Machine learning reveals features of spinon Fermi surface

TL;DR

It is demonstrated that a Quantum-Classical hybrid approach of mining sampled projective snapshots with interpretable classical machine learning can unveil signatures of seemingly featureless quantum states, and Friedel oscillations of a spinon Fermi surface are uncovered, providing support for a gapless quantum spin liquid.

Abstract

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 sampled projective snapshots with interpretable classical machine learning can unveil signatures of seemingly featureless quantum states. The Kitaev-Heisenberg model on a honeycomb lattice under external magnetic field presents an ideal system to test QuCl, where simulations have found an intermediate gapless phase (IGP) sandwiched between known phases, launching a debate over its elusive nature. We use the correlator convolutional neural network, trained on labeled projective snapshots, in conjunction with regularization path analysis to identify signatures of phases. We show that QuCl reproduces known features of established phases. Significantly, we also identify a signature of the 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 spin liquids.

Figures (6)

• Figure 1: The Kitaev-Heisenberg model and schematic of the Quantum-Classical (QuCl) approach. a) Honeycomb lattice of Kitaev model with bond-dependent interactions indicated by the three different colored bonds. b) Phase diagram of Kitaev-Heisenberg model in an external magnetic field (Eq \ref{['eq:model']}) long three axes of $K_z$, $h$, and $J$. c) A schematic description of QuCl: (i) From a pair of variational wavefunctions$|\Psi_0\rangle$ and $|\Psi_1\rangle$, labeled projective measurements ("snapshots") $B_0$ and $B_1$ are generated. (ii) The collection of labeled snapshots are used to train the correlator convolutional neural network (CCNN). (iii) The CCNN is configured with four filters $(k=0,\cdots,3)$ each with three channels, for the binary classification problem minimizing the distance between the prediction $\hat{y}$ and the label. (iv) Once the training is completed, we fix the filters and use regularization path analysis to reveal signature motifs of the two phases, 0 and 1, under consideration. The correlator weight $\beta$ onsetting upon reduction of the regularization strength $\lambda$ to a negative (positive) value signals a feature of the phase 0 (1).
• Figure 2: Gapless $\mathbb{Z}_2$ v.s. Heisenberg ordered phase. a) Ground state wavefunctions from the gapless $\mathbb{Z}_2$ phase and ferromagnetically ordered phase are obtained at the points on the $J$-axis marked by stars. The highlighted box shows the two most informative filters for the classification task. The pink and blue dots in filters correspond to projections in $z$ and $y$ basis. b) The regularization path analysis results pointing to the filters in panel a as signature motifs of the ordered phase.
• Figure 3: The chiral spin liquid v.s. the intermediate gapless phase, benchmarking the indicator of the chiral spin liquid phase. a) Ground state wavefunctions from the chiral spin liquid (CSL) phase and the intermediate gapless phase (IGP) are obtained at the points on the $h$-axis marked by stars. The highlighted box shows the most informative filter that signifies the CSL phase. Inset shows the bond anisotropy for the three colorings of bonds, e.g. $x$ refers to a $S^x_iS^x_j$ coupling. b) The plaquette operator $W_p$ is a operator defined on the six sites around a hexagonal plaquette. c) A sample snapshot from the CSL phase showing the measurement basis that makes the plaquette operator $W_p$ accessible. d) The regularization path analysis pointing to the six-point correlator of filter in panel a as indicator of the CSL.
• Figure 4: The chiral spin liquid v.s. the intermediate gapless phase, discovering the indicator of the intermediate gapless phase. a) The rotated basis vectors in relation to the cardinal axes; the external field is along $e_3$. The signature of the intermediate gapless phase (IGP) is targetted using $e_1$ basis snapshots from the same pair of wavefunctions as in \ref{['fig:csligp']}(a). b) The regularization path analysis results pointing to the filter in panel c as an indicator of the IGP. c) Most significant filter learned by the correlator convolutional neural network to be associated with the IGP. d) Fourier transform of filter in panel c, with black lines indicating first and extended Brillouin zones. Green circles mark six Bragg peaks associated with the filter tiling pattern in panel e. e) Simplest possible tiling of filter shown in panel c, resulting in a superlattice of antiferromagnetic stripes. The Bragg peaks of this tiling pattern are marked by green circles in panels d and g. f) On-site magnetization $\expval*{S^{e_1}_i}$ of the wavefunction $|\Psi_0\rangle$ in the IGP. g) Real part of Fourier transform of panel f, again with Bragg peaks of antiferromagnetic tiling marked (imaginary part is negligible). h) Real part of Fourier transform of $\expval*{S^{e_1}_i}$ of the wavefunction $|\Psi_1\rangle$ in the CSL phase showing no discernable features. i) The perpendicular magnetization $\expval*{S^{e_1}(r)}$ as a function of distance from boundary for various values of field strength $h$, showing decreasing modulation period with increasing field strength. Solid lines show fitted curve based on \ref{['eq:osc']}.
• Figure 5: Gapless v.s. gapped $\mathbb{Z}_2$ spin liquid, benchmarking distinguishing two spin liquids. a) Ground state wavefunctions from the gapless $\mathbb{Z}_2$ phase and gapped $\mathbb{Z}_2$ phase are obtained at the points on the $K_z$-axis marked by stars. The highlighted boxes shows the most informative filters that signifies the each phases. b) The regularization path analysis results associate two-point correlators of filters $0$ and $2$ to the gapped phase and that of filter $1$ to the gapless phase.