Nonperturbative Semiclassical Spin Dynamics for Ordered Quantum Magnets

The gist: The study investigates spin dynamics in ordered quantum magnets by combining truncated Hilbert space exact diagonalization (THED) and matrix product state (MPS) simulations to identify anomalous spin excitation continua as two-magnon resonances.

Introduction and Motivation

In ordered quantum magnets where interactions between elementary excitations dominate over their kinetic energy, perturbative approaches often fail, making non-perturbative methods essential to capture spectral features such as bound states and the redistribution of weight within excitation continua<ref:2508.21142#pg2>. The research focuses on the spin dynamics of the triangular-lattice antiferromagnet in its 1/3-plateau phase using THED and MPS simulations alongside benchmarking against inelastic neutron scattering (INS) data<ref:2508.21142#pg2>. This approach is crucial because strong interactions can cause quasiparticles to form bound states or resonances, substantially reshaping the structure of the multimagnon continuum<ref:2508.21142#pg3>. The goal is to extract reliable microscopic models in semiclassical regimes by investigating anomalous spin excitation continua and bound-state effects in other materials with gapped spectra<ref:2508.21142#pg4>.

Methodology

The researchers employ two complementary non-perturbative approaches:

1. Truncated Hilbert Space Exact Diagonalization (THED): This method restricts the full Hilbert space to sectors containing a limited number of magnons, capturing few-body non-perturbative effects while reducing computational complexity from exponential to polynomial scaling<ref:2508.21142#pg5>. For the spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAFM) in the presence of a magnetic field, THED includes contributions up to the twomagnon sector to capture anomalous continua observed in INS experiments, accounting for the finite single-magnon gap generated by quantum fluctuations<ref:2508.21142#pg6>.

2. Matrix Product State (MPS) Simulations: These simulations are performed on quasi-two-dimensional geometries using Density-Matrix Renormalization Group (DMRG) for the ground state and Time-Dependent Variational Principle (TDVP) for time evolution<ref:2508.21142#pg13>. The 2D lattice is mapped onto a zig-zag path on a cylinder, utilizing moderate bond dimensions to maintain accuracy<ref:2508.21142#pg14>.

Key Findings in the UUD Phase

The parallel simulations using THED and MPS confirm the key spectral features observed in INS data for KYbSe2 at 4 T. Specifically, both methods reproduce three bright branches below 1.0 meV corresponding to collective single-magnon excitations, which are described by including the leading 1/ correction to linear spin wave theory (LSWT). Crucially, the structured two-magnon continuum that cannot be captured by any finite-order 1/ expansion is accurately reproduced by both THED and MPS. The central message of the work is that THED provides complementary insights to MPS by elucidating the internal structure of the two-magnon continuum.

Interpretation and Discrepancies

The analysis reveals that the anomalous continuum in the UUD phase is a two-magnon resonance arising from hybridization between a bound state and the multi-magnon continuum. Furthermore, THED allows for uncovering bound states embedded within the continuum, which are reminiscent of bound states in the continuum (BICs) of photonic systems. The comparison shows that while THED captures sharp two-magnon branches at certain anisotropies (e.g., Δ = 5), MPS reveals a weak intensity mode around / = 1.5 in the isotropic limit Δ = 1, which is absent in THED results. Discrepancies between THED and MPS can be attributed to the truncation of mag ≥ 3 magnon states in THED and finite-size effects inherent to cylindrical geometries in MPS.

Conclusion and Broader Applicability

The combined use of THED and MPS provides a powerful framework for studying non-perturbative excitations in ordered quantum magnets with gapped single-magnon spectra<ref:2508.21142#pg4>. The results merit reexamination of previously collected INS data on triangular lattice compounds in the UUD phase<ref:2508.21142#pg5>. The THED framework is expected to be applicable to a broader class of gapped quantum magnets exhibiting semiclassical behavior, offering a computationally efficient approach for investigating non-perturbative phenomena in three dimensions<ref:2508.21142#pg6>. The dual approach provides a robust framework for extracting reliable microscopic models in semiclassical regimes<ref:2508.21142#pg2>.

Experimental Validation

The longitudinal dynamical spin structure factor (DSSF) computed via THED and MPS shows good quantitative agreement with INS data for KYbSe2, confirming the interpretation that sharp two-magnon bound states dominate the low-energy spectrum while broader features are consistent with two-magnon resonances. Similarly, for CsYbSe2 and K2Co(SeO3)2, both methods show excellent agreement when considering the strong anisotropy of the-tensor for longitudinal component extraction. The study validates that sharp two-magnon bound states dominate the low-energy spectrum, while broader features at higher energies are consistent with two-magnon resonances. The absence of spectral weight associated with the quadrupolar bound state in K2Co(SeO3)2 reinforces the conclusion that it is well described by an effective U(1)-symmetric Heisenberg model. The overall detailed comparison presented here reinforces that the combined use of THED and MPS provides a powerful framework for understanding non-perturbative spin dynamics in ordered quantum magnets<ref:2508.21142#pg4>.

How it works

The THED method diagonalizes the renormalized quadratic Hamiltonian H˜(2)UUD = H(2)lswt +H(2)nlswt, which opens a gap in the spin-wave spectrum consistent with the UUD order not breaking any continuous symmetry. To study interacting magnons, calculations are restricted to a truncated Hilbert space restricted to the mag ≤ 2 magnon subspace, consisting of single- and two-magnon states. The projected Hamiltonian in this truncated basis is given by P1,2H P1,2 = H11 H12 / H22, where the two-magnon block encodes interactions via the four-magnon vertex. MPS simulations use DMRG for the ground state and TDVP for time evolution on a cylinder geometry mapped onto a one-dimensional chain.

