Two-dimensional Heisenberg Model Research Articles

Imaginary-time relaxation quantum critical dynamics in the two-dimensional dimerized Heisenberg model

Imaginary-time relaxation quantum critical dynamics in the two-dimensional dimerized Heisenberg model

Existence of Long-Range Magnetic Order in Heisenberg Spin Nanoribbon with Edge Modification

Abstract A long-range magnetic order appears on side decorated Heisenberg spin nanoribbon at nonzero temperature, although no spontaneous magnetization exists in a one- or two-dimensional isotropic Heisenberg model at any nonzero temperature according to Mermin-Wagner theorem. By use of the spin Green’s function method, we calculated the magnetizations of Heisenberg nanoribbon decorated by side spins with single-ion anisotropy and found that the system exhibits a nonzero transition temperature, whether the decorated edge spins of the system link together or separate from each other. When the width of the nanoribbon achieves infinity limit, the transition temperatures of the system tend to a same finite constant eventually whether one edge or double edges are decorated by side spins in the nanoribbon. The results reveal that the magnetism of a low-dimensional spin system is different from that of a three-dimensional spin system. When the single-ion anisotropy of edge spins in Heisenberg spin nanoribbon can be modulated by an electric field experimentally, various useful long-range magnetic orders of the system can be obtained. The works can provide a detailed theoretical basis for designing and fabricating the next-generation low-dimensional magnetic random-access memory.

Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism.

Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

CrO<sub>2</sub> monolayer: a two-dimensional ferromagnet with high Curie temperature and half-metallicity

Owing to the complete spin-polarization of electronic states near Fermi energy, half-metallic ferromagnets, especially two-dimensional half-metallic ferromagnets, have garnered significant attention in the field of spintronics. However, the practical application of these materials is greatly hindered by their low Curie temperatures. Therefore, the exploration of high Curie temperature half-metallic ferromagnets poses a necessary and challenging task. In this study, we predict a two-dimensional transition metal oxide, CrO<sub>2</sub> monolayer, and employed first-principles calculations to investigate the crystal structure, electronic properties, magnetic ground state, and ferromagnetic phase transition. The calculations on phonon spectrum, elastic constant, and molecular dynamics simulations indicate that CrO<sub>2</sub> monolayer is dynamically, thermally, and mechanically stable. The convex hull diagram of Cr—O systems shows the hull energy of the predicted CrO<sub>2</sub> layer is only 0.18 eV, further confirming the structural stability and large possibility for experimental fabrication. More importantly, the electronic and magnetic properties of CrO<sub>2</sub> monolayer demonstrate that it is a two-dimensional ferromagnetic half-metal with wide band gap. Five <i>d</i> suborbitals are divided into E<sub>g</sub> and T<sub>2g</sub> orbitals because of the crystal field of Cr atom at the center of O tetrahedron, and the spin-polarization of E<sub>g</sub> orbitals make a major contribution to the moment around Cr atom. The ferromagnetic coupling along Cr-O-Cr chain is dominated by the superexchange interaction bridged by O 2p orbital, similar to the typical Mn-O-Mn superexchange model. The magnetic behavior of the Cr spin lattice in a CrO<sub>2</sub> monolayer is described by a two-dimensional Heisenberg model, in which the exchange coupling anisotropy is ignored and the single ion anisotropy is the main consideration. By solving the Heisenberg model using the Monte Carlo simulation method, the Curie temperature is determined to be over 400 K. The high Curie temperature ferromagnetism is rare in two-dimensional ferromagnetic materials and even rarer in semi-metallic materials, which makes it an ideal material for the fabrication of spintronic devices and the study of spin quantum effects.

Topological magnons on triangular antiferromagnetic lattices in an external field

The influence of topological magnons of the antiferromagnet in triangular lattice on Hall spin conductivity and thermodynamics is investigated. The magnons splitting due to the triangular lattice geometry leads to a complex and interesting behavior in all quantities analyzed of the system. We get the behavior of the Hall spin conductivity as a function of T as well as the specific heat, magnetization and magnetic susceptibility, where the behavior of the Hall spin conductivity decreases at low-temperature limit, dropping to zero at T→0 as expected for all bosonic system. Moreover, we get the behavior of the magnetization as a function of field h presenting problems of convergence at h→0 which is expected for the two-dimensional like Heisenberg model.

