Direct evidence for the localized single-triplet excitations and the dispersive multitriplet excitations in SrCu2(BO3)2

Field-induced bound-state condensation and spin-nematic phase in SrCu2(BO3)2 revealed by neutron scattering up to 25.9 T

In quantum magnetic materials, ordered phases induced by an applied magnetic field can be described as the Bose-Einstein condensation (BEC) of magnon excitations. In the strongly frustrated system SrCu2(BO3)2, no clear magnon BEC could be observed, pointing to an alternative mechanism, but the high fields required to probe this physics have remained a barrier to detailed investigation. Here we exploit the first purpose-built high-field neutron scattering facility to measure the spin excitations of SrCu2(BO3)2 up to 25.9 T and use cylinder matrix-product-states (MPS) calculations to reproduce the experimental spectra with high accuracy. Multiple unconventional features point to a condensation of S = 2 bound states into a spin-nematic phase, including the gradients of the one-magnon branches and the persistence of a one-magnon spin gap. This gap reflects a direct analogy with superconductivity, suggesting that the spin-nematic phase in SrCu2(BO3)2 is best understood as a condensate of bosonic Cooper pairs.

Magnetically Hidden State on the Ground Floor of the Magnetic Devil's Staircase

We investigated the low-temperature and high-field thermodynamic and ultrasonic properties of SrCu_{2}(BO_{3})_{2}, which exhibits various plateaux in its magnetization curve above 27 T, called a magnetic Devil's staircase. The results of the present study confirm that magnetic crystallization, the first step of the staircase, occurs above 27 T as a first-order transition accompanied by a sharp singularity in heat capacity C_{p} and a kink in the elastic constant. In addition, we observe a thermodynamic anomaly at lower fields around 26 T, which has not been previously detected by any magnetic probes. At low temperatures, this magnetically hidden state has a large entropy and does not exhibit Schottky-type gapped behavior, which suggests the existence of low-energy collective excitations. Based on our observations and theoretical predictions, we propose that magnetic quadrupoles form a spin-nematic state around 26 T as a hidden state on the ground floor of the magnetic Devil's staircase.

Emergent bound states and impurity pairs in chemically doped Shastry-Sutherland system

Impurities often play a defining role in the ground states of frustrated quantum magnets. Studies of their effects are crucial in understanding of the phase diagram in these materials. SrCu₂(BO₃)₂, an experimental realization of the Shastry-Sutherland (SS) lattice, provides a unique model system for such studies using both experimental and numerical approaches. Here we report effects of impurities on the crystals of bound states, and doping-induced emergent ground states in Mg-doped SrCu₂(BO₃)₂, which remain stable in high magnetic fields. Using four complementary magnetometry techniques and theoretical simulations, a rich impurity-induced phenomenology at high fields is discovered. The results demonstrate a rare example in which even a small doping concentration interacts strongly with both triplets and bound states of triplets, and thus plays a significant role in the magnetization process even at high magnetic fields. Our findings provide insights into the study of impurity effects in geometrically frustrated quantum magnets.

Exploring the impurity-induced phenomena facilitates the understanding of emergent quantum materials. Here the authors show the anomalous magnetization transitions as well as demonstrate the relation between the impurities and the excited spin states in the Mg doped Shastry-Sutherland compound SrCu₂(BO₃)₂.

Triplon band splitting and topologically protected edge states in the dimerized antiferromagnet

Search for topological materials has been actively promoted in the field of condensed matter physics for their potential application in energy-efficient information transmission and processing. Recent studies have revealed that topologically invariant states, such as edge states in topological insulators, can emerge not only in a fermionic electron system but also in a bosonic system, enabling nondissipative propagation of quasiparticles. Here we report the topologically nontrivial triplon bands measured by inelastic neutron scattering on the spin-1/2 two-dimensional dimerized antiferromagnet Ba₂CuSi₂O₆Cl₂. The excitation spectrum exhibits two triplon bands that are clearly separated by a band gap due to a small alternation in interdimer exchange interaction, consistent with a refined crystal structure. By analytically modeling the triplon dispersion, we show that Ba₂CuSi₂O₆Cl₂ is the first bosonic realization of the coupled Su-Schrieffer-Heeger model, where the presence of topologically protected edge states is prompted by a bipartite nature of the lattice.

