Abstract
Quantum magnets admit more than one classical limit and Nlevel systems with strong singleion anisotropy are expected to be described by a classical approximation based on SU(N) coherent states. Here we test this hypothesis by modeling finite temperature inelastic neutron scattering (INS) data of the effective spinone antiferromagnet Ba_{2}FeSi_{2}O_{7}. The measured dynamic structure factor is calculated with a generalized LandauLifshitz dynamics for SU(3) spins. Unlike the traditional classical limit based on SU(2) coherent states, the results obtained with classical SU(3) spins are in good agreement with the measured temperature dependent spectrum. The SU(3) approach developed here provides a general framework to understand the broad class of materials comprising weakly coupled antiferromagnetic dimers, trimers, or tetramers, and magnets with strong singleion anisotropy.
Introduction
The computation of dynamical correlation functions at finite temperature is one of the important open problems of modern quantum manybody physics. These functions are not only crucial to test models against different spectroscopic techniques, but are also critical to the development of fast machine learning tools to accelerate and enhance understanding of problems at the forefront of condensed matter physics. For instance, the inelastic neutron scattering (INS) crosssection of quantum magnets is proportional to the dynamical spin structure factor, S(Q, E). A full calculation of this dynamical correlation function is complex because of the exponential numerical cost of computing the exact Hamiltonian eigenstates as a function of the number of spins N_{s}. To surmount this challenge, classical approximations such as LandauLifshitz dynamics (LLD) have been extensively adopted and applied^{1,2,3,4,5,6} because their numerical cost becomes linear in N_{s}. Furthermore, while semiclassical approaches^{7,8,9} are only applicable at the lowest temperatures (because fluctuations around the classical ground state are assumed to be small), the classical approach, based on the LLD, can be implemented at any temperature by sampling initial states via the Metropolis algorithm (classical Monte Carlo simulation) and solving the classical equations of motion. Therefore, the LLD and its generalizations can be used to compute dynamical spin structure factors over the full temperature range. Furthermore, the possibility to determine a Hamiltonian in the more tractable classical limit is important since a classical description is expected to become a good approximation at high enough temperatures. An approach of this type can even be applied to materials that exhibit strong quantum fluctuations at low temperatures, such as spin liquid candidates^{3}.
LLD was originally introduced to describe the precession of the magnetization in a solid^{10}. This dynamics can be derived as a classical limit of quantum spin systems, whose quantum mechanical state becomes a direct product of SU(2) coherent states. The time evolution of this product state is dictated by the LLD equations. It is known, however, that Nlevel quantum mechanical systems (N = 2S + 1 for spin systems) admit more than one classical limit^{11,12,13,14,15,16,17,18}. As was pointed out in a recent work^{19}, there are large classes of lowentangled quantum magnets, such as materials with strong singleion anisotropy, for which it is necessary to use a generalized spin dynamics (GSD) which accounts for nondipolar components of spin states (nonzero expectation values of other multipoles). This GSD is also necessary to describe magnets with significant biquadratic interactions^{7,20} and those comprising weaklycoupled entangled units, such as dimers^{21,22}, trimers^{23}, and tetramers^{24,25,26,27}. The hypothesis that the present work seeks to test is that these systems are better described by direct products of SU(N) coherent states at any temperature. This implies that the traditional LLD must be extended to encompass these more general cases^{19}.
The main goal of this work is to test the aforementioned hypothesis by modeling the INS crosssection of the effective S = 1 quasi2D easyplane antiferromagnet (AFM), Ba_{2}FeSi_{2}O_{7}^{28,29}. In Ba_{2}FeSi_{2}O_{7}, a significant singleion anisotropy (D ~ 1.42 meV) induced by the large tetragonal distortion of FeO_{4} tetrahedron in conjunction with spinorbit coupling of Fe^{2+}(3d^{6}) results in an effective lowenergy threelevel manifold generated by the spin states, \(\left\vert {S}^{z}=0,\pm 1\right\rangle\). The competition with a relatively weak Heisenberg exchange interaction, places the ground state of this material (α=J/D ~ 0.187) near the quantum critical point at α_{c} ~ 0.158 that separates easyplanar AFM order from a quantum paramagnetic (QPM) phase (see Fig. 1). Since the AFM to QPM transition is driven by an enhancement of the local quadrupolar moment at the expense of the magnitude of the local dipolar moment, a proper classical description must allow for the coexistence of local dipolar and quadrupolar fluctuations, leading to transverse and longitudinal collective modes^{28}. SU(3) coherent states fulfill this condition because an SU(3) spin has 8 = 3 + 5 components that include the three components of the dipole moment and the five components of the quadupolar moment (traceless symmetric tensor)^{7,19,30,31,32}.
