Spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State

Abstract

The spin- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:math> kagome antiferromagnet (AFM) is one of the most studied models in frustrated magnetism since it is a promising candidate to host exotic spin-liquid states. However, despite numerous studies using both analytical and numerical approaches, the nature of the ground state and low-energy excitations in this system remains elusive. This challenge is related to the difficulty in determining the spin gap in various calculations. We present the results of our investigation of the kagome AFM using the recently developed group equivariant convolutional neural networks—a novel machine learning technique for studying strongly frustrated models. This approach, combined with variational Monte Carlo method, introduces significant improvement of the achievable results’ accuracy for frustrated spin systems in comparison with approaches based on other neural-network architectures. Contrary to the results obtained previously with various methods, which predicted <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi mathvariant="double-struck">Z</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:mrow> </c:math> or <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>U</f:mi> <f:mo stretchy="false">(</f:mo> <f:mn>1</f:mn> <f:mo stretchy="false">)</f:mo> </f:math> Dirac spin-liquid states, our results strongly indicate that the ground state of the kagome lattice antiferromagnet is a spinon pair density wave that does not break time-reversal symmetry or any of the lattice symmetries. The state appears due to the spinon Cooper pairing instability close to two Dirac points in the spinon energy spectrum, and it resembles the pair density wave state studied previously in the context of underdoped cuprate superconductors in connection with the pseudogap phase. This state has significantly lower energy than the lowest-energy states found by the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>S</j:mi> <j:mi>U</j:mi> <j:mo stretchy="false">(</j:mo> <j:mn>2</j:mn> <j:mo stretchy="false">)</j:mo> </j:math> symmetric density matrix renormalization group calculations and other methods.