Topological edge states can emerge not only in a fermionic electron system but also in a bosonic system. Here, Nawa et al. report topologically nontrivial triplon bands in a two-dimensional dimerized quantum antiferromagnet Ba₂CuSi₂O₆Cl₂, suggesting a bosonic realization of the coupled Su-Schrieffer-Heeger model.

Flat Bands, Indirect Gaps, and Unconventional Spin-Wave Behavior Induced by a Periodic Dzyaloshinskii-Moriya Interaction

Periodically patterned metamaterials are known for exhibiting wave properties similar to the ones observed in electronic band structures in crystal lattices. In particular, periodic ferromagnetic materials are characterized by the presence of bands and band gaps in their spin-wave spectrum at tunable GHz frequencies. Recently, the fabrication of magnets hosting Dzyaloshinskii-Moriya interactions has been pursued with high interest since properties, such as the stabilization of chiral spin textures and nonreciprocal spin-wave propagation, emerge from this antisymmetric exchange coupling. In this context, to further engineer the magnon band structure, we propose the implementation of magnonic crystals with periodic Dzyaloshinskii-Moriya interactions, which can be obtained, for instance, via patterning of periodic arrays of heavy metal wires on top of an ultrathin magnetic film. We demonstrate through theoretical calculations and micromagnetic simulations that such systems show an unusual evolution of the standing spin waves around the gaps. We also predict the emergence of indirect gaps and flat bands, effects that depend on the strength of the Dzyaloshinskii-Moriya interaction. Such phenomena, which have been previously observed in different systems, are observed here simultaneously, opening new routes towards engineered metamaterials for spin-wave-based devices.

Dynamics and Instabilities of the Shastry-Sutherland Model

We study the excitation spectrum in the dimer phase of the Shastry-Sutherland model by using an unbiased variational method that works in the thermodynamic limit. The method outputs dynamical correlation functions in all possible channels. This output is exploited to identify the order parameters with the highest susceptibility (single or multitriplon condensation in a specific channel) upon approaching a quantum phase transition in the magnetic field versus the J^{'}/J phase diagram. We find four different instabilities: antiferro spin nematic, plaquette spin nematic, stripe magnetic order, and plaquette order, two of which have been reported in previous studies.

Magnetization plateaus in the antiferromagnetic Ising chain with single-ion anisotropy and quenched disorder

We have studied the presence of plateaus on the low-temperature magnetization of an antiferromagnetic spin-1 chain, as an external uniform magnetic field is varied. A crystal-field interaction is present in the model and the exchange constants follow a random quenched (Bernoulli or Gaussian) distribution. Using a transfer-matrix technique we calculate the largest Lyapunov exponent and, from it, the magnetization at low temperatures as a function of the magnetic field, for different values of the crystal field and the width of the distributions. For the Bernoulli distribution, the number of plateaus increases, with respect to the uniform case [Litaiff et al., Solid State Commun. 147, 494 (2008)] and their presence can be linked to different ground states, when the magnetic field is varied. For the Gaussian distributions, the uniform scenario is maintained, for small widths, but the plateaus structure disappears as the width increases.

Emergence of long-range order in sheets of magnetic dimers

Quantum spins placed on the corners of a square lattice can dimerize and form singlets, which then can be transformed into a magnetic state as the interactions between dimers increase beyond threshold. This is a strictly 2D transition in theory, but real-world materials often need the third dimension to stabilize long-range order. We use high pressures to convert sheets of Cu(2+) spin 1/2 dimers from local singlets to global antiferromagnet in the model system SrCu2(BO3)2. Single-crystal neutron diffraction measurements at pressures above 5 GPa provide a direct signature of the antiferromagnetic ordered state, whereas high-resolution neutron powder and X-ray diffraction at commensurate pressures reveal a tilting of the Cu spins out of the plane with a critical exponent characteristic of 3D transitions. The addition of anisotropic, interplane, spin-orbit terms in the venerable Shastry-Sutherland Hamiltonian accounts for the influence of the third dimension.

