Heisenberg Antiferromagnet Research Articles

Emergence of spin singlets with inhomogeneous gaps in the kagome lattice Heisenberg antiferromagnets Zn-barlowite and herbertsmithite

The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in a lattice of corner-sharing triangles is a prime candidate for hosting a quantum spin liquid (QSL) ground state consisting of entangled spin singlets. But the existence of various competing states makes a convincing theoretical prediction of the QSL ground state difficult, calling for experimental clues from model materials. The kagome lattice materials Zn-barlowite ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not exhibit long range order, and they are considered the best realizations of the kagome Heisenberg antiferromagnet known to date. Here we use $^{63}$Cu nuclear quadrupole resonance combined with the inverse Laplace transform (ILT) to probe locally the inhomogeneity of delicate quantum ground states affected by disorder. We present direct evidence for the gradual emergence of spin singlets with spatially varying excitation gaps, but even at temperatures far below the super-exchange energy scale their fraction is limited to approximately 60\% of the total spins. Theoretical models need to incorporate the role of disorder to account for the observed inhomogeneously gapped behaviour.

Low-temperature thermodynamic properties of the Heisenberg antiferromagnetic chain with two easy-axis single-ion anisotropies

In this paper, the spin wave theory is applied to the one-dimensional Heisenberg antiferromagnet in the coexistence of two different anisotropies [Formula: see text] and [Formula: see text], which are separately the easy-axis single-ion anisotropies for sublattice A and sublattice B of the system. Both the ground-state and low-temperature properties of the system are strongly affected by the competition between these two anisotropies. Two kinds of the competition in terms of the deviation parameter [Formula: see text] are discussed for the uniform anisotropy taking the values of [Formula: see text] and [Formula: see text], respectively. The [Formula: see text]-dependent behaviors (such as the power, exponential and linear laws) are obtained for the total magnetization, the staggered magnetizations, the internal energy, the specific heat and the susceptibility. It is found that at zero-temperature, the interplay between these two anisotropies induces the antiferromagnetic-disorder phase transition in the small anisotropy region with [Formula: see text]. For the selected cases of [Formula: see text], our results for are in agreement with the findings obtained by the existing theories and the quantum Monte Carlo data.

Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of Representations.

Neural network quantum states (NQS) have been widely applied to spin-1/2 systems, where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here, we propose a more direct generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach, we present variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Furthermore, we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the spin-1 basis using only hidden units, compared to required in the basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the basis with only hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.

Realization of a bosonic antiferromagnet

Quantum antiferromagnets are of broad interest in condensed matter physics as they provide a platform for studying exotic many-body states including spin liquids and high-temperature superconductors. Here, we report on the creation of a one-dimensional Heisenberg antiferromagnet with ultracold bosons. In a two-component Bose-Hubbard system, we switch the sign of the spin-exchange interaction and realize the isotropic antiferromagnetic Heisenberg model in an extended 70-site chain. Starting from a low-entropy N\'eel-ordered state, we use optimized adiabatic passage to approach the bosonic antiferromagnet. We demonstrate the establishment of antiferromagnetism by probing the evolution of the staggered magnetization and spin correlations of the system. Compared with condensed matter systems, ultracold gases in optical lattices can be microscopically engineered and measured, offering significant advantages for exploring bosonic magnetism and spin dynamics.

Magnetic phases of a spin- 12 Heisenberg antiferromagnet on a frustrated bilayer

We use series expansion methods to investigate the zero-temperature phase diagram of a spin-$\frac{1}{2}$ Heisenberg frustrated bilayer model. For different parameter regions the model shows two related antiferromagnetic phases as well as a nonmagnetic dimerized phase. The phase boundaries are located to high precision and agree very well with a previous calculation using different methods. The triplon spectrum in the dimerized phase is also studied.

Low-temperature thermodynamics of the antiferromagnetic J1−J2 model: Entropy, critical points, and spin gap

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1 > 0$ between first neighbors and $J_2 = \alpha J_1$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\Delta(\alpha) > 0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\alpha = 0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\alpha$ followed by density matrix renormalization group (DMRG) calculations returns the entropy density $S(T,\alpha,N)$ and magnetic susceptibility $\chi(T,\alpha,N)$ of progressively larger systems up to $N = 96$ or 152 spins. Convergence to the thermodynamics limit, $S(T,\alpha)$ or $\chi(T,\alpha)$, is demonstrated down to $T/J \sim 0.01$ in the sectors $\alpha < 1$ and $\alpha > 1$. $S(T,\alpha)$ yields the critical points between gapless phases with $S^\prime(0,\alpha) > 0$ and gapped phases with $S^\prime(0,\alpha) = 0$. The $S^\prime(T,\alpha)$ maximum at $T^*(\alpha)$ is obtained directly in chains with large $\Delta(\alpha)$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\alpha)$ down to $T = 0$ indicates power-law deviations $T^{-\gamma(\alpha)}$ from $\exp(-\Delta(\alpha)/T)$ with exponent $\gamma(\alpha)$ that increases with $\alpha$. The $\chi(T,\alpha)$ analysis also yields power-law deviations, but with exponent $\eta(\alpha)$ that decreases with $\alpha$. $S(T,\alpha)$ and the spin density $\rho(T,\alpha) = 4T\chi(T,\alpha)$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\alpha)/\rho(T,\alpha)$ for $T < 0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\Delta(\alpha)$.

