Transfer learning of many-body electronic correlation entropy from local measurements

TL;DR

The paper tackles the challenge of measuring correlation entropy $S_{ m corr}$ in many-body electronic systems by training neural networks to infer $S_{ m corr}$ from a small set of local correlators. By employing transfer learning across variants of extended Hubbard-like Hamiltonians, the authors demonstrate accurate predictions of $S_{ m corr}$ in unseen parameter regimes and show that prediction fluctuations can reveal previously unobserved quantum phases or phase boundaries. The method relies on inter-site and density-density correlators, leveraging a central-four-site block to achieve size-independent performance while remaining robust to moderate noise. This work broadens the experimental relevance of correlation-entropy learning, enabling characterization of quantum correlations without retraining on the realized Hamiltonian and providing qualitative phase-detection signals through model uncertainty.

Abstract

The characterization of quantum correlations in many-body systems is instrumental to understanding the nature of emergent phenomena in quantum materials. The correlation entropy serves as a key metric for assessing the complexity of a quantum many-body state in interacting electronic systems. However, its determination requires the measurement of all single-particle correlators across a macroscopic sample, which can be impractical. Machine learning methods have been shown to allow learning the correlation entropy from a reduced set of measurements, yet these methods assume that the targeted system is contained in the set of training Hamiltonians. Here we show that a transfer learning strategy enables correlation entropy learning from a reduced set of measurements in families of Hamiltonians never considered in the training set. We demonstrate this transfer learning methodology in a wide variety of interacting models including local and non-local attractive and repulsive many-body interactions, long-range hopping, doping, magnetic field, and spin-orbit coupling. Furthermore, we show how this transfer learning methodology allows detecting quantum many-body phases never observed during their training set without prior knowledge about them. Our results demonstrate that correlation entropy learning can be potentially performed experimentally without requiring training in the experimentally realized Hamiltonian.

Figures (8)

• Figure 1: Schematics of the model. (a) The neural network algorithms to extract quantum correlations from local measurable correlators. (b) A neural network algorithm trained on correlators extracted from one interacting fermionic chain is used to predict the correlation entropy of another interacting fermionic chain with a different parameter regime.
• Figure 2: Phase diagrams of the correlation entropy for several interacting models. The upper panels show the true phase diagrams of (a) $\mu U$, (b) $t^{\prime}U$, (c) $VU$, (d) $hU$. The lower panels show the predicted phase diagrams of the model trained with the parameter regime $H=H(t,U,\mu>0)$ in the corresponding order. The fidelity of these predictions are all $\mathcal{F}\simeq1$. The phases identified in the diagrams can be compared to those from Refs. MOTT1968Nishimoto2008Yu2016vanDongen1997.
• Figure 3: (a) Fidelity table for the transfer learning. On each row, a neural-network algorithm trained on the Hamiltonian parameters labeled on the left, and predict on the dataset with the Hamiltonian parameters labeled on the top. (b) Histogram of (a) based on the fidelity value, and (c) shows a smaller fidelity value range.
• Figure 4: (a, b) The true and the predicted phase diagrams of $H_{J<0}$, with the algorithm trained on the data extracted from $H_{\mu>0}$. (c, d) Schematic illustration and the result of the standard deviation of the datasets $H_{J<0}$ predicted by all the trained algorithms, respectively. The total standard deviation $\overline{\sigma}$ is given as the title of (d).
• Figure 5: (a) The fidelity of the prediction of each dataset as a function of the noise rate $\omega$. The algorithm is trained on the model $H_{\mu>0}$. (b) True $Uh$ phase diagram. (c-f) Predicted $Uh$ phase diagrams at different noise rate.