In this article, we use INS to explore the temperature dependence of S(Q, E) for Ba_{2}FeSi_{2}O_{7}. The spin excitation spectra exhibit significant transfer of spectral weight across the Néel temperature (T_{N} = 5.2 K) with a continuous evolution from welldefined SU(3) spinwave modes for T < T_{N} to a diffusive resonant excitation for T > T_{N}, indicating temperature dependent spin dynamics. To account for this behavior, we generalize LLD to SU(3) coherent states. This GSD combined with Monte Carlo simulation for temperature dependent classical SU(3) coherent spin states^{19} describes the observed temperature dependence of S(Q, E) both above and below T_{N}. This result verifies the main hypothesis of this work, namely that many anisotropic magnets such as Ba_{2}FeSi_{2}O_{7} can be described by direct products of SU(N) coherent states at any temperature.
Results
Temperature dependent inelastic neutron scattering spectra
Figure 2a shows the temperature dependence of the unpolarized neutron crosssection I(Q,E) of Ba_{2}FeSi_{2}O_{7}, which is proportional to S(Q, E). Below T_{N}, the spectrum exhibits sharp spinwaves corresponding to acoustic (T_{1}) and optical (T_{2}) transverse modes. The longitudinal (L) mode is observed as a broad continuum above the T_{1}mode throughout the entire Brillouin zone (BZ) due to decay into a pair of transverse modes as described in ref. ^{28}. While the sharp spinwaves disappear above T_{N}, a broad dispersion with a finite gap emerges at the magnetic zone center (ZC), Q_{m} = (1, 0, 0.5). With increasing temperature, the gap size increases and the bandwidth becomes narrower.
To understand the diffusive spectra above T_{N}, we performed a polarized neutron scattering experiment. Figure 2c shows the neutron spin polarization dependence of I(E) at Q = (1, 0, 0). The spinflip and nonspinflip scattering crosssections are coupled to the sample magnetization and the wavevector, allowing us to extract the directional dependence of S(Q,E). The neutron spin was polarized along [0, 1, 0], which provides separate inplane (S^{⊥} = S^{xx}(Q, E) + S^{yy}(Q, E)) and outof plane (S^{∥} = S^{zz}(Q, E)) components of S(Q, E), for nonspinflip and spinflip channels, respectively. The nonspinflip channel is by far the most intense, indicating the diffusive spectra at 10 K mainly comes from the inplane components of S(Q, E).