Quasi-one-dimensional uniform spin- 12 Heisenberg antiferromagnet KNaCuP2O7 probed by P31 and Na23 NMR

We present the structural and magnetic properties of KNaCuP$_2$O$_7$ investigated via x-ray diffraction, magnetization, specific heat, and $^{31}$P NMR and $^{23}$Na NMR measurements and complementary electronic structure calculations. The temperature dependent magnetic susceptibility and $^{31}$P NMR shift could be modeled very well by the uniform spin-$1/2$ Heisenberg antiferromagnetic chain model with nearest-neighbour interaction $J/k_{\rm B}\simeq 58.7$ K. The corresponding mapping using first principles electronic structure calculations leads to $J^{\rm DFT}/k_{\rm B} \simeq 59$ K with negligibly small inter-chain couplings ($J^{\prime}/k_{\rm B}$, $J^{\prime \prime}/k_{\rm B} < 0.1$ K), further confirming that the system is indeed an one-dimensional uniform spin-$1/2$ Heisenberg antiferromagnet. The temperature-dependent unit cell volume could be described well using the Debye approximation with a Debye temperature of $\Theta_{\rm D} \simeq 294$ K, consistent with the heat capacity data. The diverging trend of the NMR spin-lattice relaxation rates ($^{31}1/T_1$ and $^{23}1/T_1$) imply the onset of a magnetic long-range-ordering at very low temperatures supporting the anticipated $T_{\rm N} \simeq 0.38$ K from the inter-chain couplings. Moreover, the NMR spin-lattice relaxation rates show the dominant contributions from uniform ($q=0$) and staggered ($q = \pm \pi/a$) spin fluctuations in the high and low temperature regimes, respectively mimicking one-dimensionality of the spin-lattice. We have also demonstrated that $^{31}1/T_1$ in high temperatures varies linearly with $1/\sqrt{H}$ reflecting the effect of spin diffusion on the dynamic susceptibility. Further, the inter-chain frustration also substantially impede the magnetic ordering rendering the spin-lattice a perfect one-dimensional uniform spin-$1/2$ Heisenberg antiferromagnet over a wide temperature range.

Spin dynamics of the antiferromagnetic Heisenberg model on a kagome bilayer

We study the spin dynamics of classical Heisenberg antiferromagnet with nearest neighbor interactions on a quasi-two-dimensional kagome bilayer. This geometrically frustrated lattice consists of two kagome layers connected by a triangular-lattice linking layer. By combining Monte Carlo with precessional spin dynamics simulations, we compute the dynamical structure factor of the classical spin liquid in kagome bilayer and investigate the thermal and dilution effects. While the low frequency and long wavelength dynamics of the cooperative paramagnetic phase is dominated by spin diffusion, weak magnon excitations persist at higher energies, giving rise the half moon pattern in the dynamical structure factor. In the presence of spin vacancies, the dynamical properties of the diluted system can be understood within the two population picture. The spin diffusion of the "correlated" spin clusters is mainly driven by the zero-energy weather-van modes, giving rise to an autocorrelation function that decays exponentially with time. On the other hand, the diffusive dynamics of the quasi-free "orphan" spins leads to a distinctive longer time power-law tail in the autocorrelation function. We discuss the implications of our work for the glassy behaviors observed in the archetypal frustrated magnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ (SCGO).

