Square-lattice Antiferromagnet Research Articles

Structure, Spin Correlations, and Magnetism of the S = 1/2 Square-Lattice Antiferromagnet Sr2CuTe1-xWxO6 (0 ≤ x ≤ 1).

Quantum spin liquids are highly entangled magnetic states with exotic properties. The S = 1/2 square-lattice Heisenberg model is one of the foundational models in frustrated magnetism with a predicted, but never observed, quantum spin liquid state. Isostructural double perovskites Sr2CuTeO6 and Sr2CuWO6 are physical realizations of this model but have distinctly different types of magnetic order and interactions due to a d10/d0 effect. Long-range magnetic order is suppressed in the solid solution Sr2CuTe1-xWxO6 in a wide region of x = 0.05-0.6, where the ground state has been proposed to be a disorder-induced spin liquid. Here, we present a comprehensive neutron scattering study of this system. We show using polarized neutron scattering that the spin liquid-like x = 0.2 and x = 0.5 samples have distinctly different local spin correlations, which suggests that they have different ground states. Low-temperature neutron diffraction measurements of the magnetically ordered W-rich samples reveal magnetic phase separation, which suggests that the previously ignored interlayer coupling between the square planes plays a role in the suppression of magnetic order at x ≈ 0.6. These results highlight the complex magnetism of Sr2CuTe1-xWxO6 and hint at a new quantum critical point between 0.2 < x < 0.4.

Impact of mixed anion ordered state on the magnetic ground states of S=1/2 square-lattice quantum spin antiferromagnets, Sr2NiO3Cl and Sr2NiO3F

The magnetic properties of the $S=1/2$ two-dimensional square-lattice antiferromagnets ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}X$ $(X=\mathrm{Cl}, \mathrm{F})$ with the trivalent nickel ions in a low-spin state were studied by magnetic susceptibility, heat capacity, neutron powder diffraction, high-field electron spin resonance (ESR), muon spin rotation and relaxation $({\ensuremath{\mu}}^{+}\mathrm{SR})$ measurements, and density functional theory (DFT) calculations. Both oxyhalides are isostructural to an ideal quantum square-lattice antiferromagnet ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$, but the chlorine/fluorine anion exclusively occupies an apical site in an ordered/disordered manner with an oxygen anion, resulting in the formation of highly distorted ${\mathrm{NiO}}_{5}X$ octahedra with an off-center nickel ion. Magnetic susceptibility measurements revealed a remarkable difference between these two compounds: the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ exhibited a broad maximum at approximately 35 K, which is typical of low-dimensional antiferromagnetic behavior. In contrast, the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$ exhibited spin-glass-like behavior below 12 K. No anomaly associated with long-range magnetic ordering was observed in the heat capacity, ESR, and neutron powder diffraction experiments. However, ${\ensuremath{\mu}}^{+}\mathrm{SR}$ measurements revealed the emergence of a static magnetic ordered state below ${T}_{\mathrm{N}}=28\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ and a short-range magnetic state below ${T}_{\mathrm{N}}=18\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$. The DFT calculations suggested that the unpaired electron occupied a ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbital, and ferromagnetic couplings between the nearest-neighbor nickel spins were energetically favored. The mechanism of ferromagnetic superexchange interactions and the reason for the difference between the magnetic ground states in these nickel oxyhalides are discussed.

Chiral Majorana fermions in two-dimensional square lattice antiferromagnet with proximity-induced superconductivity

Combination of proximity-induced superconductivity and ferromagnetic exchange field in a two-dimensional square-lattice antiferromagnet with spin–orbit coupling and nonsymmorphic symmetry can induce a topological superconductor phase with chiral Majorana edge states. The lattice model of the Bogoliubov-de Gennes (BdG) Hamiltonian was applied to study the phase diagram of bulks and chiral Majorana edge states in nanoribbons. By numerically studying the phase diagram, we found that the non-uniformity of either the superconducting pairing parameters or the exchange field at the two sublattices is necessary to induce a topological superconductor phase with chiral Majorana edge states. The BdG Chern number of certain topological superconductor phases is ±1 or ±3, such that the corresponding nanoribbons have one or three pairs of chiral Majorana edge states, respectively.