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Integration of the two-dimensional Heisenberg model by methods of differential geometry

Для интегрирования двумерной модели Гейзенберга использованы методы классической дифференциальной геометрии. Уравнения модели после преобразования годографа были записаны в терминах метрического тензора, связанного с криволинейной системой координат, и его производных. Показано, что их общее решение описывает все известные ранее точные решения, кроме плоского вихря. Предсказан и проанализирован новый тип вихревой структуры - "вихревая полоса" в двумерном ферромагнетике. Ее отличительные свойства - конечные размеры области определения, конечность полной энергии и отсутствие ядра вихря при наличии вихревой структуры.

Special transition and extraordinary phase on the surface of a two-dimensional quantum Heisenberg antiferromagnet

Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4)ys=0.86(4) and \eta_{\parallel}=-0.33(1)η∥=−0.33(1), which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.

Two-dimensional Heisenberg model with material-dependent superexchange interactions

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.

Augmenting Density Matrix Renormalization Group with Disentanglers

Density matrix renormalization group (DMRG) and its extensions in the form of matrix product states are arguably the choice for the study of one-dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two-dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. We introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as l for an l × l system). In the new method, the O(D 3) low computational cost of DMRG is kept (with an overhead of O(d 4) and d the dimension of the physical degrees of freedom). We perform benchmark calculations with this approach on the two-dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.

Provably efficient machine learning for quantum many-body problems.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. By contrast, under a widely accepted conjecture, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

Transverse instability and universal decay of spin spiral order in the Heisenberg model

We analyze the intrinsic stability of spin spiral states in the two-dimensional Heisenberg model isolated from its environment. Our analysis reveals that the SU(2) symmetric point hosts a dynamic instability that is enabled by the existence of energetically favorable transverse deformations---both in real and spin space---of the spiral order. The instability is universal in the sense that it applies to systems with any spin number, spiral wave vector, and spiral amplitude. Unlike the Landau or modulational instabilities which require impurities or periodic potential modulation of an optical lattice, quantum fluctuations alone are sufficient to trigger the transverse instability. We analytically find the most unstable mode and its growth rate, and compare our analysis with phase-space methods. By adding an easy-plane exchange coupling that reduces the Hamiltonian symmetry from SU(2) to U(1), the stability boundary is shown to continuously interpolate between the modulational instability and the transverse instability. This suggests that the transverse instability is an intrinsic mechanism that hinders long-range phase coherence even in the presence of exchange anisotropy.

P31 NMR investigation of quasi-two-dimensional magnetic correlations in T2P2S6 ( T =Mn, Ni)

We report the anomalous breakdown in the scaling of the microscopic magnetic susceptibility---as measured via the $^{31}\mathrm{P}$ nuclear magnetic resonance (NMR) shift $K$---with the bulk magnetic susceptibility $\ensuremath{\chi}$ in the paramagnetic state of ${\mathrm{Mn}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$. This anomaly occurs near ${T}_{\mathrm{max}}\ensuremath{\sim}117$ K the maximum in $\ensuremath{\chi}(T)$ and is therefore associated with the onset of quasi-two-dimensional (quasi-2D) magnetic correlations. The spin-lattice relaxation rate divided by temperature ${({T}_{1}T)}^{\ensuremath{-}1}$ in ${\mathrm{Mn}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$ exhibits broad peaklike behavior as a function of temperature, qualitatively following $\ensuremath{\chi}$, but displaying no evidence of critical slowing down above the N\'eel temperature ${T}_{N}$. In the magnetic state of ${\mathrm{Mn}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$, NMR spectra provide good evidence for ${60}^{\ensuremath{\circ}}$ rotation of stacking-fault-induced magnetic domains, as well as observation of the spin-flop transition that onsets at 4 T. The temperature-dependent critical behavior of the internal hyperfine field at the P site in ${\mathrm{Mn}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$ is consistent with previous measurements and the two-dimensional anisotropic Heisenberg model. In a sample of ${\mathrm{Ni}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$, we observe only two magnetically split resonances in the magnetic state, demonstrating that the multiple-peaked NMR spectra previously associated with ${60}^{\ensuremath{\circ}}$ rotation of stacking faults is sample-dependent. Finally, we report the observation of a spin-flop-induced splitting of the NMR spectra in ${\mathrm{Ni}}_{2}{\mathrm{P}}_{2}{\mathrm{S}}_{6}$, with an onset spin-flop field of ${\ensuremath{\mu}}_{0}{H}_{\mathrm{sf}}=14$ T.

