Antiferromagnetic Heisenberg Model Research Articles

Competing emergent Potts orders and possible nematic spin liquids in the kagome J1−J3 Heisenberg model

The extreme frustration in the kagome antiferromagnet makes it possible to realize a large number of closely competing magnetic phases either at the classical or the quantum level. Motivated by recent neutron scattering study on the kagome antiferromagnet vesignieite ${\mathrm{BaCu}}_{3}{\mathrm{V}}_{2}{\mathrm{O}}_{8}{(\mathrm{OH})}_{2}$, we have made systematic investigation of the phase diagram of the classical and quantum ${J}_{1}\ensuremath{-}{J}_{3}$ antiferromagnetic Heisenberg model defined on the kagome lattice (KAFHM). While it is shown previously that a large antiferromagnetic exchange between the third-neighboring spins can drive an emergent $q=4$ Potts order through the order-by-disorder mechanism, it is elusive what will happen in the region where the Luttinger-Tisza criteria fails to predict the classical ordering pattern. Through extensive Monte Carlo simulation, we find that such a region is characterized by competing emergent $q=3$ and $q=4$ Potts order, whose emergence are beyond the description of the conventional order-by-disorder mechanism. Our Schwinger boson mean-field calculation suggests the existence of the Potts-3 order in the ground state of quantum ${J}_{1}\ensuremath{-}{J}_{3}$ KAFHM in such an extremely frustrated region. The predicted Potts-3 state is found to take two different forms distinguished by their projective symmetry group character (PSG or quantum order), although they have exactly the same symmetry. The transition between these two quantum orders is found to occur at a ${J}_{3}$ value intriguingly close to the tricritical point in the Potts-3 phase of the classical phase diagram of the model. A comparison with the exact diagonalization result on a 36-site cluster shows that the intermediate region may indeed host a nematic spin liquid ground state featuring an anisotropic ring structure around the $\mathbf{q}=0$ point in its spinon dispersion relation. We find that such an extremely frustrated region should better be taken as the playground to study the rich competition between exotic emergent phases, rather than a burden on their theoretical analysis.

Excited-state exchange interaction in NiO determined by high-resolution resonant inelastic x-ray scattering at the Ni M2,3 edges

The electronic and magnetic excitations of bulk NiO have been determined using the ${}^{3}{A}_{2g}$ to ${}^{3}{T}_{2g}$ crystal-field transition at the Ni ${M}_{2,3}$ edges with resonant inelastic x-ray scattering at 66.3- and 67.9-eV photon energies and 33-meV spectral resolution. Unambiguous assignment of the high-energy side of this state to a spin-flip satellite is achieved. We extract an effective exchange field of $89\ifmmode\pm\else\textpm\fi{}4$ meV in the ${}^{3}{T}_{2g}$ excited final state from empirical two-peak spin-flip model. The experimental data is found consistent with crystal-field model calculations using exchange fields of 60--100 meV. Full agreement with crystal-field multiplet calculations is achieved for the incident photon energy dependence of line shapes. The lower exchange parameter in the excited state as compared to the ground-state value of 120 meV is discussed in terms of the modification of the orbital occupancy (electronic effects) and of the structural dynamics: (A) With pure electronic effects, the lower exchange energy is attributed to the reduction in effective hopping integral. (B) With no electronic effects, we use the $S=1$ Heisenberg model of antiferromagnetism to derive a second-nearest-neighbor exchange constant ${J}_{2}$ = $14.8\ifmmode\pm\else\textpm\fi{}0.6$ meV. Based on the linear correlation between ${J}_{2}$ and the lattice parameter from pressure-dependent experiments, an upper limit of 2% local Ni-O bond elongation during the femtosecond scattering duration is derived.

