Defect-Induced Spatial Dynamics

• Defect-induced spatially-resolved dynamical excitations are defined as localized, position-dependent modifications of dynamical correlations that reveal key quantum interactions.
• Advanced probing techniques like STS, INS, EELS, and nanodiffraction map energy redistribution and interference patterns in complex many-body systems.
• Machine learning-enabled Hamiltonian tomography and defect engineering are central to inferring interaction parameters and designing novel quantum materials.

Defect‐induced spatially‐resolved dynamical excitations refer to localized, position‐dependent modifications of dynamical correlation functions (or excitation spectra) in a system as a direct result of intentionally introduced or intrinsic defects. In quantum many-body systems and correlated materials, such spatially resolved dynamical signatures convey detailed information about the underlying interactions, electronic structure, and emergent collective phenomena. By exploiting defects as local probes or quantum impurities, one can map out the spatial structure of dynamical excitations, enabling both the visualization of competing correlations and, crucially, the inference or “learning” of the Hamiltonian parameters that govern complex quantum magnets and heavy-fermion compounds.

1. Fundamental Mechanisms: Defect-Induced Perturbations in Correlated Systems

In heavy-fermion Kondo lattice systems, local defects—such as Kondo holes (removed magnetic atoms) or non-magnetic impurities—perturb both the local Kondo hybridization and intersite antiferromagnetic correlations. The hybridization field $$s(\mathbf{r}) = \frac{J}{2} \sum_\alpha \langle f^\dagger_{\mathbf{r},\alpha} c_{\mathbf{r},\alpha} \rangle$$ and the correlation field $$\chi(\mathbf{r},\mathbf{r}') = \frac{I_{\mathbf{r},\mathbf{r}'}}{2} \sum_\alpha \langle f^\dagger_{\mathbf{r},\alpha} f_{\mathbf{r}',\alpha} \rangle$$ become position-dependent in the vicinity of the defect. Such perturbations generate spatial oscillations in electronic and magnetic observables, with isotropic spatial patterns in charge channels and anisotropic, rapidly decaying patterns in the magnetic channels—specifically, $$\Delta\chi(\mathbf{r},\mathbf{r}') \sim \exp(-r/\xi)$$, where the decay length $\xi$ scales with the interaction strength $I$.

The resultant changes in local density of states (LDOS) and nonlocal spin susceptibility are accessible to spectroscopic probes such as scanning tunneling spectroscopy (STS), granting direct imaging capability for real-space correlations and perturbations (Figgins et al., 2010).

2. Spatially Resolved Spectroscopy and Imaging Techniques

Spatial mapping of defect-induced excitations leverages high-resolution local spectroscopic probes with sensitivity to both electronic and magnetic correlations:

• Scanning Tunneling Spectroscopy (STS): Resolves LDOS modulations, revealing energy redistribution (e.g., at a Kondo hole, transfer of spectral weight from negative to positive energies).
• Inelastic Neutron Scattering (INS): Resolves fine structure in magnetic excitations and exchange interactions due to local distortions around defects in magnetically diluted compounds. Discrete splittings in dimer excitation spectra allow extraction of distribution functions for local exchange couplings and structural distortions, parameterized by statistical models that reflect defect propagation and spatial decay laws (constant, $1/r$, $1/r^2$ for 1D, 2D, 3D, respectively) (Furrer et al., 2014).
• Spatially Resolved Electron Energy Loss Spectroscopy (EELS): State-of-the-art instrumentation paired with frequency-resolved frozen phonon multislice simulations enables mapping of mode-specific spectral changes near defects (e.g., planar boundaries) with nanometer and even sub-atomic resolution (Zeiger et al., 2021).
• Photoluminescence Mapping and X-Ray Nanodiffraction: Applied to perovskite nanocrystal supercrystals, spatial PL/lifetime mapping, combined with nanodiffraction, links elevated misalignment, compressive strain, and loss of coherence at edges to blueshifted emission and reduced lifetimes (Lapkin et al., 2021).