Critical magnetic fluctuations in the layered ruthenates Ca2RuO4 and Ca3Ru2O7

Materials realizing the XY model in two dimensions (2D) are sparse. Here we use neutron triple-axis spectroscopy to investigate the critical static and dynamical magnetic fluctuations in the square-lattice antiferromagnets Ca$_{2}$RuO$_{4}$ and Ca$_{3}$Ru$_2$O$_{7}$. We probe the temperature-dependence of the antiferromagnetic (AFM) Bragg-intensity, the $Q$-width, the amplitude, and the energy-width of the magnetic diffuse scattering in vicinity to the N\'eel temperature $T_N$ to determine the critical behavior of the magnetic order parameter $M$, correlation length $\xi$, susceptibility $\chi$, and the characteristic energy $\Gamma$ with the corresponding critical exponents $\beta$, $\nu$, $\gamma$, and $z$, respectively. We find that the critical behaviors of the single-layer compound Ca$_{2}$RuO$_{4}$ follow universal scaling laws that are compatible with predictions of the 2D-XY model. The bilayer compound Ca$_{3}$Ru$_2$O$_{7}$ is only partly consistent with the 2D-XY theory and best described by the three-dimensional Ising (3D-I) model, which is likely a consequence of the intra-bilayer exchange interactions in combination with an orthorhombic single-ion anisotropy. Hence, our results suggest that layered ruthenates are promising solid-state platforms for research on the 2D-XY model and the effects of 3D interactions and additional spin-space anisotropies on the magnetic fluctuations.

Anticollinear order and degeneracy lifting in square lattice antiferromagnet LaSrCrO4

We report the static and dynamic magnetic properties of ${\mathrm{LaSrCrO}}_{4}$, a seemingly canonical spin-3/2 square-lattice antiferromagnet that exhibits frustration between magnetic layers---owing to their AB stacking---and offers a rare testbed to investigate accidental-degeneracy lifting in magnetism. Neutron diffraction experiments on single-crystal samples uncover a remarkable anticollinear magnetic order below ${T}_{N}$ = 170 K characterized by a N\'eel arrangement of the spins within each layer and an orthogonal arrangement between adjacent layers. To understand the origin of this unusual magnetic structure, we analyze the spin-wave excitation spectrum by means of inelastic neutron scattering and bulk measurements. A spectral gap of 0.5 meV, along with a spin-flop transition at 3.2 T, reflect the energy scale associated with the degeneracy-lifting. A minimal model to explain these observations requires both a positive biquadratic interlayer exchange and dipolar interactions, both of which are on the order of 10${}^{\ensuremath{-}4}$ meV, only a few parts per million of the dominant exchange interaction ${J}_{1}\ensuremath{\approx}11$ meV. These results provide direct evidence for the selection of a noncollinear magnetic structure by the combined effect of two distinct degeneracy lifting interactions.

Randomness and frustration in a S=12 square-lattice Heisenberg antiferromagnet

Chemical substitution in Sr${}_{2}$CuTe${}_{1\ensuremath{-}x}$W${}_{x}$O${}_{6}$ introduces a combination of randomness and frustration in the square-lattice Heisenberg quantum antiferromagnet. Neutron scattering is used to probe the ground and excited states of the mixed compounds in this series. It is demonstrated that the random network of interactions is only mildly frustrated. A semiclassical mean-field treatment of the random-bond model, which produces a ground state of small and weakly interlinked ordered patches hosting spin-wave-like excitations, agrees well with the experimental observations.

Pressure-induced phase transition in the J1−J2 square lattice antiferromagnet RbMoOPO4Cl