Transition Metal-Free Half-Metallicity in Two-Dimensional Gallium Nitride with a Quasi-Flat Band.

Two-dimensional half-metallicity without a transition metal is an attractive attribute for spintronics applications. On the basis of first-principles calculation, we revealed that a two-dimensional gallium nitride (2D-GaN), which was recently synthesized between graphene and SiC or wurtzite GaN substrate, exhibits half-metallicity due to its half-filled quasi-flat band. We found that graphene plays a crucial role in stabilizing a local octahedral structure, whose unusually high density of states due to a flat band leads to a spontaneous phase transition to its half-metallic phase from normal metal. It was also found that its half-metallicity is strongly correlated to the in-plane lattice constants and thus subjected to substrate modification. To investigate the magnetic property, we simplified its magnetic structure with a two-dimensional Heisenberg model and performed Monte Carlo simulation. Our simulation estimated its Curie temperature (TC) to be ∼165 K under a weak external magnetic field, suggesting that transition metal-free 2D-GaN exhibiting p orbital-based half-metallicity can be utilized in future spintronics.

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3)O(3) non-linear sigma model in 1+11+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+13+1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2)SU(2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to \chi_E^\text{eff} \sim 1500χEeff∼1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-TT transition and asymptotic freedom, though with a slight preference for the second.

Chebyshev expansion of spectral functions using restricted Boltzmann machines

Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional systems present insurmountable hurdles, ranging from the sign problem in quantum Monte Carlo (MC), to the entanglement area law in tensor network based methods. We hereby present a variational approach based on a Chebyshev expansion of the spectral function and a neural network representation for the wave functions. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave-functions". We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, and compare to those obtained by other methods in the literature.

Logarithmic divergent specific heat from high-temperature series expansions: Application to the two-dimensional XXZ Heisenberg model

We present an interpolation method for the specific heat $c_v(T)$, when there is a phase transition with a logarithmic singularity in $c_v$ at a critical temperature $T=T_c$. The method uses the fact that $c_v$ is constrained both by its high temperature series expansion, and just above $T_c$ by the type of singularity. We test our method on the ferro and antiferromagnetic Ising model on the two-dimensional square, triangular, honeycomb, and kagome lattices, where we find an excellent agreement with the exact solutions. We then explore the XXZ Heisenberg model, for which no exact results are available.

Turbulent relaxation after a quench in the Heisenberg model

We predict the emergence of turbulent scaling in the quench dynamics of the two-dimensional Heisenberg model for a wide range of initial conditions and model parameters. In the isotropic Heisenberg model, we find that the spin-spin correlation function exhibits universal scaling consistent with a turbulent energy cascade. When the spin rotational symmetry is broken by an easy-plane exchange anisotropy, we find a dual cascade of energy and an emergent conserved charge associated to transverse magnetization fluctuations. The scaling exponents are estimated analytically and agree with numerical simulations using phase space methods. We also define the range of initial conditions and model parameters (energy, magnetization, and spin number $S$) that lead to a turbulent cascade. The universal character of the cascade, insensitive to microscopic details or initial conditions, suggests that turbulence in spin systems can be broadly realized in cold atom and solid-state experiments.

Dielectric study and magnetic property analysis of Gd2O3 nanorods/nanowire in combination with Monte Carlo simulation

Gd2O3 nanorods/nanowires are prepared by using the two step Hydrothermal technique and formation of desired structure is substantiated from field emission scanning electron microscopy (FESEM) micrographs. The Rietveld analysis reveals that the sample crystallizes only in the cubic-C form with space group Ia3. Magnetization measurement shows the absence of any sort of magnetic ordering down to 5 K and the anti-ferromagnetic correlation is observed between the Gd3+ ions residing at two different sites. The mean exchange integrals (J) are deduced from the magnetic data and these values are used in Monte Carlo simulation based on two-dimensional classical Heisenberg model. Our simulation agrees well with experimental observation by confirming that the sample is paramagnetic in nature. Enhanced dielectric response (dielectric permittivity ~210 at room temperature) is observed in Gd2O3nanorods/nanowire. The non-Debye type of relaxation with the dominance of grain and grain boundary contribution inside the sample was confirmed from the typical Cole-Cole plot.

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.