Evolution of magnetic field induced ordering in the layered quantum Heisenberg triangular-lattice antiferromagnet Ba3CoSb2O9

Quantum fluctuations in the effective spin one-half layered structure triangular-lattice quantum Heisenberg antiferromagnet Ba$_3$CoSb$_2$O$_9$ lift the classical degeneracy of the antiferromagnetic ground state in magnetic field, producing a series of novel spin structures for magnetic fields applied within the crystallographic ab plane. Theoretically unresolved, however, are the effects of interlayer antferromagnetic coupling and transverse magnetic fields on the ground states of this system. To address these issues, we have used specific heat, neutron diffraction, thermal conductivity, and magnetic torque measurements to map out the phase diagram as a function of magnetic field intensity and orientation relative to the crystallographic ab plane. For H parallel to the ab plane, we have discovered an additional, previously unreported magnetic-field-induced phase transition at low temperature and an unexpected tetracritical point in the high field phase diagram, which - coupled with the apparent second-order nature of the phase transitions - eliminates several theoretically proposed spin structures for the high field phases. Our calorimetric measurements as a function of magnetic field orientation are in general agreement with theory for field-orientation angles close to plane parallel but diverge at angles near plane perpendicular; a predicted convergence of two phase boundaries at finite angle and a corresponding change in the order of the field induced phase transition is not observed experimentally. Our results emphasize the role of interlayer coupling in selecting and stabilizing field-induced phases, provide new guidance into the nature of the magnetic order in each phase, and reveal the need for new physics to account for the nature of magnetic ordering in this archetypal 2D spin one-half triangular lattice quantum Heisenberg antiferromagnet.

Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application, we derive site-monotonicity properties for the spin–spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates—improving previous positivity results which hold for the Cesàro sum. We also derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model and lattice permutations, thus extending the previous results of Lees and Taggi (2019).

Molecular-field-theory fits to magnetic susceptibilities of antiferromagnetic GdCu2Si2, CuO, LiCrO2, and [formula omitted]-CaCr2O4 single crystals below their Néel temperatures

A recently-developed molecular field theory (MFT) has been used to fit single-crystal magnetic susceptibility χ versus temperature T data below the respective antiferromagnetic ordering temperature TN for a variety of collinear and coplanar noncollinear Heisenberg antiferromagnets. The spins in the system are assumed to interact by Heisenberg exchange and to be identical and crystallographically equivalent. The fitting parameters for χ(T) of collinear antiferromagnets are measurable quantities: the Weiss temperature θp in the Curie–Weiss law, TN,χ(TN), and the spin S. For coplanar noncollinear helix and cycloid structures, an additional fitting parameter is the turn angle between layers of ferromagnetically-aligned spins. Here MFT fits to anisotropic χ(T) data from the literature for single crystals of the collinear antiferromagnets GdCu2Si2 and CuO and the noncollinear antiferromagnets LiCrO2 with a 120° cycloidal structure and α-CaCr2O4 with a 120° helical structure below their respective Néel temperatures are presented. The MFT fit to the anisotropic χ(T⩽TN) data for CuO is poor, whereas the fits to the data for GdCu2Si2, LiCrO2, and α-CaCr2O4 are quite good. The poor fit for CuO is attributed to the influence of strong quantum fluctuations associated with the small Cu spin and the quasi-one-dimensional magnetism that are not taken into account by the MFT. The magnetic contribution to the zero-field heat capacity of the collinear antiferromagnet GdNiGe3 at T⩽TN is also fitted by the MFT.

Antiferromagnets with random vacancies and substitutional spins on the triangular lattice

We discuss theoretically static and dynamical properties of $XY$ and Heisenberg antiferromagnets on triangular lattice with random vacancies and substitutional spins. It is shown that the distortion of $120^\circ$ magnetic order produced by a single defect is described by electrostatic equations for a field of an electrically neutral complex of six charges located around the impurity. The first finite term in the multipole expansion of this field is the octupole moment which decays as $1/r^3$ with the distance $r$. The linearity of equations allows to describe analytically the distortion of the long-range magnetic order at a small concentration $c$ of defects. We obtain analytically renormalization of the elastic neutron scattering cross section and the magnon spectrum $\epsilon_{\bf k}$ in the leading order in $c$. We find that the scattering on impurities renormalizes weakly the bare spectrum $\epsilon_{\bf k}\propto k$ at $k\gg\sqrt c$. However the renormalization is substantial of the long-wavelength magnon spectrum at $k\ll\sqrt c$: $\epsilon_{\bf k}\propto \sqrt{c /\ln(1/k)}$ at $k\to0$ and there is a parametrically large region in which magnons with not too small momenta are overdamped and localized. This strong modification of the long-wavelength spectrum leads to the stabilization of the slightly distorted magnetic long-range order at $T<T_N\sim S^2J/\ln(1/c)$ and to the considerable change in the density of states and in the specific heat. The overdamped modes arise also in quasi-2D spin systems on a stacked triangular lattice.

Detecting Many-Body Bell Nonlocality by Solving Ising Models.

