate determination of critical anisotropy, above which out-of-plane vortices become stable. In particular, we explore the vortex formation of the Gaussian-type interaction model and determine the critical anisotropy accurately for square, hexagonal, and triangular lattices.

We investigate the two-dimensional classical Heisenberg model with a nonlinear nearest-neighbor interaction V(s,s') = 2K[(1+s x s')/2]p. The analogous nonlinear interaction for the XY model was introduced by Domany, Schick, and Swendsen, who find that for large p the Kosterlitz-Thouless transition is preempted by a first-order transition. Here we show that, whereas the standard (p = 1) Heisenberg model has no phase transition, for large enough p a first-order transition appears. Both phases have only short-range order, but with a correlation length that jumps at the transition.

Binding and unbinding of vortices drives Kosterlitz-Thouless phase transition in two-dimensional XY model. Here we investigate whether similar mechanism works in two-dimensional Heisenberg model, by using the fluctuation of skyrmion number inside a loop to characterize the nature of binding vs. unbinding of defects. Through Monte Carlo simulations, we find that the fluctuation is proportional to the perimeter of the loop at low temperatures while it is proportional to the area of the loop at high temperatures, implying binding of the defects at low temperatures and unbinding at high temperatures.

We have studied the relation between the resonating-valence-bond (RVB) state and the exact ground state for the two-dimensional antiferromagnetic (AF) Heisenberg model on a few small lattices. We describe a method to obtain the state of minimum energy within a family of RVB states. Based on the insight provided by the RVB states, and supported by our numerical results, we argue in favor of the existence of long-range order at T=0 for the two-dimensional AF Heisenberg model.

The authors present numerical simulation studies on the two-dimensional easy-plane classical ferromagnetic Heisenberg model with fourfold in-plane symmetry breaking. Continuum limit equations of motion are obtained and some non-linear particular solutions are discussed. Their simulation data are compared with experimental data obtained for Rb2CrCl4. They find evidence for two transition temperatures in this model.

Using a collective-mode Monte Carlo method (the Wolff-Swendsen-Wang algorithm), we compute the spin stiffness of the two-dimensional classical Heisenberg model. We show that it is the relevant physical quantity to investigate the behaviour of the model in the very low-temperature range inaccessible to previous studies based on correlation length and susceptibility calculations.

The low energy behaviour of the two-dimensional antiferromagnetic Heisenberg model is studied in the sector with total spins S = 0,1,2 by means of a renormalization group procedure, which generates a recursion formula for the interaction matrix ΔS (n+1) of 4 neighbouring “n clusters” of size 2n × 2n, n = 1,2,3,... from the corresponding quantities ΔS (n). Conservation of total spin S is implemented explicitly and plays an important role. It is shown, how the ground state energies ES (n+1), S = 0,1,2 approach each other for increasing n, i.e. system size. The most relevant couplings in the interaction matrices are generated by the transitions 〈S’,m’;n+1|Sq *|S,m;n+1〉 between the ground states |S,m;n+1〉 (m = -S,...,S) on an (n+1)-cluster of size 2n+1 × 2n+1, mediated by the staggered spin operator Sq *.

The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI$_3$ and Cr$_2$Ge$_2$Te$_6$, and the 2D ferromagnetic metals, such as Fe$_3$GeTe$_2$ and MnSe$_2$, have been obtained in recent experiments and attracted a lot of attentions. The superexchange interaction has been suggested to dominate the magnetic interactions in these 2D magnetic systems. In the usual theoretical studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences. Here, we propose a method to determine the expression of the 2D Heisenberg models by counting the possible superexchange paths with the density functional theory (DFT) and Wannier function calculations. With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange coupling constants for the 2D ferromagnetic metal Cr$_3$Te$_6$, which is very different for the crystal structure of Cr atoms in Cr$_3$Te$_6$. The calculated Curie temperature Tc = 328 K is close to the Tc = 344 K of 2D Cr$_3$Te$_6$ reported in recent experiment. In addition, we predict two stable 2D ferromagnetic semiconductors Cr$_3$O$_6$ and Mn$_3$O$_6$ sharing the same crystal structure of Cr$_3$Te$_6$. The similar Heisenberg models are obtained for 2D Cr$_3$O$_6$ and Mn$_3$O$_6$, where the calculated Tc is 218 K and 208 K, respectively. Our method offers a general approach to determine the expression of Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis for further studies.

The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.

Multilayer Pd/Pd(1.2 at. % Fe)/Pd thin films were prepared by dc magnetron sputtering and characterized with dc SQUID magnetometry. Evidence for two-dimensional (2D) Heisenberg and 2D XY model behaviors was found in two different regions of temperature. In these temperature regimes, our data agree with the predicted form for magnetization M and susceptibility χ as functions of field and temperature. Both model’s susceptibilities were found to fit our data. M and χ exhibit discontinuities at magnetic layer thickness-dependent temperatures which separate the different model behaviors.

By introducing a trial action ${\mathit{S}}_{0}$, the thermodynamic properties of the two-dimensional (2D) and 3D quantum and classical antiferromagnetic Heisenberg model are studied and compared analytically with the variational cumulant expansion method. The free energy of the two models are expanded up to the fourth order and the critical temperature ${\mathit{T}}_{\mathit{N}}$ is given for each order and for different lattice structures. On the simple cubic lattice and square lattice, the sublattice magnetization ${\mathit{M}}_{\mathit{s}}$ and the staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{s}}$ are calculated.

The nearest-neighbor quantum-antiferromagnetic (AF) Heisenberg model for spin 1/2 on a two-dimensional square lattice is studied in the auxiliary-fermion representation. Expressing spin operators by canonical fermionic particles requires a constraint on the fermion charge Q=1 on each lattice site, which is imposed approximately through the thermal average. The resulting interacting fermion system is first treated in mean-field theory (MFT), which yields an AF ordered ground state and spin waves in quantitative agreement with conventional spin-wave theory. At finite temperature a self-consistent approximation beyond mean field is required in order to fulfill the Mermin-Wagner theorem. We first discuss a fully self-consistent approximation, where fermions are renormalized due to fluctuations of their spin density, in close analogy to FLEX. While static properties like the correlation length come out correctly, the dynamical response lacks the magnon-like peaks which would reflect the appearance of short-range order at low T. This drawback, which is caused by overdamping, is overcome in a `minimal self-consistent approximation' (MSCA), which we derive from the equations of motion. The MSCA features dynamical scaling at small energy and temperature and is qualitatively correct both in the regime of order-parameter relaxation at long wavelengths and in the short-range-order regime. We also discuss the impact of vertex corrections and the problem of pseudo-gap formation in the single-particle density of states due to long-range fluctuations. Finally we show that the (short-range) magnetic order in MFT and MSCA helps to fulfill the constraint on the local fermion occupancy.

The Raman spectrum of the in-phase flux state of the Wigner-Jordan spinless fermion, i.e., a flux state with in-phase orbital currents circulating around each elementary plaquette of the underlying lattice, of the two-dimensional spin-1/2 antiferromagnetic Heisenberg model is calculated and is found to be in excellent agreement with the experimental Raman spectrum of ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$. The exchange parameter J of the Heisenberg model is found to be 1060 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ from a comparison of the theoretical calculation with the experimental data. Our analysis also suggests that the short-wavelength excitations are fermion-like.