Limitations and Future Directions

Discrepancies between THED and MPS can arise from the truncation of mag ≥ 3 magnon states in THED or finite-size effects in MPS calculations, such as those introduced by cylindrical geometries. Extending the current THED implementation to include the four-magnon sector is a promising avenue for future work to further reduce truncation errors and clarify the origin of these discrepancies. Efforts in advancing MPS algorithms may enable simulations on wider cylinders, thereby reducing the finite-size effects introduced by the cylindrical geometry. The study suggests that in situ correlation measurements could directly detect this real-space, time-resolved evolution of the quantum states.

U(1) Symmetry Breaking Term

The U(1) symmetry can be explicitly broken by introducing an additional anisotropy term in the Hamiltonian H = Õ⟨,⟩ ((+ 1+) + Δ) - Õ. The parameter Δ quantifies the strength of U(1) symmetry breaking, enabling quadrupolar excitations with Δ = −2 to hybridize with dipolar modes. Breaking U(1) symmetry allows the quadrupolar bound state to acquire dipolar character, leading to finite spectral weight and making it detectable in INS measurements. The analysis shows that K2Co(SeO3)2 preserves U(1) symmetry to a very good approximation, supporting the conclusion that it is accurately described by the proposed Hamiltonian.

Real Space Wave Function Interpretability

The THED method offers interpretability through direct access to eigenstates via the squared overlap between an eigenstate and a real-space wave function. For bound states, the probability of finding the second spin flip decays exponentially beyond a characteristic binding length, whereas for resonant states, this probability remains finite even at large separations. This distinction is made clearer in angle-integrated plots where bound states exhibit clear exponential decay within the finite cluster while resonance states display extended profiles, indicating delocalization. The analysis of spin flip evolution shows that for the Heisenberg point (Δ = 1), two spin flips form a transient bound pair that persists over a finite lifetime before decaying into two independent magnons.

Post-Processing

A post-processing correction involves dividing the obtained correlation functions by an extracted amplification curve to remove the spurious recurrence of magnetization order observed during time evolution. This procedure yields a significantly cleaner spectral function for all data presented in the main text and in Sec. B 1 of this supplementary material.

1. Improved AI can perform non-perturbative calculations of spin dynamics in ordered quantum magnets using Truncated Hilbert Space Exact Diagonalization (THED). This allows it to capture spectral features like bound states and anomalous scattering continua that perturbative methods miss, providing a robust framework for investigating gapped spectra.

2. Improved AI can cross-validate numerical results by combining THED with Matrix Product State (MPS) simulations against experimental data from inelastic neutron scattering (INS). This dual approach ensures reliable microscopic models by identifying features like the two-magnon resonance arising from hybridization between bound states and the continuum.

3. Improved AI can perform direct interpretation of excitation dynamics through real-space wave functions obtained from THED. It can analyze the squared overlap between this eigenstate and the real-space wave function to distinguish between bound states (which show exponential decay) and resonant states (which display extended profiles).

4. Improved AI can systematically investigate how truncation errors affect results by extending THED to include higher magnon sectors, such as three- and four-magnon states. This allows for a more complete understanding of spectral weight redistribution and clarifies discrepancies between different theoretical methods.

5. Improved AI can assess the impact of computational geometry on MPS calculations by comparing results from cylinders with different aspect ratios. It can identify how the cylindrical geometry can induce spectral-weight splittings in these cases, highlighting finite-size effects that need to be accounted for.

6. Improved AI can constrain microscopic Hamiltonian parameters directly from spectroscopic data like INS intensities using the extracted longitudinal dynamical spin structure factor (DSSF). This enables the extraction of parameters like exchange anisotropy by comparing simulated data with the measured S (,) with spins polarized along the-axis.

7. Improved AI can identify and correct numerical artifacts in time evolution simulations, such as the unphysical recurrence of magnetization order. It can apply a post-processing correction by dividing correlation functions by the extracted amplification curve to yield a cleaner spectral function.

In ordered quantum magnets where interactions between elementary excitations dominate over their kinetic energy, perturbative approaches often fail, making non-perturbative methods essential to capture spectral features such as bound states and the redistribution of weight within excitation continua. Although an increasing number of experiments report anomalous spin excitation continua in such systems, their microscopic interpretation remains an open challenge. Here, we investigate the spin dynamics of the triangular-lattice antiferromagnet in its 1/3-plateau phase using two complementary non-perturbative approaches: exact diagonalization in a truncated Hilbert space for a gas of elementary excitations (THED) and matrix product state (MPS) simulations. Alongside cross-validation between these methods, we benchmark our results against inelastic neutron scattering (INS) data. The THED analysis confirms the presence of two-magnon bound states and identifies the anomalous scattering continuum observed in both MPS and INS as a two-magnon resonance, arising from hybridization between the bound state and the two-magnon continuum. Furthermore, THED reveals bound states overlapping with the continuum, enriching the interpretation of continuum anomalies. More broadly, THED provides a robust framework for investigating anomalous spin excitation continua and bound-state effects in other materials with gapped spectra. Its combination of accuracy and computational efficiency makes it a powerful tool for extracting reliable microscopic models in semiclassical regimes.