Calculation of generalized spin dynamics for SU(3) spin
To account for the measured spectra at finite temperatures, we performed GSD calculations. The lowenergy effective Hamiltonian for Ba_{2}FeSi_{2}O_{7} is \({{{\mathcal{H}}}}={\sum }_{{{{\boldsymbol{r}}}},{{{\boldsymbol{\delta }}}}}{S}_{{{{\boldsymbol{r}}}}}^{\mu }{{{{\mathcal{J}}}}}_{{{{\boldsymbol{\delta }}}}}^{\mu \nu }{S}_{{{{\boldsymbol{r}}}}+{{{\boldsymbol{\delta }}}}}^{\nu }+D{\sum }_{{{{\boldsymbol{r}}}}}{({S}_{{{{\boldsymbol{r}}}}}^{z})}^{2}\)^{28}, with the convention of summation over repeated indices μ, ν = {x, y, z} and δ runs over the neighboring bonds with finite exchange interaction. This Hamiltonian can be recast in terms of SU(3) generators \({\hat{O}}_{{{{\boldsymbol{r}}}}}^{\eta }\)^{19}
where the exchange tensor \({{{{\mathcal{J}}}}}_{{{{\boldsymbol{\delta }}}}}^{\eta \gamma }={\delta }_{\eta \gamma }{J}_{{{{\boldsymbol{\delta }}}}}({\delta }_{\eta 1}+{\delta }_{\eta 2}+{\delta }_{\eta 3}{{{\Delta }}}_{{{{\boldsymbol{\delta }}}}})\) with \({J}_{{{{\boldsymbol{\delta }}}}}=J(J^{\prime} )\), \({{{\Delta }}}_{{{{\boldsymbol{\delta }}}}}={{\Delta }}({{\Delta }}^{\prime} )\) for nearestneighbor intralayer (interlayer) bonds and 1 ≤ η, γ ≤ 8. The generators \({\hat{O}}_{{{{\boldsymbol{r}}}}}^{13}\) correspond to dipolar operators \(({S}_{{{{\boldsymbol{r}}}}}^{x},{S}_{{{{\boldsymbol{r}}}}}^{y},{S}_{{{{\boldsymbol{r}}}}}^{z})\) and \({\hat{O}}^{48}\) are the quadrupolar operators (bilinear traceless forms of the dipolar operators). See ref. ^{19} for the matrix representations. We note that the singleion anisotropy becomes an external (quadrupolar) field that is linearly coupled to the SU(3) spins.
After taking the classical limit of Eq. (1) using SU(3) coherent states^{19}, \({\hat{O}}_{{{{\boldsymbol{r}}}}}^{\eta }\to {o}_{{{{\boldsymbol{r}}}}}^{\eta }\equiv \left\langle {Z}_{{{{\boldsymbol{r}}}}}\right\vert {\hat{O}}_{{{{\boldsymbol{r}}}}}^{\eta }\left\vert {Z}_{{{{\boldsymbol{r}}}}}\right\rangle\), we obtain the classical equation of motion (EOM) of the SU(3) spins
where f_{ηγλ} are the SU(3) structure constants: \([{\hat{O}}_{{{{\boldsymbol{r}}}}}^{\eta },{\hat{O}}_{{{{\boldsymbol{r}}}}}^{\gamma }]=i{f}_{\eta \gamma \lambda }{\hat{O}}_{{{{\boldsymbol{r}}}}}^{\lambda }\). To compute S(Q, E) at finite temperature, the initial conditions of Eq. (2) are sampled with the standard MetropolisHastings Monte Carlo (MC) algorithm from the CP^{2} manifold (classical phase space) of SU(3) coherent states (see Supplementary Note 2 for detailed information). The numerical integration methods for the classical EOM (2) are explained in Supplementary Note 3 and ref. ^{33}. The INS intensity I(Q, E) is obtained from the Fourier transform of the classical dipolar operators \({o}_{{{{\boldsymbol{r}}}}}^{\mu }(t)\,\mu =1,2,3\) (see Supplementary Note 2). For the calculation, we used J = 0.266 meV, \(J^{\prime} =0.1J^{\prime}\), and D = 1.42 meV from ref. ^{28} and finite lattices consisting of 24 × 24 × 12 sites. Additionally, for low values of the spin S, the Néel temperature of the classical spin Hamiltonian is significantly lower than the Néel temperature of the quantum mechanical Hamiltonian because quantum fluctuations further increase the energy gain of the ordered state relative to a disordered state. This is a wellknown fact for the traditional classical limit based on SU(2) coherent states which remains true for the more general case that we are considering here. Hence the classical approximation used here underestimates the value of Néel temperature, \({T}_{{{{\rm{N}}}}}^{{{{\rm{cl}}}}}=1.38\)K, compared to experimental value T_{N} by a factor of ~ 3.75. Therefore in Fig. 2a, b we compare the measured and calculated spectra at the same values of T/T_{N} and \({T}^{{{{\rm{cl}}}}}/{T}_{{{{\rm{N}}}}}^{{{{\rm{cl}}}}}\), respectively.