Elastically induced magnetization at ultrafast time scales in a chiral helimagnet

The extreme frustration in the Kagome antiferromagnet makes it possible to realize a large number of closely competing magnetic phases either at the classical or the quantum level. Motivated by recent neutron scattering study on the Kagome antiferromagnet vesignieite BaCu$_{3}$V$_{2}$O$_{8}$(OH)$_{2}$, we have made systematic investigation of the phase diagram of the classical and quantum $J_{1}-J_{3}$ antiferromagnetic Heisenberg model defined on the Kagome lattice(KAFHM). While it is shown previously that a large antiferromagnetic exchange between the third-neighboring spins can drive an emergent $q=4$ Potts order through the order-by-disorder mechanism, it is elusive what will happen in the so called "grey region" where the Luttinger-Tisza criteria fails to predict the classical ordering pattern. Through extensive Monte Carlo simulation, we find that the "grey region" is characterized by competing emergent $q=3$ and $q=4$ Potts order, whose emergence are beyond the description of the conventional order-by-disorder mechanism. Our Schwinger Boson mean field calculation in the "grey region" confirms the existence of the Potts-3 order in the ground state of quantum $J_{1}-J_{3}$ KAFHM. The predicted Potts-3 state is found to take two different forms distinguished by their projective symmetry group character(PSG or quantum order), although they have exactly the same symmetry. A comparison with the exact diagonalization result on a 36-site cluster shows that the "grey region" may indeed host a nematic spin liquid ground state featuring an anisotropic ring structure around the $\mathbf{q}=0$ point in its spinon dispersion relation. We find that such "grey region" of extremely frustrated magnets should better be taken as the playground to study the rich competition between exotic emergent phases, rather than a burden on their theoretical analysis.

Assessment of the Variational Quantum Eigensolver: Application to the Heisenberg Model

We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versus gradient-free optimizers. Extrapolation to the thermodynamic limit accurately yields the analytical value for the ground state energy, given by the Bethe ansatz. We predict that a fully functional quantum computer with 100 qubits can calculate the ground state energy with a relatively small error.

Characterizing topological excitations of a long-range Heisenberg model with trapped ions

Realizing and characterizing interacting topological phases in synthetic quantum systems is a formidable challenge. Here, we propose a Floquet protocol to realize the antiferromagnetic Heisenberg model with power-law decaying interactions. Based on analytical and numerical arguments, we show that this model features a quantum phase transition from a liquid to a valence bond solid that spontaneously breaks lattice translational symmetry and is reminiscent of the Majumdar-Ghosh state. The different phases can be probed dynamically by measuring the evolution of a fully dimerized state. We moreover introduce an interferometric protocol to characterize the topological excitations and the bulk topological invariants of the interacting many-body system.

Magnetoelectric coupling on fused azulene oligomers

The global magnetic phase diagram for fused azulene oligomers is obtained by using a fermionic Hubbard model, which is an intermediate between the molecular Pariser-Parr-Pople empiric Hamiltonian and the spin-$1/2$ antiferromagnetic Heisenberg model. We employ the density matrix renormalization group (DMRG) approach to explore the ground state properties of azulene-like molecules as a function of the electronic correlation and the oligomer size. It is shown that, depending on the length of the oligomer, fused azulene transitions from a singlet ($S=0$) to a higher-spin ($S=1,2$) ground state. Near the quantum magnetic phase transition the electric dipole moment, present on fused azulene molecules, couples with the magnetic moment leading to a divergent magnetoelectric susceptibility at the boundary lines of the magnetic phase diagram. These spontaneous electric and magnetic polarizations, together with the magnetoelectric coupling between them, indeed corroborate that these fused azulene oligomers can be viewed as a purely organic multiferroic material, being a magnetoelectric molecule.

Magnetic properties of a capped kagome molecule with 60 quantum spins

We compute ground-state properties of the isotropic, antiferromagnetic Heisenberg model on the sodalite cage geometry. This is a 60-spin spherical molecule with 24 vertex-sharing tetrahedra which can be regarded as a molecular analogue of a capped kagome lattice and which has been synthesized with high-spin rare-earth atoms. Here, we focus on the S=1/2S=1/2 case where quantum effects are strongest. We employ the SU(2)-symmetric density-matrix renormalization group (DMRG). We find a threefold degenerate ground state that breaks the spatial symmetry and that splits up the molecule into three large parts which are almost decoupled from each other. This stands in sharp contrast to the behaviour of most known spherical molecules. On a methodological level, the disconnection leads to ``glassy dynamics’’ within the DMRG that cannot be targeted via standard techniques. In the presence of finite magnetic fields, we find broad magnetization plateaus at 4/5, 3/5, and 1/5 of the saturation, which one can understand in terms of localized magnons, singlets, and doublets which are again nearly decoupled from each other. At the saturation field, the zero-point entropy is S=\ln(182)\approx 5.2S=ln(182)≈5.2 in units of the Boltzmann constant.