Bell nonlocality represents the ultimate consequence of quantum entanglement, fundamentally undermining the classical tenet that spatially separated degrees of freedom possess objective attributes independently of the act of their measurement. Despite its importance, probing Bell nonlocality in many-body systems is considered to be a formidable challenge, with a computational cost scaling exponentially with system size. Here we propose and validate an efficient variational scheme, based on the solution of inverse classical Ising problems, which in polynomial time can probe whether an arbitrary set of quantum data is compatible with a local theory; and, if not, it delivers the many-body Bell inequality most strongly violated by the quantum data. We use our approach to unveil new many-body Bell inequalities, violated by suitable measurements on paradigmatic quantum states (the low-energy states of Heisenberg antiferromagnets), paving the way to systematic Bell tests in the many-body realm.

Possible Inversion Symmetry Breaking in the S=1/2 Pyrochlore Heisenberg Magnet.

We address the ground-state properties of the long-standing and much-studied three-dimensional quantum spin liquid candidate, the S=1/2 pyrochlore Heisenberg antiferromagnet. By using SU(2) density-matrix renormalization group (DMRG), we are able to access cluster sizes of up to 128 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground-state energy, E_{0}/N_{sites}=-0.490(6)J, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures.

Incoherent transport in a classical spin liquid

We study the energy and spin transport of the classical spin liquid hosted by the pyrochlore Heisenberg antiferromagnet in the large $S$ limit. Molecular dynamics calculation suggests that both the energy and spin diffusion constants approach finite limits as the temperature tends to zero. We explain our results in terms of an effective disorder model, where the energy/spin-carrying normal modes propagate in a quasi-static disordered spin background. The finite zero temperature limits of the diffusion constants are then naturally understood as a result of the finite mean free path of the normal modes due to the effective disorder.

The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet

We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial D 3h spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.

Phase competition in frustrated anisotropic antiferromagnet in strong magnetic field

We discuss theoretically a frustrated Heisenberg antiferromagnet in magnetic field close to the saturation one. It is demonstrated that a small biaxial anisotropy and/or the magnetic dipolar interaction produce a delicate balance between phases with a commensurate canted, incommensurate helical (conical), and fan spin orderings. As a result, different sequences of phase transitions are realized depending on values of these small anisotropic interactions. We derive analytical expressions for critical fields and ground-state energies of the phases which are in a quantitative agreement with our and previous Monte-Carlo simulations.

Formation and growth of skyrmion crystal phase in a frustrated Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interaction

We study the formation and growth of a skyrmion crystal (SkX) phase in a frustrated antiferromagnetic Heisenberg model on a triangular lattice with Dzyaloshinskii-Moriya interaction (DMI) in the presence of an external magnetic field. We build phase diagrams in the temperature-field parameter plane, featuring first-order phase transitions and tricritical points, and study their evolution with the increasing DMI strength. It is found that already relatively small DMI intensity can lead to the appearance of the SkX phase at low temperatures in the vicinity of the meeting point of the remaining helical, coplanar up-up-down, and canted V-like ordered phases. By means of a parallel tempering (PT) Monte Carlo algorithm we find that the minimum value of DMI, at which the skyrmion phase emerges, Dt≈0.02, is one order of magnitude smaller than previously reported. We demonstrate the efficiency of PT in reliable and precise location of the phase boundaries between different phases separated by either very large (strong first-order transition) or barely detectable energy barriers, where the standard Monte Carlo approach may easily fail to find a stable solution.

Quantum disordered state in the J1−J2 square-lattice antiferromagnet Sr2Cu(Te0.95W0.05)O6

The $B$-site ordered double perovskites ${\mathrm{Sr}}_{2}\mathrm{Cu}({\mathrm{Te}}_{1\ensuremath{-}x}{\mathrm{W}}_{x}){\mathrm{O}}_{6}$ provide an excellent arena for investigating exotic phases expected for the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square-lattice Heisenberg antiferromagnet. Here, combining magnetic susceptibility and specific-heat measurements with electron spin resonance (ESR) and muon spin rotation/relaxation $(\ensuremath{\mu}\mathrm{SR})$ techniques, we explore a spin-liquid-like state in the vicinity of the N\'eel critical end point $(x=0.05--0.1)$. The specific heat and the ESR and muon relaxation rates give evidence for an energy hierarchy of low-energy excitations, reminiscent of randomness-induced singlet states. In addition, the weak transverse $\ensuremath{\mu}\mathrm{SR}$ data show a fraction of frozen magnetic moments in the random-singlet background. The origin of a random-singlet-like state near the phase boundary is discussed in terms of concomitant exchange randomness and local strain generated by the ${\mathrm{W}}^{6+}\text{\ensuremath{-}}\mathrm{for}\text{\ensuremath{-}}{\mathrm{Te}}^{6+}$ substitution.

Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networks

The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2J1−J2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45J2/J1≤0.45. The unrestricted iPEPS simulations reveal an U(1)U(1) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1)J2/J1≈0.46(1). The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.