We report results of magnetization and $^{31}$P NMR measurements under high pressure up to 6.4~GPa on RbMoOPO$_4$Cl, which is a frustrated square-lattice antiferromagnet with competing nearest-neighbor and next-nearest-neighbor interactions. Anomalies in the pressure dependences of the NMR shift and the transferred hyperfine coupling constants indicate a structural phase transition at 2.6~GPa, which is likely to break mirror symmetry and triggers significant change of the exchange interactions. In fact, the NMR spectra in magnetically ordered states reveal a change from the columnar antiferromagnetic (CAF) order below 3.3~GPa to the N\'{e}el antiferromagnetic (NAF) order above 3.9~GPa. The spin lattice relaxation rate $1/T_1$ also indicates a change of dominant magnetic fluctuations from CAF-type to NAF-type with pressure. Although the NMR spectra in the intermediate pressure region between 3.3 and 3.9 GPa show coexistence of the CAF and NAF phases, a certain component of $1/T_1$ shows paramagnetic behavior with persistent spin fluctuations, leaving possibility for a quantum disordered phase. The easy-plane anisotropy of spin fluctuations with unusual nonmonotonic temperature dependence at ambient pressure gets reversed to the Ising anisotropy at high pressures. This unexpected anisotropic behavior for a spin 1/2 system may be ascribed to the strong spin-orbit coupling of Mo-4$d$ electrons.

Spin order and entropy in antiferromagnetic films subjected to magnetic fields

Using systematic effective field theory, we explore the properties of antiferromagnetic films subjected to magnetic and staggered fields that are either mutually aligned or mutually orthogonal. We provide low-temperature series for the entropy density in either case up to two-loop order. Invoking staggered, uniform and sublattice magnetizations of the bipartite antiferromagnet, we investigate the subtle order–disorder phenomena in the spin arrangement, induced by temperature, magnetic and staggered fields—some of which are quite counterintuitive. In the figures we focus on the spin- square-lattice antiferromagnet, but our results are valid for any other bipartite two-dimensional lattice.

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.

Quantum versus thermal fluctuations in the fcc antiferromagnet: Alternative routes to order by disorder

In frustrated magnetic systems with competing interactions fluctuations can lift the residual accidental degeneracy. We argue that the state selection may have different outcomes for quantum and thermal order by disorder. As an example, we consider the semiclassical Heisenberg fcc antiferromagnet with only the nearest-neighbor interactions. Zero-point oscillations select the type 3 collinear antiferromagnetic state at T=0. Thermal fluctuations favor instead the type 1 antiferromagnetic structure. The opposite tendencies result in a finite-temperature transition between the two collinear states. Competition between effects of quantum and thermal order by disorder is a general phenomenon and is also realized in the J1-J2 square-lattice antiferromagnet at the critical point J2 = 0.5 J1.

Enhanced thermal Hall effect in the square-lattice Néel state

Recent experiments on several cuprate compounds have identified an enhanced thermal Hall response in the pseudogap phase. Most strikingly, this enhancement persists even in the undoped system, which challenges our understanding of the insulating parent compounds. To explain these surprising observations, we study the quantum phase transition of a square-lattice antiferromagnet from a confining N\'eel state to a state with coexisting N\'eel and semion topological order. The transition is driven by an applied magnetic field and involves no change in the symmetry of the state. The critical point is described by a strongly-coupled conformal field theory with an emergent global $SO(3)$ symmetry. The field theory has four different formulations in terms of $SU(2)$ or $U(1)$ gauge theories, which are all related by dualities; we relate all four theories to the lattice degrees of freedom. We show how proximity of the confining N\'eel state to the critical point can explain the enhanced thermal Hall effect seen in experiment.

Spin-wave thermodynamics of square-lattice antiferromagnets revisited

Modifying the conventional spin-wave theory in a novel manner based on the Wick decomposition, we present an elaborate thermodynamics of square-lattice quantum antiferromagnets. Our scheme is no longer accompanied by the notorious problem of an artificial transition to the paramagnetic state inherent in modified spin waves in the Hartree-Fock approximation. In the cases of spin $\frac{1}{2}$ and spin $1$, various modified-spin-wave findings for the internal energy, specific heat, static uniform susceptibility, and dynamic structure factor are not only numerically compared with quantum Monte Carlo calculations and Lanczos exact diagonalizations but also analytically expanded into low-temperature series. Modified spin waves interacting via the Wick decomposition provide reliable thermodynamics over the whole temperature range of absolute zero to infinity. Adding higher-order spin couplings such as ring exchange interaction to the naivest Heisenberg Hamiltonian, we precisely reproduce inelastic-neutron-scattering measurements of the high-temperature-superconductor-parent antiferromagnet $\mathrm{La}_2\mathrm{CuO}_4$. Modifying Dyson-Maleev bosons combined with auxiliary pseudofermions also yields thermodynamics of square-lattice antiferromagnets free from thermal breakdown, but it is less precise unless temperature is sufficiently low. Applying all the schemes to layered antiferromagnets as well, we discuss the advantages and disadvantages of modified spin-wave and combined boson-pseudofermion representations.