On the fly swapping algorithm for ordering of degrees of freedom in density matrix renormalization group

Density matrix renormalization group (DMRG) and its time-dependent variants have found widespread applications in quantum chemistry, including ab initio electronic structure of complex bio-molecules, spectroscopy for molecular aggregates, and charge transport in bulk organic semiconductors. The underlying wavefunction ansatz for DMRG, matrix product state (MPS), requires mapping degrees of freedom (DOF) into a one-dimensional topology. DOF ordering becomes a crucial factor for DMRG accuracy. In this work, we propose swapping neighboring DOFs during the DMRG sweeps for DOF ordering, which we term ‘on the fly swapping’ (OFS) algorithm. We show that OFS is universal for both static and time-dependent DMRG with minimum computational overhead. Examples are given for one dimensional antiferromagnetic Heisenberg model, ab initio electronic structure of N2 molecule, and the S1/S2 internal conversion dynamics of pyrazine molecule. It is found that OFS can indeed improve accuracy by finding better DOF ordering in all cases.

Effect of hole doping on the 120 degree order in the triangular lattice Hubbard model: a Hartree–Fock revisit

We revisit the unrestricted Hartree Fock study on the evolution of the ground state of the Hubbard model on the triangular lattice with hole doping. At half-filling, it is known that the ground state of the Hubbard model on triangular lattice develops a 120 degree coplanar order at half-filling in the strong interaction limit, i.e., in the spin 1/2 anti-ferromagnetic Heisenberg model on the triangular lattice. The ground state property in the doped case is still in controversy even though extensive studies were performed in the past. Within Hartree Fock theory, we find that the 120 degree order persists from zero doping to about 0.3 hole doping. At 1/3 hole doping, a three-sublattice collinear order emerges in which the doped hole is concentrated on one of the three sublattices with antiferromagnetic Neel order on the remaining two sublattices, which forms a honeycomb lattice. Between the 120 degree order and 1/3 doping region, a phase separation occurs in which the 120 degree order coexists with the collinear anti-ferromagnetic order in different regions of the system. The collinear phase extends from 1/3 doping to about 0.41 doping, beyond which the ground state is paramagnetic with uniform electron density. The phase diagram from Hartree Fock could provide guidance for the future study of the doped Hubbard model on triangular lattice with more sophisticated many-body approaches.

Ground-state and thermodynamic properties of the spin-$\frac{1}{2}$ Heisenberg model on the anisotropic triangular lattice

The spin-\frac1212 triangular lattice antiferromagnetic Heisenberg model has been for a long time the prototypical model of magnetic frustration. However, only very recently this model has been proposed to be realized in the Ba_88CoNb_66O_{24}24 compound. The ground-state and thermodynamic properties are evaluated from a high-temperature series expansion interpolation method, called ``entropy method’’, and compared to experiments. We find a ground-state energy e_0 = -0.5445(2)e0=−0.5445(2) in perfect agreement with exact diagonalization results. We also calculate the specific heat and entropy at all temperatures, finding a good agreement with the latest experiments, and evaluate which further interactions could improve the comparison. We explore the spatially anisotropic triangular lattice and provide evidence that supports the existence of a gapped spin liquid between the square and triangular lattices.

Ultrafast dynamics of entanglement in Heisenberg antiferromagnets

We investigate entanglement dynamics in the antiferromagnetic Heisenberg model in two dimensions following a spatially anisotropic quench of the exchange interactions. Opposed to established results in one dimension, the magnon quasiparticles show an initial growth of entanglement dynamics that does not depend on the system size and is governed by the oscillation period of the exchange interaction. We ascribe this to the dominance of the intrinsic entanglement of short wavelength nonpropagating magnon pairs, which also leads to a competition between area-law and volume-law contribution in the entanglement dynamics. Furthermore, by adopting the neural-network quantum states, we provide numerical evidence that this behavior survives even in the presence of strong magnon-magnon interactions, suggesting intriguing avenues for manipulating entanglement dynamics in quantum materials.