Comparison of measured and calculated spectra
Below T_{N}, the calculated spectrum exhibits T_{1}, T_{2} and Lmodes (see Fig. 2), where the calculated intensities by the GSD are multiplied by the classical to quantum correspondence factor for a harmonic oscillator \({\beta }^{{{{\rm{cl}}}}}E/(1{e}^{{\beta }^{{{{\rm{cl}}}}}E})\) with β^{cl} = 1/k_{B}T^{cl} ^{34}. Since the GSD calculation at lowtemperatures coincides with the generalized linear spinwave calculation^{28}, the decay and renormalization of the T_{2} and Lmodes observed at 1.6 K (Fig. 2a) are not captured by this classical approximation. Capturing these feature requires the nonlinear approach described in ref. ^{28}. Above T_{N}, the GSD calculation reproduces the gapped nature of the spectrum representing a resonant excitation between \(\left\vert {S}^{z}=0\right\rangle\) and \(\left\vert \pm 1\right\rangle\) states with a finite dispersion due to the exchange interaction. In the classical description, this diffusive mode originates from the combined effect of the “external SU(3) field” D that induces a precession of each SU(3) moment with frequency D/ℏ (center of the peak) and the random molecular field due to the exchange interaction with the fluctuating neighboring moments that determines the width of the peak. When J ≪ T, the spectrum thus becomes a dispersionless broad peak centered around an energy Δ_{para} ≃ D. The computed spectra reproduces the main characteristics of the observed dispersions and bandwidth. Since the \(\left\vert {S}^{z}=0\right\rangle\) and \(\left\vert \pm 1\right\rangle\) states are connected by the components that are transverse to the zaxis, S^{± }= S^{x} ± iS^{y}, the corresponding intensity of S(Q, E) should appear in the channel S^{⊥} = S^{xx}(Q, E)+S^{yy}(Q, E) (see Fig. 2d), which is qualitatively in good agreement with the polarization dependence of the measured S(Q, E).
The detailed spectral change across T_{N} is shown in Fig. 3 which compares the measured and calculated constant momentum scans at the ZC with varying temperature. For T < T_{N}, we consider a wave vector Q = (1, 0, 0.2) that is close but not exactly equal to the ZC Q = (1, 0, 0.5) in order to avoid the large tail of the magnetic Bragg reflection as well as the technical challenges associated with calculating the spectrum at the ZC. In this case, the T_{1} (Goldstone) mode becomes visible because of its finite energy at Q = (1, 0, 0.2) due to the nonzero [0, 0, L]dispersion produced by the small interlayer coupling J_{inter}. As a result, the three T_{1}, T_{2}, L modes are observed in the spectrum (see Fig. 3a). While the T_{1} and T_{2} transverse modes remain nearly unchanged with increasing temperature, the energy of the Lmode decreases and the mode becomes broader and indistinguishable from the quasielastic scattering near T_{N}. Above T_{N}, the quasielastic scattering continuously evolves into a broad peak centered at finite energy (Δ_{para}), whose energy increases gradually with the temperature (see Fig. 3c). To extract the spectral weight of the resonant excitation above T_{N} and the Lmode below T_{N}, the data were fitted with a double Lorentzian function associated with a damped harmonicoscillator (DLDHO),
that provides a simplified description of the contribution of an overdamped mode^{35,36}. The n(E) + 1 is the Bose factor, and A, Δ, and Γ indicate the intensity, energy, and linewidth of the peak, respectively. The extracted spectral weights and parameters of the Lmode and the resonant excitations are indicated by the shaded regions in Fig. 3a–c, and summarized in Fig. 3d, e, respectively.
Figure 3b, c show a comparison of the GSD calculation with the INS data across T_{N}. Remarkably, the spectral weight for the Lmode is enhanced and shifts to lowenergy with increasing temperature, which is consistent with the data. Above T_{N}, the GSD calculation gives a diffusive resonant peakshaped spectrum centered at the energy Δ_{para} that approaches D for T ≫ T_{N}. We note that the traditional LLD based on SU(2) coherent states cannot explain this gapped diffusive mode as well as the Lmode, leading to incorrect results in the hightemperature limit^{19}. However, we notice that the calculated spectrum underestimates the width of the mode at temperatures T ≳ T_{N}. This discrepancy arises from an inadequate normalization of the SU(3) spins at T ≳ T_{N}^{19}, similar to the issue raised in traditional SU(2) LLD^{37}.