Magnetic interactions in the S = 1/2 square-lattice antiferromagnets Ba2CuTeO6 and Ba2CuWO6: parent phases of a possible spin liquid.

The isostructural double perovskites Ba2CuTeO6 and Ba2CuWO6 are shown by theory and experiment to be frustrated square-lattice antiferromagnets with opposing dominant magnetic interactions. This is driven by differences in orbital hybridisation of Te6+ and W6+. A spin-liquid-like ground state is predicted for Ba2Cu(Te1-xWx)O6 solid solution similar to recent observations in Sr2Cu(Te1-xWx)O6.

Magnetism of theN=42kagome lattice antiferromagnet

For the paradigmatic frustrated spin-half Heisenberg antiferromagnet on the kagome lattice we performed large-scale numerical investigation of thermodynamic functions by means of the finite-temperature Lanczos method for system sizes of up to N=42. We present the dependence of magnetization as well as specific heat on temperature and external field and show in particular that a finite-size scaling of specific heat supports the appearance of a low-temperature shoulder below the major maximum. This seems to be the result of a counterintuitive motion of the density of singlet states towards higher energies. Other interesting features that we discuss are the asymmetric melting of the 1/3 magnetization plateau as well the field dependence of the specific heat that exhibits characteristic features caused by the existence of a flat one-magnon band. By comparison with the unfrustrated square-lattice antiferromagnet the tremendous role of frustration in a wide temperature range is illustrated.

Giant Pressure Dependence and Dimensionality Switching in a Metal-Organic Quantum Antiferromagnet.

We report an extraordinary pressure dependence of the magnetic interactions in the metal-organic system [CuF_{2}(H_{2}O)_{2}]_{2}pyrazine. At zero pressure, this material realizes a quasi-two-dimensional spin-1/2 square-lattice Heisenberg antiferromagnet. By high-pressure, high-field susceptibility measurements we show that the dominant exchange parameter is reduced continuously by a factor of 2 on compression. Above 18kbar, a phase transition occurs, inducing an orbital re-ordering that switches the dimensionality, transforming the quasi-two-dimensional lattice into weakly coupled chains. We explain the microscopic mechanisms for both phenomena by combining detailed x-ray and neutron diffraction studies with quantitative modeling using spin-polarized density functional theory.

Tuning the S=1/2 square-lattice antiferromagnet Sr2Cu(Te1−xWx)O6 from Néel order to quantum disorder to columnar order

The spin-1/2 square-lattice Heisenberg model is predicted to have a quantum disordered ground state when magnetic frustration is maximized by competing nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ interactions ($J_2/J_1 \approx 0.5$). The double perovskites Sr$_2$CuTeO$_6$ and Sr$_2$CuWO$_6$ are isostructural spin-1/2 square-lattice antiferromagnets with N\'eel ($J_1$ dominates) and columnar ($J_2$ dominates) magnetic order, respectively. Here we characterize the full isostructural solid solution series Sr$_2$Cu(Te$_{1-x}$W$_x$)O$_6$ ($0 \leq x \leq 1$) tunable from N\'eel order to quantum disorder to columnar order. A spin-liquid-like ground state was previously observed for the $x$ = 0.5 phase, but we show that the magnetic order is suppressed below 1.5 K in a much wider region of $x \approx$ 0.1-0.6. This coincides with significant $T$-linear terms in the low-temperature specific heat. However, density functional theory calculations predict most of the materials are not in the highly frustrated $J_2/J_1 \approx 0.5$ region square-lattice Heisenberg model. Thus, a combination of both magnetic frustration and quenched disorder is the likely origin of the spin-liquid-like state in $x$ = 0.5.