Phase diagrams, quantum correlations and critical phenomena of antiferromagnetic Heisenberg model on diamond-type hierarchical lattices

The spin-1/2 antiferromagnetic (AF) Heisenberg systems are studied on three typical diamond-type hierarchical lattices (systems A, B and C) with fractal dimensions 1.63, 2 and 2.58, respectively, and the phase diagrams, critical phenomena and quantum correlations are calculated by a combination of the equivalent transformation and real-space renormalization group methods. We find that there exist a reentrant behavior for system A and a finite temperature phase transition in the isotropic Heisenberg limit for system C (not for system B). Unlike the ferromagnetic case, the Néel temperatures of AF systems A and B are inversely proportional to (when Δ → Δc) and ln Δ (when Δ → 0), respectively. And we also find that there is a turning point of quantum correlation in the isotropic Heisenberg limit Δ = 0 where there is a ‘peak’ of the contour and no matter how large the size of system is, quantum correlation will change to zero in the Ising limit for the three systems. The quantum correlation decreases with the increase of lattice size L and it is almost zero when L ⩾ 30 for system A, and for systems B and C, they still exist when L is larger than that of system A. Moreover, we discuss the effects of quantum fluctuation and analyze the errors of results in the above three systems, which are induced by the noncommutativity.

Energy Scale Deformation on Regular Polyhedra

A variant of energy scale deformation is considered for the S = 1/2 antiferromagnetic Heisenberg model on polyhedra. The deformation is induced by the perturbations to the uniform Hamiltonian, whose coefficients are determined by the bond coordinates. On the tetrahedral, octahedral, and cubic clusters, the perturbative terms do not affect the ground state of the uniform Hamiltonian when they are sufficiently small. On the other hand, for the icosahedral and dodecahedral clusters, it is numerically confirmed that the ground state of the uniform Hamiltonian is almost insensitive to the perturbations unless they lead to a discontinuous change in the ground state. The obtained results suggest the existence of a generalization of sine-square deformation in higher dimensions.

Probing ground-state properties of the kagome antiferromagnetic Heisenberg model using the variational quantum eigensolver

Finding and probing the ground states of spin lattices, such as the antiferromagnetic Heisenberg model on the kagome lattice (KAFH), is a very challenging problem on classical computers and only possible for relatively small systems. We propose using the variational quantum eigensolver (VQE) to find the ground state of the KAFH on a quantum computer. We find efficient ansatz circuits and show how physically interesting observables can be measured efficiently. To investigate the expressiveness and scaling of our ansatz circuits we used classical, exact simulations of VQE for the KAFH for different lattices ranging from 8 to 24 qubits. We find that the fidelity with the ground state approaches one exponentially in the circuit depth for all lattices considered, except for a 24-qubit lattice with an almost degenerate ground state. We conclude that VQE circuits that are able to represent the ground state of the KAFH on lattices inaccessible to exact diagonalization techniques may be achievable on near-term quantum hardware. However, for large systems, circuits with many variational parameters are needed to achieve high fidelity with the ground state.

Quantum Correlation, Entanglement in the Two-Dimensional Heisenberg Antiferromagnetic Model on Kagome Lattice

Quantum Correlation, Entanglement in the Two-Dimensional Heisenberg Antiferromagnetic Model on Kagome Lattice

S = 1 antiferromagnetic electron-spin systems on hydrogenated phenalenyl-tessellation molecules for material-based quantum-computation resources

A resource state for measurement-based quantum computation is proposed using a material design of S = 1 antiferromagnetic spin chains. Specifying hydrogen adsorption positions on polymerized phenalenyl-tessellation molecules gives rise to formation of graphene zero modes that produce local S = 1 spins or S = 1/2 spins in the required order through exchange interactions. When the S = 1 antiferromagnetic Heisenberg models serve as quantum-computation resources, hydrogen adatoms inducing zero modes can also work as local electron-spin probes in nuclear spin spectroscopy, which could be used for controlling and measuring local spins.