This discrepancy in the linewidth can be removed in the hightemperature limit by applying an adequate renormalization of the SU(3) spins \({o}_{{{{\boldsymbol{r}}}}}^{\eta }\to \kappa {o}_{{{{\boldsymbol{r}}}}}^{\eta }\), with κ = 2 in the hightemperature (T ≫ T_{N}) limit, as described in Ref. ^{19}. This renormalization guarantees that the SU(3) S(Q, E) satisfies the exact sumrule in the highT limit. Properly renormalizing the spin has the additional virtue of bringing the theoretical Néel temperature to \({T}_{{{{\rm{N}}}}}^{{{{\rm{cl}}}}}\,\simeq 7.5\)K (κ = 2) closer to the experimental value T_{N} = 5.2 K. Figure 4a shows the comparison between the new calculations including the renormalization factor and the data measured at the same temperature T^{cl} = T (experiment). This comparison reveals a better agreement in spectral shape at T_{N} ≪ T than those without renormalization factor (see Fig. 3c). As expected, significant deviations in lowenergy spectrum are observed at T = 10 K and 12 K because they are relatively close to the \({T}_{{{{\rm{N}}}}}^{{{{\rm{cl}}}}}\) which deviates from the experimental T_{N}. In other words, since the energy Δ_{para} of the diffusive peak goes to zero at T_{N}, its position is shifted to the lower energy relative to the measured peak at T = 10 K. The emergence of a quasielastic peak (centered at E = 0) in the theoretical calculation indicates proximity to the \({T}_{{{{\rm{N}}}}}^{{{{\rm{cl}}}}}\). Figure 4b, c show the computed INS crosssection, I(Q, E), at T = 10 K and 40 K along the same direction in reciprocal space that is presented in Fig. 2a. The T = 10 K and 40 K spectra with renormalized spin provide a good description of the measured spectra at these temperatures.
Discussion
In summary, the GSD based on SU(3) spin provides a good approximation of the measured INS crosssection over a broad temperature range. The most quantitative deviations are observed at very lowtemperatures T ≪ T_{N} and close to T_{N}. The former case is due to the requirement of a oneloop quantum correction to account for the decay of the Lmode^{28}, and the latter is due to the expected discrepancy between the experimental and rescaled values of T_{N} originated from the renormalization factor κ = 2 to the classical SU(3) spins. This renormalization factor arises from enforcing the sumrule in the infinite Tlimit. Similarly, the GSD of unrenormalized classical SU(3) (κ = 1) leads to the correct sumrule in the zero temperature limit after quantizing the normal modes. Therefore, the correct scaling factor should be defined as a function κ(T) that monotonically interpolates between the two limiting cases κ(0) = 1 and κ(∞) = 2.
The verification of the main hypothesis of this work has very important consequences for the characterization of quantum magnets. For instance, while INS is an ideally suited technique for extracting models from data, the solution of this “inverse scattering problem" requires the development of fast solvers of the direct problem (inferring the INS crosssection of a given model). A crucial advantage of the GSD demonstrated here is that the cost of the simulations scales linearly in the system size, while the computation cost of exact dynamics grows exponentially^{33}, making it an ideal solver for attacking the inverse scattering problem with machinelearningbased approaches^{4}. Moreover, since the GSD can reproduce the INS data in the hightemperature regime, the method can still be applied to quantum magnets that exhibit longrange entanglement at low enough temperatures, but undergo a quantum to classical crossover above a certain temperature T_{QC}^{3,38}. Finally, we note that the SU(N) approach described here is also relevant to the broad class of materials comprising weakly coupled antiferromagnetic magnets including dimers, trimers, or tetramers as well as magnets with strong singleion anisotropy, where similar effects may be anticipated.
