Dirac-Type Nodal Spin Liquid Revealed by Refined Quantum Many-Body Solver Using Neural-Network Wave Function, Correlation Ratio, and Level Spectroscopy

Yusuke Nomura,Masatoshi Imada

doi:10.1103/physrevx.11.031034

Abstract

Pursuing fractionalized particles that do not bear properties of conventional measurable objects, exemplified by bare particles in the vacuum such as electrons and elementary excitations such as magnons, is a challenge in physics. Here we show that a machine-learning method for quantum many-body systems that has achieved state-of-the-art accuracy reveals the existence of a quantum spin liquid (QSL) phase in the region $0.49\lesssim J_2/J_1\lesssim0.54$ convincingly in spin-1/2 frustrated Heisenberg model with the nearest and next-nearest neighbor exchanges, $J_1$ and $J_2$, respectively, on the square lattice. This is achieved by combining with the cutting-edge computational schemes known as the correlation ratio and level spectroscopy methods to mitigate the finite-size effects. The quantitative one-to-one correspondence between the correlations in the ground state and the excitation spectra enables the reliable identification and estimation of the QSL and its nature. The spin excitation spectra containing both singlet and triplet gapless Dirac-like dispersions signal the emergence of gapless fractionalized spin-1/2 Dirac-type spinons in the distinctive QSL phase. Unexplored critical behavior with coexisting and dual power-law decays of N\'{e}el antiferromagnetic and dimer correlations is revealed. The power-law decay exponents of the two correlations differently vary with $J_2/J_1$ in the QSL phase and thus have different values except for a single point satisfying the symmetry of the two correlations. The isomorph of excitations with the cuprate $d$-wave superconductors implies a tight connection between the present QSL and superconductivity. This achievement demonstrates that the quantum-state representation using machine learning techniques, which had mostly been limited to benchmarks, is a promising tool for investigating grand challenges in quantum many-body physics.

Highlights

Collective excitations such as magnons and phonons consist of many elementary particles and provide us with fundamental understanding beyond the noninteracting picture, where the spontaneous symmetry breaking and associated Nambu-Goldstone bosons are required in many cases
We have confirmed that the restricted Boltzmann machine (RBM) þ PP achieves state-ofthe-art accuracy among machine-learning-based methods and among all available numerical methods
We have found that the RBM þ PP represents excited states with unprecedented accuracy (Fig. 11)

Summary

Introduction

Collective excitations such as magnons and phonons consist of many elementary particles and provide us with fundamental understanding beyond the noninteracting picture, where the spontaneous symmetry breaking and associated Nambu-Goldstone bosons are required in many cases. Elementary particles themselves can often be viewed as a bound state of more elementary objects, namely, the fractionalized particles, and such exotic particles emerge through the deconfinement. Though the electron is an elementary particle in the vacuum, such deconfinement of electrons can be seen at low energies in specific circumstances followed by the ground-state structure of materials. The expectation would be that the emergent particles arising from the fractionalization still have particle character as low-energy excitations distinct from the elementary particles in the vacuum and the collective excitations in the

Methods

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Discussion

Conclusion