With rapid progress in simulation of strongly interacting quantum
Hamiltonians, the challenge in characterizing unknown phases becomes a
bottleneck for scientific progress. We demonstrate that a Quantum-Classical
hybrid approach (QuCl) of mining the projective snapshots with interpretable
classical machine learning, can unveil new signatures of seemingly featureless
quantum states. The Kitaev-Heisenberg model on a honeycomb lattice with
bond-dependent frustrated interactions presents an ideal system to test QuCl.
The model hosts a wealth of quantum spin liquid states: gapped and gapless
$\mathbb{Z}_2$ spin liquids, and a chiral spin liquid (CSL) phase in a small
external magnetic field. Recently, various simulations have found a new
intermediate gapless phase (IGP), sandwiched between the CSL and a partially
polarized phase, launching a debate over its elusive nature. We reveal
signatures of phases in the model by contrasting two phases pairwise using an
interpretable neural network, the correlator convolutional neural network
(CCNN). We train the CCNN with a labeled collection of sampled projective
measurements and reveal signatures of each phase through regularization path
analysis. We show that QuCl reproduces known features of established spin
liquid phases and ordered phases. Most significantly, we identify a signature
motif of the field-induced IGP in the spin channel perpendicular to the field
direction, which we interpret as a signature of Friedel oscillations of gapless
spinons forming a Fermi surface. Our predictions can guide future experimental
searches for $U(1)$ spin liquids.

我们使用量子 - 经典混合方法和可解释的机器学习方法，探测无特征量子态的新迹象，并在模拟六角格子上 Kitaev - Heisenberg 模型中明确表示出磁场引起间隔相的特征振荡。