Correlated decay of triplet excitations in the Shastry-Sutherland compound SrCu2(BO3)2

The temperature dependence of the gapped triplet excitations (triplons) in the 2D Shastry-Sutherland quantum magnet SrCu(2)(BO(3))(2) is studied by means of inelastic neutron scattering. The excitation amplitude rapidly decreases as a function of temperature, while the integrated spectral weight can be explained by an isolated dimer model up to 10 K. Analyzing this anomalous spectral line shape in terms of damped harmonic oscillators shows that the observed damping is due to a two-component process: one component remains sharp and resolution limited while the second broadens. We explain the underlying mechanism through a simple yet quantitatively accurate model of correlated decay of triplons: an excited triplon is long lived if no thermally populated triplons are nearby but decays quickly if there are. The phenomenon is a direct consequence of frustration induced triplon localization in the Shastry-Sutherland lattice.

Crystals of bound states in the magnetization plateaus of the Shastry-Sutherland model

Using infinite projected entangled-pair states, we show that the Shastry-Sutherland model in an external magnetic field has low-magnetization plateaus which, in contrast to previous predictions, correspond to crystals of bound states of triplets, and not to crystals of triplets. The first sizable plateaus appear at magnetization 1/8, 2/15 and 1/6, in agreement with experiments on the orthogonal-dimer antiferromagnet SrCu2(BO3)2, and they can be naturally understood as regular patterns of bound states, including the intriguing 2/15 one. We also show that, even in a confined geometry, two triplets bind into a localized bound state with Sz=2. Finally, we discuss the role of competing domain-wall and supersolid phases, as well as that of additional anisotropic interactions.

Continuous and discontinuous quantum phase transitions in a model two-dimensional magnet

The Shasty-Sutherland model, which consists of a set of spin 1/2 dimers on a 2D square lattice, is simple and soluble but captures a central theme of condensed matter physics by sitting precariously on the quantum edge between isolated, gapped excitations and collective, ordered ground states. We compress the model Shastry-Sutherland material, SrCu(2)(BO(3))(2), in a diamond anvil cell at cryogenic temperatures to continuously tune the coupling energies and induce changes in state. High-resolution X-ray measurements exploit what emerges as a remarkably strong spin-lattice coupling to both monitor the magnetic behavior and the absence or presence of structural discontinuities. In the low-pressure spin-singlet regime, the onset of magnetism results in an expansion of the lattice with decreasing temperature, which permits a determination of the pressure-dependent energy gap and the almost isotropic spin-lattice coupling energies. The singlet-triplet gap energy is suppressed continuously with increasing pressure, vanishing completely by 2 GPa. This continuous quantum phase transition is followed by a structural distortion at higher pressure.

Bose-Einstein condensation of magnons in polycrystalline gadolinium with nano-size grains

We report the observation of Bose-Einstein condensation (BEC) of magnons in nanocrystalline Gd. Employing a self-consistent approach, the variations with magnetic field (H) of the BEC transition temperature, T(c)(H), and the volume, V (H), over which the condensate wavefunction retains its phase coherence, the temperature and magnetic field variations of the chemical potential, μ(T, H), and the average occupation number for the ground state, [linear span]n(0)(T, H)[linear span], are accurately determined from the magnetization, M(T, H), and specific heat, C(T, H), data. The variation of T(c) with magnetic field has the functional form T(c)(H) = T(c)(H = 0) + aH(2/3) that is characteristic of BEC. In conformity with the predictions of BEC theory (i) for T ≤ T(c), the condensate fraction [linear span]n(0)(T, H)[linear span]/[linear span]n(0)(T = 1.8 K, H)[linear span] at constant H scales with the reduced temperature as [T/T(c)(H)](3/2), (ii) in the limit H−>0, μ(T, H) ͠= 0 for T ≤ T(c) and abruptly falls to large negative values as the temperature exceeds T(c), and (iii) the magnetic-field-induced change in magnon entropy, deduced from both M(T, H) and C(T, H), follows the T(3/2) power law at low temperatures T<<T(p)(*) and goes through a peak at T(p)(*).