Spin-liquid-like state in a spin-1/2 square-lattice antiferromagnet perovskite induced by d10\u2013d0 cation mixing

A quantum spin liquid state has long been predicted to arise in spin-1/2 Heisenberg square-lattice antiferromagnets at the boundary region between Néel (nearest-neighbor interaction dominates) and columnar (next-nearest-neighbor interaction dominates) antiferromagnetic order. However, there are no known compounds in this region. Here we use d10–d0 cation mixing to tune the magnetic interactions on the square lattice while simultaneously introducing disorder. We find spin-liquid-like behavior in the double perovskite Sr2Cu(Te0.5W0.5)O6, where the isostructural end phases Sr2CuTeO6 and Sr2CuWO6 are Néel and columnar type antiferromagnets, respectively. We show that magnetism in Sr2Cu(Te0.5W0.5)O6 is entirely dynamic down to 19 mK. Additionally, we observe at low temperatures for Sr2Cu(Te0.5W0.5)O6—similar to several spin liquid candidates—a plateau in muon spin relaxation rate and a strong T-linear dependence in specific heat. Our observations for Sr2Cu(Te0.5W0.5)O6 highlight the role of disorder in addition to magnetic frustration in spin liquid physics.

Mutually attracting spin waves in the square-lattice quantum antiferromagnet

Spin waves (magnons) in two dimensions are the potential glue in high-temperature superconductors so that their quantitative understanding is mandatory. Yet even for the fundamental case of the undoped Heisenberg model on the square lattice a consistent picture is still lacking. Significant spectral continua are taken as evidence of the existence of fractional excitations (spinons), but descriptions in terms of spinons fail to show the established absence of an energy gap. Here a fully consistent picture of the dynamics in the square-lattice quantum antiferromagnet is provided which agrees with the experimental findings. The key step is to capture (i) the strong attractive interaction between the spin waves and (ii) the vertex corrections of the observables.

Nearly Deconfined Spinon Excitations in the Square-Lattice Spin- 1/2 Heisenberg Antiferromagnet

We study the dynamic spin structure factor of the spin-$1/2$ square-lattice Heisenberg antiferromagnet and of the $J$-$Q$ model (with 4-spin interactions $Q$ and Heisenberg exchange $J$). Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with QMC simulations, we can treat the sharp ($\delta$-function) contribution from spinwaves (magnons) and a continuum at higher energy. The results for the Heisenberg model agree with neutron scattering experiments on Cu(DCOO)$_2$$\cdot$4D$_2$O, where a broad spectral-weight continuum at $q=(\pi,0)$ was interpreted as deconfined spinons. Our results at $(\pi,0)$ show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at $q=(\pi/2,\pi/2)$ (as also seen experimentally). Turning on $Q$, we observe a rapid reduction of the $(\pi,0)$ magnon weight to zero, well before the deconfined quantum phase transition into a spontaneously dimerized state. We re-interpret the picture of deconfined spinons at $(\pi,0)$ in the experiments as nearly deconfined spinons---a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile $(\pi,0)$-magnon in the Heisenberg model and its depletion in the $J$-$Q$ model, we introduce an effective model in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy with the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model reproduces the $(\pi,0)$ and $(\pi/2,\pi/2)$ features of the Heisenberg model. It can also account for the rapid loss of the $(\pi,0)$ magnon with increasing $Q$ and a remarkable persistence of a large magnon pole at $q=(\pi/2,\pi/2)$ even at the deconfined critical point.

Magnetic excitations from the two-dimensional interpenetrating Cu framework in Ba2Cu3O4Cl2

We report detailed neutron scattering studies on Ba$_2$Cu$_3$O$_4$Cl$_2$. The compound consists of two interpenetrating sublattices of Cu, labeled as Cu$_{\rm A}$ and Cu$_{\rm B}$, each of which forms a square-lattice Heisenberg antiferromagnet. The two sublattices order at different temperatures and effective exchange couplings within the sublattices differ by an order of magnitude. This yields an inelastic neutron spectrum of the Cu$_{\rm A}$ sublattice extending up to 300 meV and a much weaker dispersion of Cu$_{\rm B}$ going up to around 20 meV. Using a single-band Hubbard model we derive an effective spin Hamiltonian. From this, we find that linear spin-wave theory gives a good description to the magnetic spectrum. In addition, a magnetic field of 10 T is found to produce effects on the Cu$_{\rm B}$ dispersion that cannot be explained by conventional spin-wave theory.