3. Theoretical Frameworks for Dynamical Responses to Defects

Defect-induced local excitations encode information about underlying many-body Hamiltonians:

• Heavy Fermion Models: Quantities such as $s(\mathbf{r})$, $\chi(\mathbf{r},\mathbf{r}')$, and $$n_c(\mathbf{r}) = \int d\omega\, n_F(\omega)\, N_c(\mathbf{r},\omega)$$, respond nonlinearly to local perturbations, exposing the competition between Kondo screening and antiferromagnetic correlations. Periodic placement of defects leads to non-linear quantum interference, reconstructing the free energy landscape and inducing first-order phase transitions to inhomogeneous ground states when $I > I_c$ (Figgins et al., 2010).
• Spin Models and Hamiltonian Learning: The spatial and frequency-resolved mapping of dynamical spin correlators, $$\mathbf{S}_n^{aa}(\omega) = \langle GS | S_n^a \delta(\omega - H + E_{GS}) S_n^a | GS \rangle$$, upon defect or impurity placement, captures the localized response sensitive to various exchange couplings (Heisenberg $J_1$, $J_2$, $J_3$; anisotropy $J_Z$; and Dzyaloshinskii-Moriya interaction $J_{DMI}$) (Karjalainen et al., 21 Oct 2025). The signature patterns in these correlators, as measured by STM (via $dI/dV \sim \int_0^V S(\omega) d\omega$), provide the raw data for Hamiltonian inference.

4. Machine Learning–Enabled Tomography of Spin Hamiltonians

Sophisticated supervised learning frameworks are constructed to extract Hamiltonian parameters from the spatially and energetically resolved excitation data:

• Input Features: Cumulative dynamical correlators, possibly from multiple impurity configurations, constitute the input space.
• Network Architecture: Principal component dimensionality reduction, followed by dense, ReLU-activated layers and regularization via dropout, map the input correlators to Hamiltonian parameter outputs ($J_2$, $J_Z$, $J_3$, $J_{DMI}$ etc.).
• Training & Fidelity: Training data are generated from exact diagonalization for randomized parameters. Noise robustness is achieved by introducing frequency-dependent Gaussian noise into the input spectra. The performance is measured via a fidelity metric $\mathcal{F}$, constructed from the covariance of predicted and true parameters.
• Physical Grounding: The defect-induced, spatially resolved dynamical spectra are mathematically and physically sensitive to the underlying interactions, so the network can robustly recover subtle couplings, including antisymmetric Dzyaloshinskii-Moriya terms, from local data. (Karjalainen et al., 21 Oct 2025)

5. Nonlinear and Interference Effects in Defect Ensembles

Nonlinear interference is central when multiple defects are introduced:

• In Kondo lattice systems, non-additive interference between spatially extended perturbations of $s(\mathbf{r})$ and $\chi(\mathbf{r},\mathbf{r}')$ leads to emergent collective behavior not reducible to single-defect physics, including first-order transitions to inhomogeneous ground states (Figgins et al., 2010).
• In Hamiltonian learning via impurity tomography, variation of impurity placements and separation systematically alters the dynamical excitation landscape, granting access to interaction range and anisotropy in complex spin models (Karjalainen et al., 21 Oct 2025).

6. Implications for Quantum Materials Design and Quantum Simulation

The spatially resolved response to defects enables several key advances:

• Material Characterization: Direct measurement of local correlation lengths, anisotropies, and the spatial extent of hybridization or magnetic order.
• Defect Engineering: Deliberate manipulation of defect patterns (e.g., periodic arrays of Kondo holes or engineered impurity placement) can drive controllable transitions and ground state reorganizations, or act as design tools for emergent quantum states.
• Quantum Device Optimization: For systems such as perovskite superfluorescent emitters or qubit materials based on solid-state defects, minimizing strain and misalignment (as made visible by local optical and structural probes) is vital for maximizing performance (Lapkin et al., 2021).
• Quantum Simulation: The combination of spatially resolved spectroscopy and machine learning-based Hamiltonian inference provides a robust path to reconstruct complex many-body models realized in artificial lattices, facilitating the design and testing of novel quantum simulators (Karjalainen et al., 21 Oct 2025, Karjalainen et al., 2022).

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Summary Table: Key Constructs in Defect-Induced Spatially Resolved Dynamical Excitations

Observable/Feature	Measurement/Model	Reveals
$s(\mathbf{r})$, $\chi(\mathbf{r},\mathbf{r}')$	STS, AFM, theory	Hybridization, AF order, correlation lengths
$S_n^{aa}(\omega)$	STM $dI/dV$, machine learning	Spin interaction strengths (Hamiltonian), anisotropy, DMI
PL blueshift/lifetime	Photoluminescence, nanodiffraction	Strain, coherence, defect location
Dynamical structure factor	INS, EELS, multislice simulation	Local phonon/magnon modes, vibrational defects
Nonlinear defect interference	Theory, simulation	Emergence of novel inhomogeneous states

The field of defect-induced spatially resolved dynamical excitations, underpinned by advanced experimental imaging approaches, sophisticated theoretical modeling, and machine learning-enabled Hamiltonian extraction, is establishing defects as a versatile and quantitative probe of correlations, ground states, and emergent phases in strongly interacting and quantum-disordered systems.