Heavy fermion compounds on the geometrically frustrated Shastry-Sutherland lattice

We present measurements of the basic properties of Ce(2)Ge(2)Mg, Yb(2)Pt(2)Pb and Ce(2)Pt(2)Pb, which are members of a new class of geometrically frustrated magnets R(2)T(2)X (R = rare earth, T = transition metal, X = main group). Here, the moment-bearing R atoms are confined to layers where they are arranged in the Shastry-Sutherland lattice. Magnetic susceptibility and specific heat measurements indicate that Ce(2)Ge(2)Mg orders antiferromagnetically at 9.4 K and Yb(2)Pt(2)Pb at 2.07 K. No long ranged order is observed in Ce(2)Pt(2)Pb above 0.05 K. Analysis of Schottky peaks in the specific heat indicates that all three compounds have doublet ground states that are well separated in energy from the excited states of the crystal-field-split manifold. Electrical resistivity measurements show that Ce(2)Ge(2)Mg and Yb(2)Pt(2)Pb are excellent metals with small residual resistivities. However, the measured resistivity of Ce(2)Pt(2)Pb is large and almost temperature-independent, suggesting that strong disorder or perhaps strong quantum critical fluctuations saturate the quasiparticle scattering in this compound. The magnetic entropy develops very slowly above the onset of antiferromagnetic order and we discuss the possibility that a nonordered fluid of dimerized moments exists above T(N) in Ce(2)Ge(2)Mg and Yb(2)Pt(2)Pb, and for a wide range of temperatures in Ce(2)Pt(2)Pb, which appears to be close to a frustration-driven quantum critical point.

Singlet-triplet excitations in the unconventional spin-Peierls TiOBr compound

We have performed time-of-flight neutron scattering measurements on powder samples of the unconventional spin-Peierls compound TiOBr using the fine-resolution Fermi chopper spectrometer (SEQUOIA) at the Spallation Neutron Source at Oak Ridge National Laboratory. These measurements reveal two branches of magnetic excitations within the commensurate and incommensurate spin-Peierls phases, which we associate with n=1 and n=2 triplet excitations out of the singlet ground state. These results represent the first direct measurement of the singlet-triplet energy gap in TiOBr, which has a value of E(g)=21.2±1.0  meV.

NMR evidence for the persistence of a spin superlattice beyond the 1/8 magnetization plateau in SrCu2(BO3)_{2}

We present 11B NMR studies of the 2D frustrated dimer spin system SrCu2(BO3)_{2} in the field range 27-31 T covering the upper phase boundary of the 1/8 magnetization plateau, identified at 28.4 T. Our data provide a clear evidence that above 28.4 T the spin superlattice of the 1/8 plateau is modified but does not melt even though the magnetization increases. Although this is precisely what is expected for a supersolid phase, the microscopic nature of this new phase is much more complex. We discuss the field-temperature phase diagram on the basis of our NMR data.

In-gap spin excitations and finite triplet lifetimes in the dilute singlet ground state system SrCu(2-x)Mgx(BO3)2

High resolution neutron scattering measurements on a single crystal of SrCu(2-x)Mgx(BO3)2 with x approximately 0.05 reveal the presence of new spin excitations within the gap of this quasi-two-dimensional, singlet ground state system. The application of a magnetic field induces Zeeman-split states associated with S=1/2 unpaired spins which are antiferromagnetically correlated with the bulk singlet. Substantial broadening of both the one- and two-triplet excitations in the doped single crystal is observed, as compared with pure SrCu2(BO3)2. Theoretical calculations using a variational algorithm and a single quenched magnetic vacancy on an infinite lattice are shown to qualitatively account for these effects.

Resonance Raman spectra of n-pi* singlet-triplet transition of p-benzoquinone at low concentrations

A weak visible absorption spectrum of p-benzoquinone (p-BQ) in CS2 due to n-pi* singlet-triplet transition was measured. Using the resonance Raman (RR) effect in liquid-core optical fiber (LCOF), we have obtained the 514.5 nm excited RR spectra of p-benzoquinone near 1445 cm(-1) and have demonstrated that the new characteristic RR band is attributed to the symmetric C=O stretch (nu(C=O)) of n-pi* singlet-triplet transition of p-BQ. The effect of solution concentration on the RR band was investigated at very low concentrations. The RR peak spreads toward short wavelength side with decreasing solution concentration ranging from 10(-7) to 10(-11) mol L(-1), whereas the blue-shift isn't obvious when the concentration is, at single molecule level, lower than 10(-11) mol L(-1). Our result is useful for single molecule detection to some extent.