Methods
Inelastic neutron scattering experiment
For the INS experiments, the same single crystal of Ba_{2}FeSi_{2}O_{7} (mass: 2.13 g) as was used in ref. ^{28} was aligned on an aluminum plate with an [H, 0, L] horizontal scattering plane. Unpolarized INS data were collected using the cold neutron tripleaxis spectrometer (CTAX) at the High Flux Isotope Reactor (HFIR) and the hybrid spectrometer (HYSPEC) at the Spallation Neutron Source (SNS) located at Oak Ridge National Laboratory^{39}. A liquid helium cryostat was used to control temperature for both experiments. At CTAX, the initial neutron energy was selected using a PG (002) monochromator, and the final neutron energy was fixed to E_{f} = 3.0 meV by a PG (002) analyzer. The horizontal collimation was guide − open\(40^{\prime} 120^{\prime}\), which provides an energy resolution with full width half maximum (FWHM) = 0.1 and 0.18 meV for E = 0 and 2.5 meV, respectively. For the HYSPEC experiment, E_{i} = 9 meV and a Fermi chopper frequency of 300 Hz were used, which gives FWHM = 0.28 meV and 0.19 meV of energy resolution at E = 0 and 2.5 meV, respectively. Measurements were performed by rotating the sample from − 50° to 170° with 1° steps. Data were integrated over K = [ − 0.16, 0.16] and L = [L − 0.1, L + 0.1], and symmetrized over positive and negative H.
We also performed polarized neutron scattering measurement as part of the HYSPEC experiment using XYZpolarization analysis which is the same configuration as the experiment in ref. ^{40}. The Xaxis is defined along Q = [1, 0, 0] for the scattering wave vector, the Zaxis is defined along [0, 1, 0] perpendicular to the scattering plane, and the Yaxis is defined along the direction [0, 0, 1] perpendicular to the X and Zaxes. In the experiment, the neutron was polarized along the Zdirection, and the nonspinflip and spinflip scattering crosssections provide \({I}_{{{{\rm{n}}}}}(E)+{I}_{{{{\rm{mag}}}}}^{{{{\rm{Z}}}}}(E)\) and \({I}_{{{{\rm{mag}}}}}^{{{{\rm{Y}}}}}(E)\), respectively^{40,41}. The I_{n}(E) indicates nonmagnetic structure factor and \({I}_{{{{\rm{mag}}}}}^{\alpha }(E)\), where α ∈ {X, Y, Z}, is the component dependent magnetic structure factor. The measured spinflip and nonspinflip crosssections provide distinct \({I}_{{{{\rm{mag}}}}}^{{{{\rm{Z}}}}}(E)\) and \({I}_{{{{\rm{mag}}}}}^{{{{\rm{Y}}}}}(E)\), which correspond to S^{xx+yy}(E) and S^{zz}(E), respectively, on the crystal axis where x∥a, y∥b, and z∥c in the tetragonal crystal structure. All of data sets were reduced and analyzed using the MANTID^{42} and DAVE^{43} software packages.
Data availability
Data are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. This research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, DOE Office of Science User Facilities operated by the Oak Ridge National Laboratory (ORNL). The work at Max Planck POSTECH/Korea Research Initiative was supported by the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (2020M3H4A2084417 and 2022M3H4A1A04074153) The work at Rutgers University was supported by the DOE under Grant No. DOE: DEFG0207ER46382. D.D., K.B., and C.D.B. acknowledge support from U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DESC0022311.
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S.H.D., H.Z., C.D.B., and A.D.C. conceived the project. T.H.J., S.W.C., and J.H.P. provided single crystals. S.H.D., T.J.W., T.H., V.O.G., and A.D.C. performed INS experiments. S.H.D., and A.D.C. analyzed the neutron data. H.Z., D.A.D, K.B., and C.D.B. constructed theoretical model and calculations. S.H.D., H.Z., C.D.B., and A.D.C. wrote the manuscript with input from all authors.
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Do, SH., Zhang, H., Dahlbom, D.A. et al. Understanding temperaturedependent SU(3) spin dynamics in the S = 1 antiferromagnet Ba_{2}FeSi_{2}O_{7}. npj Quantum Mater. 8, 5 (2023). https://doi.org/10.1038/s41535022005267
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DOI: https://doi.org/10.1038/s41535022005267
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