Stability of skyrmion crystal phase in antiferromagnetic triangular lattice with DMI and single-ion anisotropy

We study a frustrated antiferromagnetic Heisenberg model on a triangular lattice with the Dzyaloshinskii–Moriya interaction (DMI) in the presence of an external magnetic field and focus on the effects of the single-ion anisotropy on the overall phase diagram topology and the skyrmion lattice (SkX) phase stability. Phase diagrams in the temperature-field plane for both easy-axis and easy-plane anisotropy of a varying strength are constructed in the regimes of a moderate and strong DMI by parallel tempering Monte Carlo simulations. For the considered range of parameters the phase diagrams featuring up to four ordered phases are identified. The SkX phase is found to be stabilized within some temperature and field window for sufficiently large DMI and not too strong anisotropy. We demonstrate that both types of the anisotropy can have beneficial effects on the SkX phase stabilization. Namely, in the systems with moderate DMI a small easy-plane anisotropy extends the field window of the SkX phase appearance by shifting its lower bound to smaller values and in the systems with larger DMI a small easy-axis anisotropy shifts the entire field range of the SkX phase stabilization towards smaller values. The Metropolis dynamics is employed to probe the persistence of the SkX phase upon quenching the field to zero or small finite values. It is confirmed that low temperatures, small DMI and small easy-plane anisotropy values support the skyrmions persistence, i.e., increase their lifetime.

Dimerized nature of magnetic interactions in the S = 1/2 quantum antiferromagnet Cu(en)2SO4

Specific heat, magnetic susceptibility and magnetization of Cu(en)2SO4 single crystal were investigated in the field applied along the b-axis. The analysis of the susceptibility within the Curie-Weiss law indicates the antiferromagnetic nature of the exchange coupling between Cu(II) ions with zJ/kB = −6.4 K. Considering the structure of this material, the formation of magnetic dimers can be expected. Very good description of susceptibility data was achieved using the Heisenberg antiferromagnetic dimer model with intra-dimer coupling J/kB = −5.52 K, effective inter-dimer coupling J'/kB = −0.3 K and gb = 2.12. The energy gap in the spin excitation spectrum was estimated Δ/kB ≈ 11 K and corresponding critical field Bcb ≈ 7.8 T required for closing the gap. The dimerized nature of magnetic interactions was also confirmed by specific heat and isothermal magnetization data which manifest qualitatively the same behavior as the aforementioned dimer model. The possibility of further low-temperature experiments in the closed gap regime is discussed to obtain more detailed concept of magnetic interactions in this interesting compound.

Entanglement-based tensor-network strong-disorder renormalization group

We propose an entanglement-based algorithm of the tensor-network strong-disorder renormalization group (tSDRG) method for quantum spin systems with quenched randomness. In contrast to the previous tSDRG algorithm based on the energy spectrum of renormalized block Hamiltonians, we directly utilizes the entanglement structure associated with the blocks to be renormalized. We examine accuracy of the new algorithm for the random antiferromagnetic Heisenberg models on the one-dimensional, triangular, and square lattices. We then find that the entanglement-based tSDRG achieves better accuracy than the previous one for the square lattice model with weak randomness, while it is less efficient for the one-dimensional and triangular lattice models particularly in the strong randomness region. The theoretical background and possible improvements of the algorithm are also discussed.

Geometric frustration on the trillium lattice in a magnetic metal–organic framework

In the dense metal-organic framework Na[Mn(HCOO)$_3$], Mn$^{2+}$ ions ($S=\frac{5}{2}$) occupy the nodes of a `trillium' hyperkagome net. We show that this material exhibits a variety of behaviour characteristic of geometric frustration: the N\'eel transition is suppressed well below the characteristic magnetic interaction strength; short-range magnetic order persists far above the N\'eel temperature; and the magnetic susceptibility exhibits a pseudo-plateau at $\frac{1}{3}$-saturation magnetisation. We demonstrate that a simple nearest-neighbour Heisenberg antiferromagnet model accounts quantitatively for each observation, and hence Na[Mn(HCOO)$_3$] is the first experimental realisation of this model on the trillium net. We develop a mapping between this trillium model and that on the two-dimensional Shastry-Sutherland lattice, and demonstrate how both link geometric frustration within the classical spin liquid regime to a strong magnetocaloric response at low fields.