High-resolution study of spin excitations in the singlet ground state of SrCu2(BO3)2

High-resolution, inelastic neutron scattering measurements on SrCu2(BO3)2, a realization of the Shastry-Sutherland model for two-dimensional Heisenberg antiferromagnets, reveal the dispersion of the three single triplet excitations continuously across the (H,0) direction within its tetragonal basal plane. These measurements also show distinct Q dependencies for the single and multiple triplet excitations, and that these excitations are largely dispersionless perpendicular to this plane. The temperature dependence of the intensities of these excitations is well described as the complement of the dc susceptibility of SrCu2(BO3)2.

Competing orders and superconductivity in the doped mott insulator on the Shastry-Sutherland lattice

Quantum antiferromagnets on geometrically frustrated lattices often allow a number of unusual paramagnetic ground states. The fate of these Mott insulators upon doping is an important issue that may shed some light on the high T(c) cuprate problem. We consider the doped Mott insulator on the Shastry-Sutherland lattice via the t-J model. The U(1) slave-boson mean-field theory reveals the strong competition between different broken symmetry states. It is found that, in some ranges of doping, there exist superconducting phases with or without coexisting translational-symmetry-breaking orders such as the staggered flux or dimerization. Our results will be directly relevant to SrCu2(BO3)(2) when this material is doped in future.

Dynamic structure factor of the two-dimensional Shastry-Sutherland model

We calculate the two-triplon contribution to the dynamic structure factor of the two-dimensional Shastry-Sutherland model, realized in SrCu2(BO3)(2), by means of perturbative continuous unitary transformations. For realistic parameters we find flat bound two-triplon bands. These bands show large weight in the structure factor depending strongly on momentum. So our findings permit a quantitative understanding of high precision inelastic neutron scattering experiments.

Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3

Bose-Einstein condensation denotes the formation of a collective quantum ground state of identical particles with integer spin or intrinsic angular momentum. In magnetic insulators, the magnetic properties are due to the unpaired shell electrons that have half-integer spin. However, in some such compounds (KCuCl3 and TlCuCl3), two Cu2+ ions are antiferromagnetically coupled to form a dimer in a crystalline network: the dimer ground state is a spin singlet (total spin zero), separated by an energy gap from the excited triplet state (total spin one). In these dimer compounds, Bose-Einstein condensation becomes theoretically possible. At a critical external magnetic field, the energy of one of the Zeeman split triplet components (a type of boson) intersects the ground-state singlet, resulting in long-range magnetic order; this transition represents a quantum critical point at which Bose-Einstein condensation occurs. Here we report an experimental investigation of the excitation spectrum in such a field-induced magnetically ordered state, using inelastic neutron scattering measurements of TlCuCl3 single crystals. We verify unambiguously the theoretically predicted gapless Goldstone mode characteristic of the Bose-Einstein condensation of the triplet states.

Magnetic superstructure in the two-dimensional quantum antiferromagnet SrCu2(BO3)2

We report the observation of magnetic superstructure in a magnetization plateau state of SrCu2(BO3)2, a frustrated quasi-two-dimensional quantum spin system. The Cu and B nuclear magnetic resonance (NMR) spectra at 35 millikelvin indicate an apparently discontinuous phase transition from uniform magnetization to a modulated superstructure near 27 tesla, above which a magnetization plateau at 1/8 of the full saturation has been observed. Comparison of the Cu NMR spectrum and the theoretical analysis of a Heisenberg spin model demonstrates the crystallization of itinerant triplets in the plateau phase within a large rhomboid unit cell (16 spins per layer) showing oscillations of the spin polarization. Thus, we are now in possession of an interesting model system to study a localization transition of strongly interacting quantum particles.

Evidence for an unconventional magnetic instability in the spin-tetrahedra system Cu(2)Te(2)O(5)Br(2)

Thermodynamic experiments as well as Raman scattering have been used to study the magnetic instabilities in the spin-tetrahedra systems Cu(2)Te(2)O(5)X(2), X = Cl and Br. While the phase transition observed in the Cl system at T(N) = 18.2 K is consistent with 3D antiferromagnetic ordering, the phase transition at T(o) = 11.4 K in the Br system has several unusual features. We propose an explanation in terms of weakly coupled tetrahedra with a singlet-triplet gap and low-lying singlets.

