Hamiltonian learning quantum magnets with dynamical impurity tomography

TL;DR

The paper tackles the problem of extracting a full quantum spin Hamiltonian from spatially and spectrally resolved spin excitations measured by scanning probes. It introduces impurity tomography with two learning strategies, SingleImp and MultiImp, that feed dynamical correlators into supervised neural networks to recover parameters such as $J_1$, $J_2$, $J_3$, $J_Z$, and $J_{\mathrm{DMI}}$. Results show robustness to realistic noise, with MultiImp yielding higher fidelities across parameters and improved reconstructed spectra via $\text{DMRG-KPM}$, highlighting the method's practical viability for complex quantum magnets. The approach provides a scalable route to quantify intricate spin Hamiltonians and can be extended to other strongly correlated states, with data and code made publicly available.

Abstract

Nanoscale engineered spin systems, ranging from spins on surfaces to nanographenes, provide flexible platforms to realize entangled quantum magnets from a bottom up approach. However, assessing the quantum many-body Hamiltonian realized in a specific experiment remains an exceptional open challenge, due to the difficulty of disentangling competing terms accounting for the many-body excitations. Here, we demonstrate a machine learning strategy to learn a quantum many-body spin Hamiltonian from scanning spectroscopy measurements of spin excitations. Our methodology leverages the spatially-resolved reconstruction of the many-body excitations induced by depositing quantum impurities next to the quantum magnet. We demonstrate that our algorithm allows us to predict long-range Heisenberg exchange interactions, anisotropic exchange, as well as antisymmetric Dzyaloshinskii-Moriya interaction, including in the presence of sizable noise. Our methodology establishes defect-induced spatially-resolved dynamical excitations in quantum magnets as a powerful strategy to understand the nature of quantum spin many-body models.

Figures (5)

• Figure 1: Hamiltonian learning with impurity tomography. Two strategies are shown, (a) SingleImp that uses only single impurity placement, and (b) MultiImp several impurity configurations, both single and several impurities, with variable distance between impurities. Dynamical correlators of the spin chain are computed or measured and passed to a machine learning model. The trained neural networks then reconstructs the corresponding Hamiltonian of the spin chain.
• Figure 2: Impurity configurations and impact in the many-body excitations. Impurities are added to quantum spin model in different locations, triggering different many-body reconstructions depending on the Hamiltonian. Examples show how different dominating parameters affect on the appearance of the dynamical correlators. (a) Only one impurity is placed next to the spin chain. (b) Two impurities are placed one spin apart. (c) Two impurities are placed two spins apart.
• Figure 3: Noiseless Hamiltonian learning. Comparison between true dynamical correlators (a,d), and dynamical correlators obtained from the parameters by the Hamiltonian learning algorithms (b,c,e,f), using the SingleImp (b,e) and MultiImp (c,f) algorithms. Panels (a,b,c) show the $S_{xx}$ dynamical correlator, and $d,e,f$ the $S_{zz}$ dynamical correlator. Predictions are made under no-noise conditions of $\chi=0.0$ and with impurity coupling of $\lambda=0.122$. We have observed that for pristine dynamical correlators, both networks perform similarly.
• Figure 4: Noisy Hamiltonian learning. Many-body dynamical correlators, comparing true (a,d) and predicted many-body excitations (b,c,e,f), obtained by SingleImp (b,e) and MultiImp (c,f) algorithms. Predictions are made under noise conditions of $\chi=1.0$ and with impurity coupling of $\lambda=0.122$. Panels (a,b,c) show the $S_{xx}$ dynamical correlator, and $d,e,f$ the $S_{zz}$ dynamical correlator. It is observed that MultiImp network performs a more faithful prediction, in particular leading to spectral functions that agree with the original ones in small features that SingleImp does not account for.
• Figure 5: Fidelities of Hamiltonian learning as a function of increasing amount of noise $\chi$. The NN SingleImp is trained with single impurity dataset Fig \ref{['Fig:SingleImp and MultiImp']} (a) and the MultiImp is trained with concatenated datasets from multiple impurity placements Fig \ref{['Fig:SingleImp and MultiImp']} (b). Error bars indicate the standard deviation of fidelity values obtained over ten stochastic runs.