Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model

Abstract

<jats:title>Abstract</jats:title> <jats:p>In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. <jats:inline-formula> <jats:tex-math><?CDATA $3 \times 3$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:mn>3</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mlstac3ddfieqn1.gif" xlink:type="simple" /> </jats:inline-formula> with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of <jats:inline-formula> <jats:tex-math><?CDATA $0.09\%$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>0.09</mml:mn> <mml:mi mathvariant="normal">%</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mlstac3ddfieqn2.gif" xlink:type="simple" /> </jats:inline-formula>. Moreover, the wave-functions we find have <jats:inline-formula> <jats:tex-math><?CDATA $99.89\%$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>99.89</mml:mn> <mml:mi mathvariant="normal">%</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mlstac3ddfieqn3.gif" xlink:type="simple" /> </jats:inline-formula> overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.</jats:p>