Dzyaloshinski-Moriya interaction in the 2D spin gap system SrCu(2)(BO(3))(2)

The Dzyaloshinski-Moriya interaction partially lifts the magnetic frustration of the spin-1/2 oxide SrCu(2)(BO(3))(2). It explains the fine structure of the excited triplet state and its unusual magnetic field dependence, as observed in previous ESR and new neutron inelastic scattering experiments. We claim that it is mainly responsible for the dispersion. We propose also a new mechanism for the observed ESR transitions forbidden by standard selection rules, which relies on an instantaneous Dzyaloshinski-Moriya interaction induced by spin-phonon couplings.

Magnetization plateaus of SrCu(2)(BO(3))(2) from a Chern-Simons theory

The antiferromagnetic Heisenberg model on the frustrated Shastry-Sutherland lattice is studied by a mapping onto spinless fermions carrying one quantum of statistical flux. Using a mean-field approximation these fermions populate the bands of a generalized Hofstadter problem. Their filling leads to the magnetization curve. For SrCu(2)(BO(3))(2) we reproduce plateaus at 1/3 and 1/4 of the saturation moment and predict a new one at 1/2. Gaussian fluctuations of the gauge field are shown to be massive at these plateau values.

Strong damping of phononic heat current by magnetic excitations in SrCu2(BO3)(2)

Measurements of the thermal conductivity as a function of temperature and magnetic field in the 2D dimer spin system SrCu2(BO3)(2) are presented. In zero magnetic field the thermal conductivity along and perpendicular to the magnetic planes shows a pronounced double-peak structure as a function of temperature. The low-temperature maximum is drastically suppressed with increasing magnetic field. Our quantitative analysis reveals that the heat current is due to phonons and that the double-peak structure arises from pronounced resonant scattering of phonons by magnetic excitations.

Soft acoustic modes in the two-dimensional spin system SrCu2(BO3)(2)

SrCu2(BO3)(2) is a two-dimensional dimerized quantum spin system which is close to a quantum critical point. The sound velocity for the longitudinal and transverse acoustic modes shows strong spin-lattice effects. The shear c(66) mode exhibits a pronounced softening of 4.5% as a function of temperature and softens more than 25% in fields up to 50 T. This huge effect occurs in the vicinity of the magnetization plateaus m/m(0) = 1/4 and 1/3. We can analyze quantitatively the temperature dependence of all measured elastic modes c(11), c(44), and c(66) with an exchange striction mechanism. The soft c(66) mode with B(2g) symmetry enables us to predict the possible symmetry of the condensed triplets in some plateaus.

Low-lying magnetic excitation of the Shastry-Sutherland model

By using perturbation calculation and numerical diagonalization, the low-energy spin dynamics of the Shastry-Sutherland model is investigated with particular attention to the two-particle coherent motion. In addition to spin-singlet- and triplet-bound states, we find novel branches of coherent motion of a bound quintet pair, which are usually unstable because of repulsion. Unusual dispersion observed in neutron-scattering measurements is explained by the present theory. The importance of the effects of phonons is also pointed out.

Dispersion and symmetry of bound states in the shastry-sutherland model

Bound states made from two triplet excitations on the Shastry-Sutherland lattice are investigated. Based on the perturbative unitary transformation by flow equations quantitative properties like dispersions and qualitative properties like symmetries are determined. The high order results [up to (J2/J1)(14)] permit one to fix the parameters of SrCu2(BO3)(2) precisely: J1 = 6.16(10) meV, x J2/J1 = 0.603(3), J( perpendicular) = 1.3(2) meV. At the border of the magnetic Brillouin zone a general double degeneracy is derived. An unexpected instability in the triplet channel at x = 0.63 indicates a transition towards another phase. The possible nature of this phase is discussed.

Collective singlet excitations and evolution of raman spectral weights in the 2D spin dimer compound SrCu2(BO3)(2)

Raman light scattering of the two-dimensional quantum spin system SrCu2(BO3)(2) shows a rich structure in the magnetic excitation spectrum, including several well-defined bound state modes at low temperature, and a scattering continuum and quasielastic light scattering contributions at high temperature. The key to the understanding of the unique features of SrCu2(BO3)(2) is the presence of strong interactions between well-localized triplet excitations in the network of orthogonal spin dimers realized in this compound.